Introduction to Abstract Algebra/Problem set 1

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Problem Set #1: Introduction to Abstract Algebra.

As you work through these problems, think about the logical steps you are using.  You should know if your proof is correct or not if you have a reason for every step.

1: If&lt;math&gt;B \subset A&lt;/math&gt;, prove &lt;math&gt;A \cup B = A&lt;/math&gt; and conversely 

2: Prove &lt;math&gt;( A \cap B ) \cap C = A \cap ( B \cap C )&lt;/math&gt;

3: Prove &lt;math&gt;A \cup ( B \cap C ) = ( A \cup B ) \cap ( A \cup C )&lt;/math&gt;

4: Define the set C' to be the ''complement'' of the set C relative to a defined universal set U for which &lt;math&gt;C \subset U&lt;/math&gt;  such that &lt;math&gt;\forall c \in C, c \notin C'&lt;/math&gt;.  Prove that &lt;math&gt;( A \cap B )' = A' \cup B'&lt;/math&gt; and &lt;math&gt;( A \cup B )' = A' \cap B'&lt;/math&gt;

[[Category:Abstract Algebra]]</text>
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== Time to heal ==
As many of you know, I am an educator at Brown. I'd like to inform the community that the [[w:2025 Brown University shooting|2025 Brown University shooting]] occurred in my office building. I've given lectures in the auditorium where the tragedy happened. I'm unable to respond to on-wiki communication at this time. I need to provide and receive support from my colleagues and students who are struggling to cope with this horrific event. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 23:07, 16 December 2025 (UTC)

:We just prayed the ''[[w:janazah|janazah]]'' prayer for Mukhammad Aziz Umurzokov here in Virginia. Very heartbreaking. My thoughts are with you, the victims, and the Brown University community. Please take your time and heal, and I'm hoping that the preparator of this senseless violence is apprehend as soon as possible. All the best, —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 01:24, 17 December 2025 (UTC)

::I greatly appreciate your support and concern. Thank you for reaching out to me in this difficult time. It is deeply meaningful to me. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 20:23, 17 December 2025 (UTC)

== [[Wikiversity:Candidates for Custodianship/Dan Polansky]] ==

Hello, a Wikiversity contributor has been nominated for curatorship by one of our bureaucrats. The discussion is open for more than 2 weeks. Another bureaucrat has supported custodianship ([[Special:Diff/2642118]]). There are some discussion participants who made edits after the opening of the discussion ([[Special:Diff/2641936]], [[Special:Diff/2645506]]), but there are no major objections. Please consider the closure of this discussion to grant (temporary) curatorship or custodianship. Thank you very much for your attention. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 03:25, 20 August 2024 (UTC)
:Thank you for the heads up. I will take a look at this over the weekend. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 23:41, 21 August 2024 (UTC)
:Hi Mike, Wondering if you could close this nomination when you get a chance? Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:10, 9 September 2024 (UTC)

== Email sent! ==

{{you've got mail}}
—[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 17:54, 13 November 2025 (UTC)

== Emergency removal of my tools ==
You have emergency removed my curator tools and opened [[Wikiversity:Community Review/Dan Polansky]]. I have asked elsewhere: &quot;I responded there. Can you clarify why you removed the rights before the discussion rather than after the discussion? Did I misuse the rights in any way, or was there a risk of my misuse of the rights?&quot; Would you be able and willing to provide an answer? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 11:41, 19 November 2025 (UTC)

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== 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! ==
&lt;div style=&quot;border: 3px solid #FFD700; background-color: #FFFAF0; padding:0.2em 0.4em; height:auto; min-height:173px; border-radius:1em; {{box-shadow|0.1em|0.1em|0.5em|rgba(0,0,0,0.75)}}&lt;!--
--&gt;&quot; class=&quot;plainlinks&quot;&gt;
[[File:Everlasting Fireworks looped.gif|left|x173px]][[File:Happy new year 01.svg|x173px|right]]
{{Paragraph break}}
{{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;</text>
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    <title>User:Jtneill/Publications</title>
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      <text bytes="7446" sha1="4td0jlg5lj0bk8xph2swgbn7pboxe1g" xml:space="preserve">See also: [[User:Jtneill/Research|Research]]
==[[../Research/Profiles|Profiles]]==
{{../Research/Profiles}}
&lt;!-- ==2026==
--&gt;
==2025==
{{Hanging indent|1=
Brichacek, A., Neill, J. T., Murray, K., Rieger, E., Watsford, C. (2025). Body Image Flexibility and Inflexibility Scale (BIFIS). In W. Ramseyer Winter, T. L. Tylka, &amp; A. M. Landor (Eds.), ''Handbook of body image-related measures''. Cambridge University Press (pp. 118–121). https://doi.org/10.1017/9781009398275.039

{{User:Jtneill/Publications/2025/Body}}&lt;!--
Neill, J. T., Herbert, S., Hartley, R., &amp; D'Cunha, N. (in preparation). ''Art for Wellbeing at the National Gallery of Australia: Thematic analysis of participant and staff perspectives''.

Lozancic Babic, V. &amp; Neill, J. T. ...  --&gt;
}}

==2024==
{{Hanging indent|1=

Black, H. M., &amp; Neill, J. T. (2024). Wellbeing through nature: A qualitative exploration of psychosocial aspects of a Landcare ACT nature-connection program. ''Journal of Outdoor and Environmental Education''. https://doi.org/10.1007/s42322-024-00184-2

Boerma, M., Beel, N., Neill, J. T., Jeffries, C., Krishnamoorthy, G., &amp; Guerri-Guttenberg, J. (2024). Male-friendly counselling for young men: a thematic analysis of client and caregiver experiences of Menslink counselling. ''Australian Psychologist'', 1–12. https://doi.org/10.1080/00050067.2024.2378119 ([https://www.researchgate.net/publication/385649063_Male-friendly_counselling_for_young_men_a_thematic_analysis_of_client_and_caregiver_experiences_of_Menslink_counselling#fullTextFileContent pdf])

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (accepted). The Body Image Flexibility and Inflexibility Scale (BIFIS). In V. Ramseyer Winter, T.  Tylka, &amp; A. Landor (Eds.), ''Handbook of body image-related measures'' (pp. *–*). Cambridge University Press.

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2024). The distinct affect regulation functions of body image flexibility and inflexibility: A prospective study in adolescents and emerging adults. ''Body Image'', ''50'', 101726. https://doi.org/10.1016/j.bodyim.2024.101726

{{User:Jtneill/Publications/2024/Collaborative}}

Neill, J. T. &amp; Black. H. (2024). ''Landcare ACT Wellbeing through Nature program evaluation: Final report''. University of Canberra, Australia.

{{User:Jtneill/Publications/2024/Rich}}

}}

==2023==
{{Hanging indent|1=
Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2023). Ways of responding to body image threats: Development of the Body Image Flexibility and Inflexibility Scale for Youth. ''Journal of Contextual Behavioral Science'', ''30'', 31–40. https://doi.org/10.1016/j.jcbs.2023.08.007

{{/2023/WIL}}

Ross, B. M., &amp; Neill, J. T. (2023). Exploring the relationship between mental health, drug use, personality, and attitudes towards psilocybin-assisted therapy. ''[https://akjournals.com/view/journals/2054/2054-overview.xml Journal of Psychedelic Studies]'', ''7''(2). https://doi.org/10.1556/2054.2023.00264, 114–118.

==2022==
{{Hanging indent|1=

Neill, J. T., Goch, I., Sullivan, A., &amp; Simons, M. (2022). The role of burn camp in the recovery of young people from burn injury: A qualitative study using long-term follow-up interviews with parents and participants. ''Burns'', ''48''(5), 1139–1148. https://doi.org/10.1016/j.burns.2021.09.020

Stevenson, D. J., Neill, J. T., Ball, K., Smith, R., &amp; Shores, M. C. (2022). How do preschool to year 6 educators prevent and cope with occupational violence from students? ''Australian Journal of Education'', ''66''(2), 154–170. https://doi.org/10.1177/00049441221092472. [https://www.teachermagazine.com/au_en/articles/the-research-files-episode-77-coping-with-violence-from-students Podcast].

}}

==2021==
{{Hanging indent|1=Brichacek, A. L., Murray, K., Neill, J. T., &amp; Rieger, E. (2021). Contextual behavioral approaches to understanding body image threats and coping in youth: A qualitative study. ''Journal of Adolescent Research'', ''39''(2), 328–360. https://doi.org/10.1177/07435584211007851}}

==2016==
{{Hanging indent|1=

Bowen, D. J., &amp; Neill, J. T. (2016). Effects of the PCYC Catalyst outdoor adventure intervention program on youths' life skills, mental health, and delinquent behaviour. ''International Journal of Adolescence and Youth'', ''21''(1), 34–55. https://doi.org/10.1080/02673843.2015.1027716

{{/2016/Internationalisation}}}}

==2013==
{{Hanging indent|1=
{{/2013/Promoting}}

{{/2013/Teaching}}
}}

==2011==
{{Hanging indent|1=

Gray, T. L. &amp; Neill, J. T. (2011). Program evaluation. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 164–182). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.010

Neill, J. T., &amp; Gray, T. L. (2011). Technology, risk and outdoor programming. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 132–149). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.008
}}

==2010==
Mackay, G. J., &amp; Neill, J. T. (2010). The effect of “green exercise” on state anxiety and the role of exercise duration, intensity, and greenness: A quasi-experimental study. ''Psychology of Sport and Exercise'', ''11''(3), 238–245. https://doi.org/10.1016/j.psychsport.2010.01.002

==2008==
{{Hanging indent|1=Neill, J. T. (2008). Enhancing life effectiveness: The impacts of outdoor education programs. [Unpublished doctoral dissertation]. University of Western Sydney. https://researchdirect.westernsydney.edu.au/islandora/object/uws:6441/}}

==2002==
{{/2002/Dramaturgy}}

==1997==
Hattie, J., Marsh, H. W., Neill, J. T., &amp; Richards, G. E. (1997). Adventure education and Outward Bound: Out-of-class experiences that make a lasting difference. ''Review of Educational Research'', ''67''(1), 43-87. https://doi.org/10.3102/00346543067001043

==Reports==
{{Hanging indent|1=
Neill, J. T. &amp; Bowen, D. J. (2014). ''[https://drive.google.com/file/d/0B2N4zSp4hmN9WUF3bzhuZ3JoNGM/view?usp=sharing&amp;resourcekey=0-y0ZTjcdhHXqQKNtz50BW0A Research evaluation of PCYC Bornhoffen Catalyst intervention programs for youth-at-risk &lt;nowiki&gt;[&lt;/nowiki&gt;2012-2013&lt;nowiki&gt;]&lt;/nowiki&gt;]''. University of Canberra.
}}

==Theses==
* [[User:Jtneill/PhD|PhD]]
&lt;!-- 
==Published==
* [http://www.wilderdom.com/JamesNeill/JamesNeillpublications.htm Articles &amp; presentations by James Neill]
--&gt;

==Ideas / In progress==
* [[User:Jtneill/4 pillars of free and open teaching|4 pillars of free and open teaching]]
* Some international trends in outdoor education - Past, present, and future
* Ingando camp (life effectiveness)
* Life Effectiveness Questionnaire psychometrics
* OE outcomes (longitudinal study)
* Adolescent Coping Scale psychometrics
* Resilience Scale psychometrics
* Overview of Outdoor Education Theory and/or Research
* Overview of Outdoor Education in Australia
* Overview of Adventure Therapy Theory and/or Education
* Past Trends and Future Directions for Outdoor Education
* Psychological Aspects of Outdoor Education
* Outdoor Education and Modern Technology
* Outdoor Education and Environmental Sustainability

==See also==
* [[User:Jtneill/Presentations]]</text>
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      <text bytes="7447" sha1="i1xloeam0wlbuaykiorkrqj4zsyb1pw" xml:space="preserve">See also: [[User:Jtneill/Research|Research]]
==[[../Research/Profiles|Profiles]]==
{{../Research/Profiles}}
&lt;!-- ==2026==
--&gt;
==2025==
{{Hanging indent|1=
Brichacek, A., Neill, J. T., Murray, K., Rieger, E., Watsford, C. (2025). Body Image Flexibility and Inflexibility Scale (BIFIS). In W. Ramseyer Winter, T. L. Tylka, &amp; A. M. Landor (Eds.), ''Handbook of body image-related measures''. Cambridge University Press (pp. 118–121). https://doi.org/10.1017/9781009398275.039

{{User:Jtneill/Publications/2025/Body}}&lt;!--
Neill, J. T., Herbert, S., Hartley, R., &amp; D'Cunha, N. (in preparation). ''Art for Wellbeing at the National Gallery of Australia: Thematic analysis of participant and staff perspectives''.

Lozancic Babic, V. &amp; Neill, J. T. ...  --&gt;
}}

==2024==
{{Hanging indent|1=

Black, H. M., &amp; Neill, J. T. (2024). Wellbeing through nature: A qualitative exploration of psychosocial aspects of a Landcare ACT nature-connection program. ''Journal of Outdoor and Environmental Education''. https://doi.org/10.1007/s42322-024-00184-2

Boerma, M., Beel, N., Neill, J. T., Jeffries, C., Krishnamoorthy, G., &amp; Guerri-Guttenberg, J. (2024). Male-friendly counselling for young men: a thematic analysis of client and caregiver experiences of Menslink counselling. ''Australian Psychologist'', 1–12. https://doi.org/10.1080/00050067.2024.2378119 ([https://www.researchgate.net/publication/385649063_Male-friendly_counselling_for_young_men_a_thematic_analysis_of_client_and_caregiver_experiences_of_Menslink_counselling#fullTextFileContent pdf])

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (accepted). The Body Image Flexibility and Inflexibility Scale (BIFIS). In V. Ramseyer Winter, T.  Tylka, &amp; A. Landor (Eds.), ''Handbook of body image-related measures'' (pp. *–*). Cambridge University Press.

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2024). The distinct affect regulation functions of body image flexibility and inflexibility: A prospective study in adolescents and emerging adults. ''Body Image'', ''50'', 101726. https://doi.org/10.1016/j.bodyim.2024.101726

{{User:Jtneill/Publications/2024/Collaborative}}

Neill, J. T. &amp; Black. H. (2024). ''Landcare ACT Wellbeing through Nature program evaluation: Final report''. University of Canberra, Australia.

{{User:Jtneill/Publications/2024/Rich}}

}}

==2023==
{{Hanging indent|1=
Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2023). Ways of responding to body image threats: Development of the Body Image Flexibility and Inflexibility Scale for Youth. ''Journal of Contextual Behavioral Science'', ''30'', 31–40. https://doi.org/10.1016/j.jcbs.2023.08.007

{{/2023/WIL}}

Ross, B. M., &amp; Neill, J. T. (2023). Exploring the relationship between mental health, drug use, personality, and attitudes towards psilocybin-assisted therapy. ''[https://akjournals.com/view/journals/2054/2054-overview.xml Journal of Psychedelic Studies]'', ''7''(2), 114–118. https://doi.org/10.1556/2054.2023.00264}}

==2022==
{{Hanging indent|1=

Neill, J. T., Goch, I., Sullivan, A., &amp; Simons, M. (2022). The role of burn camp in the recovery of young people from burn injury: A qualitative study using long-term follow-up interviews with parents and participants. ''Burns'', ''48''(5), 1139–1148. https://doi.org/10.1016/j.burns.2021.09.020

Stevenson, D. J., Neill, J. T., Ball, K., Smith, R., &amp; Shores, M. C. (2022). How do preschool to year 6 educators prevent and cope with occupational violence from students? ''Australian Journal of Education'', ''66''(2), 154–170. https://doi.org/10.1177/00049441221092472. [https://www.teachermagazine.com/au_en/articles/the-research-files-episode-77-coping-with-violence-from-students Podcast].

}}

==2021==
{{Hanging indent|1=Brichacek, A. L., Murray, K., Neill, J. T., &amp; Rieger, E. (2021). Contextual behavioral approaches to understanding body image threats and coping in youth: A qualitative study. ''Journal of Adolescent Research'', ''39''(2), 328–360. https://doi.org/10.1177/07435584211007851}}

==2016==
{{Hanging indent|1=

Bowen, D. J., &amp; Neill, J. T. (2016). Effects of the PCYC Catalyst outdoor adventure intervention program on youths' life skills, mental health, and delinquent behaviour. ''International Journal of Adolescence and Youth'', ''21''(1), 34–55. https://doi.org/10.1080/02673843.2015.1027716

{{/2016/Internationalisation}}}}

==2013==
{{Hanging indent|1=
{{/2013/Promoting}}

{{/2013/Teaching}}
}}

==2011==
{{Hanging indent|1=

Gray, T. L. &amp; Neill, J. T. (2011). Program evaluation. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 164–182). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.010

Neill, J. T., &amp; Gray, T. L. (2011). Technology, risk and outdoor programming. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 132–149). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.008
}}

==2010==
Mackay, G. J., &amp; Neill, J. T. (2010). The effect of “green exercise” on state anxiety and the role of exercise duration, intensity, and greenness: A quasi-experimental study. ''Psychology of Sport and Exercise'', ''11''(3), 238–245. https://doi.org/10.1016/j.psychsport.2010.01.002

==2008==
{{Hanging indent|1=Neill, J. T. (2008). Enhancing life effectiveness: The impacts of outdoor education programs. [Unpublished doctoral dissertation]. University of Western Sydney. https://researchdirect.westernsydney.edu.au/islandora/object/uws:6441/}}

==2002==
{{/2002/Dramaturgy}}

==1997==
Hattie, J., Marsh, H. W., Neill, J. T., &amp; Richards, G. E. (1997). Adventure education and Outward Bound: Out-of-class experiences that make a lasting difference. ''Review of Educational Research'', ''67''(1), 43-87. https://doi.org/10.3102/00346543067001043

==Reports==
{{Hanging indent|1=
Neill, J. T. &amp; Bowen, D. J. (2014). ''[https://drive.google.com/file/d/0B2N4zSp4hmN9WUF3bzhuZ3JoNGM/view?usp=sharing&amp;resourcekey=0-y0ZTjcdhHXqQKNtz50BW0A Research evaluation of PCYC Bornhoffen Catalyst intervention programs for youth-at-risk &lt;nowiki&gt;[&lt;/nowiki&gt;2012-2013&lt;nowiki&gt;]&lt;/nowiki&gt;]''. University of Canberra.
}}

==Theses==
* [[User:Jtneill/PhD|PhD]]
&lt;!-- 
==Published==
* [http://www.wilderdom.com/JamesNeill/JamesNeillpublications.htm Articles &amp; presentations by James Neill]
--&gt;

==Ideas / In progress==
* [[User:Jtneill/4 pillars of free and open teaching|4 pillars of free and open teaching]]
* Some international trends in outdoor education - Past, present, and future
* Ingando camp (life effectiveness)
* Life Effectiveness Questionnaire psychometrics
* OE outcomes (longitudinal study)
* Adolescent Coping Scale psychometrics
* Resilience Scale psychometrics
* Overview of Outdoor Education Theory and/or Research
* Overview of Outdoor Education in Australia
* Overview of Adventure Therapy Theory and/or Education
* Past Trends and Future Directions for Outdoor Education
* Psychological Aspects of Outdoor Education
* Outdoor Education and Modern Technology
* Outdoor Education and Environmental Sustainability

==See also==
* [[User:Jtneill/Presentations]]</text>
      <sha1>i1xloeam0wlbuaykiorkrqj4zsyb1pw</sha1>
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    <revision>
      <id>2806947</id>
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        <id>10242</id>
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      <text bytes="7545" sha1="dg79p8a754ahhabmycgmyk1i60msbvy" xml:space="preserve">See also: [[User:Jtneill/Research|Research]]
==[[../Research/Profiles|Profiles]]==
{{../Research/Profiles}}
&lt;!-- ==2026==
--&gt;
==2025==
{{Hanging indent|1=
Brichacek, A., Neill, J. T., Murray, K., Rieger, E., Watsford, C. (2025). Body Image Flexibility and Inflexibility Scale (BIFIS). In W. Ramseyer Winter, T. L. Tylka, &amp; A. M. Landor (Eds.), ''Handbook of body image-related measures''. Cambridge University Press (pp. 118–121). https://doi.org/10.1017/9781009398275.039

{{User:Jtneill/Publications/2025/Body}}&lt;!--
Neill, J. T., Herbert, S., Hartley, R., &amp; D'Cunha, N. (in preparation). ''Art for Wellbeing at the National Gallery of Australia: Thematic analysis of participant and staff perspectives''.

Lozancic Babic, V. &amp; Neill, J. T. ...  --&gt;
}}

==2024==
{{Hanging indent|1=

Black, H. M., &amp; Neill, J. T. (2024). Wellbeing through nature: A qualitative exploration of psychosocial aspects of a Landcare ACT nature-connection program. ''Journal of Outdoor and Environmental Education''. https://doi.org/10.1007/s42322-024-00184-2

Boerma, M., Beel, N., Neill, J. T., Jeffries, C., Krishnamoorthy, G., &amp; Guerri-Guttenberg, J. (2024). Male-friendly counselling for young men: a thematic analysis of client and caregiver experiences of Menslink counselling. ''Australian Psychologist'', 1–12. https://doi.org/10.1080/00050067.2024.2378119 ([https://www.researchgate.net/publication/385649063_Male-friendly_counselling_for_young_men_a_thematic_analysis_of_client_and_caregiver_experiences_of_Menslink_counselling#fullTextFileContent pdf])

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (accepted). The Body Image Flexibility and Inflexibility Scale (BIFIS). In V. Ramseyer Winter, T.  Tylka, &amp; A. Landor (Eds.), ''Handbook of body image-related measures'' (pp. *–*). Cambridge University Press.

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2024). The distinct affect regulation functions of body image flexibility and inflexibility: A prospective study in adolescents and emerging adults. ''Body Image'', ''50'', 101726. https://doi.org/10.1016/j.bodyim.2024.101726

{{User:Jtneill/Publications/2024/Collaborative}}

Neill, J. T. &amp; Black. H. (2024). ''[https://landcareact.org.au/wp-content/uploads/2026/01/Wellbeing-through-Nature-Final-Report.pdf Landcare ACT Wellbeing through Nature program evaluation: Final report'']. University of Canberra, Australia.

{{User:Jtneill/Publications/2024/Rich}}

}}

==2023==
{{Hanging indent|1=
Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2023). Ways of responding to body image threats: Development of the Body Image Flexibility and Inflexibility Scale for Youth. ''Journal of Contextual Behavioral Science'', ''30'', 31–40. https://doi.org/10.1016/j.jcbs.2023.08.007

{{/2023/WIL}}

Ross, B. M., &amp; Neill, J. T. (2023). Exploring the relationship between mental health, drug use, personality, and attitudes towards psilocybin-assisted therapy. ''[https://akjournals.com/view/journals/2054/2054-overview.xml Journal of Psychedelic Studies]'', ''7''(2), 114–118. https://doi.org/10.1556/2054.2023.00264}}

==2022==
{{Hanging indent|1=

Neill, J. T., Goch, I., Sullivan, A., &amp; Simons, M. (2022). The role of burn camp in the recovery of young people from burn injury: A qualitative study using long-term follow-up interviews with parents and participants. ''Burns'', ''48''(5), 1139–1148. https://doi.org/10.1016/j.burns.2021.09.020

Stevenson, D. J., Neill, J. T., Ball, K., Smith, R., &amp; Shores, M. C. (2022). How do preschool to year 6 educators prevent and cope with occupational violence from students? ''Australian Journal of Education'', ''66''(2), 154–170. https://doi.org/10.1177/00049441221092472. [https://www.teachermagazine.com/au_en/articles/the-research-files-episode-77-coping-with-violence-from-students Podcast].

}}

==2021==
{{Hanging indent|1=Brichacek, A. L., Murray, K., Neill, J. T., &amp; Rieger, E. (2021). Contextual behavioral approaches to understanding body image threats and coping in youth: A qualitative study. ''Journal of Adolescent Research'', ''39''(2), 328–360. https://doi.org/10.1177/07435584211007851}}

==2016==
{{Hanging indent|1=

Bowen, D. J., &amp; Neill, J. T. (2016). Effects of the PCYC Catalyst outdoor adventure intervention program on youths' life skills, mental health, and delinquent behaviour. ''International Journal of Adolescence and Youth'', ''21''(1), 34–55. https://doi.org/10.1080/02673843.2015.1027716

{{/2016/Internationalisation}}}}

==2013==
{{Hanging indent|1=
{{/2013/Promoting}}

{{/2013/Teaching}}
}}

==2011==
{{Hanging indent|1=

Gray, T. L. &amp; Neill, J. T. (2011). Program evaluation. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 164–182). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.010

Neill, J. T., &amp; Gray, T. L. (2011). Technology, risk and outdoor programming. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 132–149). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.008
}}

==2010==
Mackay, G. J., &amp; Neill, J. T. (2010). The effect of “green exercise” on state anxiety and the role of exercise duration, intensity, and greenness: A quasi-experimental study. ''Psychology of Sport and Exercise'', ''11''(3), 238–245. https://doi.org/10.1016/j.psychsport.2010.01.002

==2008==
{{Hanging indent|1=Neill, J. T. (2008). Enhancing life effectiveness: The impacts of outdoor education programs. [Unpublished doctoral dissertation]. University of Western Sydney. https://researchdirect.westernsydney.edu.au/islandora/object/uws:6441/}}

==2002==
{{/2002/Dramaturgy}}

==1997==
Hattie, J., Marsh, H. W., Neill, J. T., &amp; Richards, G. E. (1997). Adventure education and Outward Bound: Out-of-class experiences that make a lasting difference. ''Review of Educational Research'', ''67''(1), 43-87. https://doi.org/10.3102/00346543067001043

==Reports==
{{Hanging indent|1=
Neill, J. T. &amp; Bowen, D. J. (2014). ''[https://drive.google.com/file/d/0B2N4zSp4hmN9WUF3bzhuZ3JoNGM/view?usp=sharing&amp;resourcekey=0-y0ZTjcdhHXqQKNtz50BW0A Research evaluation of PCYC Bornhoffen Catalyst intervention programs for youth-at-risk &lt;nowiki&gt;[&lt;/nowiki&gt;2012-2013&lt;nowiki&gt;]&lt;/nowiki&gt;]''. University of Canberra.
}}

==Theses==
* [[User:Jtneill/PhD|PhD]]
&lt;!-- 
==Published==
* [http://www.wilderdom.com/JamesNeill/JamesNeillpublications.htm Articles &amp; presentations by James Neill]
--&gt;

==Ideas / In progress==
* [[User:Jtneill/4 pillars of free and open teaching|4 pillars of free and open teaching]]
* Some international trends in outdoor education - Past, present, and future
* Ingando camp (life effectiveness)
* Life Effectiveness Questionnaire psychometrics
* OE outcomes (longitudinal study)
* Adolescent Coping Scale psychometrics
* Resilience Scale psychometrics
* Overview of Outdoor Education Theory and/or Research
* Overview of Outdoor Education in Australia
* Overview of Adventure Therapy Theory and/or Education
* Past Trends and Future Directions for Outdoor Education
* Psychological Aspects of Outdoor Education
* Outdoor Education and Modern Technology
* Outdoor Education and Environmental Sustainability

==See also==
* [[User:Jtneill/Presentations]]</text>
      <sha1>dg79p8a754ahhabmycgmyk1i60msbvy</sha1>
    </revision>
    <revision>
      <id>2806948</id>
      <parentid>2806947</parentid>
      <timestamp>2026-04-29T04:24:37Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
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      <comment>+ 2020</comment>
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      <format>text/x-wiki</format>
      <text bytes="7955" sha1="2jhz9reca6ehsqucdn0q22ij53y2t5f" xml:space="preserve">See also: [[User:Jtneill/Research|Research]]
==[[../Research/Profiles|Profiles]]==
{{../Research/Profiles}}
&lt;!-- ==2026==
--&gt;
==2025==
{{Hanging indent|1=
Brichacek, A., Neill, J. T., Murray, K., Rieger, E., Watsford, C. (2025). Body Image Flexibility and Inflexibility Scale (BIFIS). In W. Ramseyer Winter, T. L. Tylka, &amp; A. M. Landor (Eds.), ''Handbook of body image-related measures''. Cambridge University Press (pp. 118–121). https://doi.org/10.1017/9781009398275.039

{{User:Jtneill/Publications/2025/Body}}&lt;!--
Neill, J. T., Herbert, S., Hartley, R., &amp; D'Cunha, N. (in preparation). ''Art for Wellbeing at the National Gallery of Australia: Thematic analysis of participant and staff perspectives''.

Lozancic Babic, V. &amp; Neill, J. T. ...  --&gt;
}}

==2024==
{{Hanging indent|1=

Black, H. M., &amp; Neill, J. T. (2024). Wellbeing through nature: A qualitative exploration of psychosocial aspects of a Landcare ACT nature-connection program. ''Journal of Outdoor and Environmental Education''. https://doi.org/10.1007/s42322-024-00184-2

Boerma, M., Beel, N., Neill, J. T., Jeffries, C., Krishnamoorthy, G., &amp; Guerri-Guttenberg, J. (2024). Male-friendly counselling for young men: a thematic analysis of client and caregiver experiences of Menslink counselling. ''Australian Psychologist'', 1–12. https://doi.org/10.1080/00050067.2024.2378119 ([https://www.researchgate.net/publication/385649063_Male-friendly_counselling_for_young_men_a_thematic_analysis_of_client_and_caregiver_experiences_of_Menslink_counselling#fullTextFileContent pdf])

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (accepted). The Body Image Flexibility and Inflexibility Scale (BIFIS). In V. Ramseyer Winter, T.  Tylka, &amp; A. Landor (Eds.), ''Handbook of body image-related measures'' (pp. *–*). Cambridge University Press.

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2024). The distinct affect regulation functions of body image flexibility and inflexibility: A prospective study in adolescents and emerging adults. ''Body Image'', ''50'', 101726. https://doi.org/10.1016/j.bodyim.2024.101726

{{User:Jtneill/Publications/2024/Collaborative}}

Neill, J. T. &amp; Black. H. (2024). ''[https://landcareact.org.au/wp-content/uploads/2026/01/Wellbeing-through-Nature-Final-Report.pdf Landcare ACT Wellbeing through Nature program evaluation: Final report'']. University of Canberra, Australia.

{{User:Jtneill/Publications/2024/Rich}}

}}

==2023==
{{Hanging indent|1=
Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2023). Ways of responding to body image threats: Development of the Body Image Flexibility and Inflexibility Scale for Youth. ''Journal of Contextual Behavioral Science'', ''30'', 31–40. https://doi.org/10.1016/j.jcbs.2023.08.007

{{/2023/WIL}}

Ross, B. M., &amp; Neill, J. T. (2023). Exploring the relationship between mental health, drug use, personality, and attitudes towards psilocybin-assisted therapy. ''[https://akjournals.com/view/journals/2054/2054-overview.xml Journal of Psychedelic Studies]'', ''7''(2), 114–118. https://doi.org/10.1556/2054.2023.00264}}

==2022==
{{Hanging indent|1=

Neill, J. T., Goch, I., Sullivan, A., &amp; Simons, M. (2022). The role of burn camp in the recovery of young people from burn injury: A qualitative study using long-term follow-up interviews with parents and participants. ''Burns'', ''48''(5), 1139–1148. https://doi.org/10.1016/j.burns.2021.09.020

Stevenson, D. J., Neill, J. T., Ball, K., Smith, R., &amp; Shores, M. C. (2022). How do preschool to year 6 educators prevent and cope with occupational violence from students? ''Australian Journal of Education'', ''66''(2), 154–170. https://doi.org/10.1177/00049441221092472. [https://www.teachermagazine.com/au_en/articles/the-research-files-episode-77-coping-with-violence-from-students Podcast].

}}

==2021==
{{Hanging indent|1=Brichacek, A. L., Murray, K., Neill, J. T., &amp; Rieger, E. (2021). Contextual behavioral approaches to understanding body image threats and coping in youth: A qualitative study. ''Journal of Adolescent Research'', ''39''(2), 328–360. https://doi.org/10.1177/07435584211007851}}

==2020==
{{Hanging indent|1=Boerma, M., &amp; Neill, D. J. (2020). The role of grit and self-control in university student academic achievement and satisfaction. ''College Student Journal''.

Boerma, M., Neill, J., &amp; Brown, P. (2020). Perseverance of Effort Moderates The Relationship Between Psychological Distress And Life Satisfaction. ''European Journal Of Applied Positive Psychology'', ''4''(16), 1–11.}}

==2016==
{{Hanging indent|1=

Bowen, D. J., &amp; Neill, J. T. (2016). Effects of the PCYC Catalyst outdoor adventure intervention program on youths' life skills, mental health, and delinquent behaviour. ''International Journal of Adolescence and Youth'', ''21''(1), 34–55. https://doi.org/10.1080/02673843.2015.1027716

{{/2016/Internationalisation}}}}

==2013==
{{Hanging indent|1=
{{/2013/Promoting}}

{{/2013/Teaching}}
}}

==2011==
{{Hanging indent|1=

Gray, T. L. &amp; Neill, J. T. (2011). Program evaluation. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 164–182). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.010

Neill, J. T., &amp; Gray, T. L. (2011). Technology, risk and outdoor programming. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 132–149). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.008
}}

==2010==
Mackay, G. J., &amp; Neill, J. T. (2010). The effect of “green exercise” on state anxiety and the role of exercise duration, intensity, and greenness: A quasi-experimental study. ''Psychology of Sport and Exercise'', ''11''(3), 238–245. https://doi.org/10.1016/j.psychsport.2010.01.002

==2008==
{{Hanging indent|1=Neill, J. T. (2008). Enhancing life effectiveness: The impacts of outdoor education programs. [Unpublished doctoral dissertation]. University of Western Sydney. https://researchdirect.westernsydney.edu.au/islandora/object/uws:6441/}}

==2002==
{{/2002/Dramaturgy}}

==1997==
Hattie, J., Marsh, H. W., Neill, J. T., &amp; Richards, G. E. (1997). Adventure education and Outward Bound: Out-of-class experiences that make a lasting difference. ''Review of Educational Research'', ''67''(1), 43-87. https://doi.org/10.3102/00346543067001043

==Reports==
{{Hanging indent|1=
Neill, J. T. &amp; Bowen, D. J. (2014). ''[https://drive.google.com/file/d/0B2N4zSp4hmN9WUF3bzhuZ3JoNGM/view?usp=sharing&amp;resourcekey=0-y0ZTjcdhHXqQKNtz50BW0A Research evaluation of PCYC Bornhoffen Catalyst intervention programs for youth-at-risk &lt;nowiki&gt;[&lt;/nowiki&gt;2012-2013&lt;nowiki&gt;]&lt;/nowiki&gt;]''. University of Canberra.
}}

==Theses==
* [[User:Jtneill/PhD|PhD]]
&lt;!-- 
==Published==
* [http://www.wilderdom.com/JamesNeill/JamesNeillpublications.htm Articles &amp; presentations by James Neill]
--&gt;

==Ideas / In progress==
* [[User:Jtneill/4 pillars of free and open teaching|4 pillars of free and open teaching]]
* Some international trends in outdoor education - Past, present, and future
* Ingando camp (life effectiveness)
* Life Effectiveness Questionnaire psychometrics
* OE outcomes (longitudinal study)
* Adolescent Coping Scale psychometrics
* Resilience Scale psychometrics
* Overview of Outdoor Education Theory and/or Research
* Overview of Outdoor Education in Australia
* Overview of Adventure Therapy Theory and/or Education
* Past Trends and Future Directions for Outdoor Education
* Psychological Aspects of Outdoor Education
* Outdoor Education and Modern Technology
* Outdoor Education and Environmental Sustainability

==See also==
* [[User:Jtneill/Presentations]]</text>
      <sha1>2jhz9reca6ehsqucdn0q22ij53y2t5f</sha1>
    </revision>
    <revision>
      <id>2806950</id>
      <parentid>2806948</parentid>
      <timestamp>2026-04-29T04:25:33Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
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      <minor />
      <comment>/* 2020 */</comment>
      <origin>2806950</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7955" sha1="cj6n02ewde2g7tpgaivt691lwywwl8b" xml:space="preserve">See also: [[User:Jtneill/Research|Research]]
==[[../Research/Profiles|Profiles]]==
{{../Research/Profiles}}
&lt;!-- ==2026==
--&gt;
==2025==
{{Hanging indent|1=
Brichacek, A., Neill, J. T., Murray, K., Rieger, E., Watsford, C. (2025). Body Image Flexibility and Inflexibility Scale (BIFIS). In W. Ramseyer Winter, T. L. Tylka, &amp; A. M. Landor (Eds.), ''Handbook of body image-related measures''. Cambridge University Press (pp. 118–121). https://doi.org/10.1017/9781009398275.039

{{User:Jtneill/Publications/2025/Body}}&lt;!--
Neill, J. T., Herbert, S., Hartley, R., &amp; D'Cunha, N. (in preparation). ''Art for Wellbeing at the National Gallery of Australia: Thematic analysis of participant and staff perspectives''.

Lozancic Babic, V. &amp; Neill, J. T. ...  --&gt;
}}

==2024==
{{Hanging indent|1=

Black, H. M., &amp; Neill, J. T. (2024). Wellbeing through nature: A qualitative exploration of psychosocial aspects of a Landcare ACT nature-connection program. ''Journal of Outdoor and Environmental Education''. https://doi.org/10.1007/s42322-024-00184-2

Boerma, M., Beel, N., Neill, J. T., Jeffries, C., Krishnamoorthy, G., &amp; Guerri-Guttenberg, J. (2024). Male-friendly counselling for young men: a thematic analysis of client and caregiver experiences of Menslink counselling. ''Australian Psychologist'', 1–12. https://doi.org/10.1080/00050067.2024.2378119 ([https://www.researchgate.net/publication/385649063_Male-friendly_counselling_for_young_men_a_thematic_analysis_of_client_and_caregiver_experiences_of_Menslink_counselling#fullTextFileContent pdf])

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (accepted). The Body Image Flexibility and Inflexibility Scale (BIFIS). In V. Ramseyer Winter, T.  Tylka, &amp; A. Landor (Eds.), ''Handbook of body image-related measures'' (pp. *–*). Cambridge University Press.

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2024). The distinct affect regulation functions of body image flexibility and inflexibility: A prospective study in adolescents and emerging adults. ''Body Image'', ''50'', 101726. https://doi.org/10.1016/j.bodyim.2024.101726

{{User:Jtneill/Publications/2024/Collaborative}}

Neill, J. T. &amp; Black. H. (2024). ''[https://landcareact.org.au/wp-content/uploads/2026/01/Wellbeing-through-Nature-Final-Report.pdf Landcare ACT Wellbeing through Nature program evaluation: Final report'']. University of Canberra, Australia.

{{User:Jtneill/Publications/2024/Rich}}

}}

==2023==
{{Hanging indent|1=
Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2023). Ways of responding to body image threats: Development of the Body Image Flexibility and Inflexibility Scale for Youth. ''Journal of Contextual Behavioral Science'', ''30'', 31–40. https://doi.org/10.1016/j.jcbs.2023.08.007

{{/2023/WIL}}

Ross, B. M., &amp; Neill, J. T. (2023). Exploring the relationship between mental health, drug use, personality, and attitudes towards psilocybin-assisted therapy. ''[https://akjournals.com/view/journals/2054/2054-overview.xml Journal of Psychedelic Studies]'', ''7''(2), 114–118. https://doi.org/10.1556/2054.2023.00264}}

==2022==
{{Hanging indent|1=

Neill, J. T., Goch, I., Sullivan, A., &amp; Simons, M. (2022). The role of burn camp in the recovery of young people from burn injury: A qualitative study using long-term follow-up interviews with parents and participants. ''Burns'', ''48''(5), 1139–1148. https://doi.org/10.1016/j.burns.2021.09.020

Stevenson, D. J., Neill, J. T., Ball, K., Smith, R., &amp; Shores, M. C. (2022). How do preschool to year 6 educators prevent and cope with occupational violence from students? ''Australian Journal of Education'', ''66''(2), 154–170. https://doi.org/10.1177/00049441221092472. [https://www.teachermagazine.com/au_en/articles/the-research-files-episode-77-coping-with-violence-from-students Podcast].

}}

==2021==
{{Hanging indent|1=Brichacek, A. L., Murray, K., Neill, J. T., &amp; Rieger, E. (2021). Contextual behavioral approaches to understanding body image threats and coping in youth: A qualitative study. ''Journal of Adolescent Research'', ''39''(2), 328–360. https://doi.org/10.1177/07435584211007851}}

==2020==
{{Hanging indent|1=Boerma, M., &amp; Neill, D. J. (2020). The role of grit and self-control in university student academic achievement and satisfaction. ''College Student Journal''.

Boerma, M., Neill, J., &amp; Brown, P. (2020). Perseverance of effort moderates the relationship between psychological distress and life satisfaction. ''European Journal Of Applied Positive Psychology'', ''4''(16), 1–11.}}

==2016==
{{Hanging indent|1=

Bowen, D. J., &amp; Neill, J. T. (2016). Effects of the PCYC Catalyst outdoor adventure intervention program on youths' life skills, mental health, and delinquent behaviour. ''International Journal of Adolescence and Youth'', ''21''(1), 34–55. https://doi.org/10.1080/02673843.2015.1027716

{{/2016/Internationalisation}}}}

==2013==
{{Hanging indent|1=
{{/2013/Promoting}}

{{/2013/Teaching}}
}}

==2011==
{{Hanging indent|1=

Gray, T. L. &amp; Neill, J. T. (2011). Program evaluation. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 164–182). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.010

Neill, J. T., &amp; Gray, T. L. (2011). Technology, risk and outdoor programming. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 132–149). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.008
}}

==2010==
Mackay, G. J., &amp; Neill, J. T. (2010). The effect of “green exercise” on state anxiety and the role of exercise duration, intensity, and greenness: A quasi-experimental study. ''Psychology of Sport and Exercise'', ''11''(3), 238–245. https://doi.org/10.1016/j.psychsport.2010.01.002

==2008==
{{Hanging indent|1=Neill, J. T. (2008). Enhancing life effectiveness: The impacts of outdoor education programs. [Unpublished doctoral dissertation]. University of Western Sydney. https://researchdirect.westernsydney.edu.au/islandora/object/uws:6441/}}

==2002==
{{/2002/Dramaturgy}}

==1997==
Hattie, J., Marsh, H. W., Neill, J. T., &amp; Richards, G. E. (1997). Adventure education and Outward Bound: Out-of-class experiences that make a lasting difference. ''Review of Educational Research'', ''67''(1), 43-87. https://doi.org/10.3102/00346543067001043

==Reports==
{{Hanging indent|1=
Neill, J. T. &amp; Bowen, D. J. (2014). ''[https://drive.google.com/file/d/0B2N4zSp4hmN9WUF3bzhuZ3JoNGM/view?usp=sharing&amp;resourcekey=0-y0ZTjcdhHXqQKNtz50BW0A Research evaluation of PCYC Bornhoffen Catalyst intervention programs for youth-at-risk &lt;nowiki&gt;[&lt;/nowiki&gt;2012-2013&lt;nowiki&gt;]&lt;/nowiki&gt;]''. University of Canberra.
}}

==Theses==
* [[User:Jtneill/PhD|PhD]]
&lt;!-- 
==Published==
* [http://www.wilderdom.com/JamesNeill/JamesNeillpublications.htm Articles &amp; presentations by James Neill]
--&gt;

==Ideas / In progress==
* [[User:Jtneill/4 pillars of free and open teaching|4 pillars of free and open teaching]]
* Some international trends in outdoor education - Past, present, and future
* Ingando camp (life effectiveness)
* Life Effectiveness Questionnaire psychometrics
* OE outcomes (longitudinal study)
* Adolescent Coping Scale psychometrics
* Resilience Scale psychometrics
* Overview of Outdoor Education Theory and/or Research
* Overview of Outdoor Education in Australia
* Overview of Adventure Therapy Theory and/or Education
* Past Trends and Future Directions for Outdoor Education
* Psychological Aspects of Outdoor Education
* Outdoor Education and Modern Technology
* Outdoor Education and Environmental Sustainability

==See also==
* [[User:Jtneill/Presentations]]</text>
      <sha1>cj6n02ewde2g7tpgaivt691lwywwl8b</sha1>
    </revision>
    <revision>
      <id>2806951</id>
      <parentid>2806950</parentid>
      <timestamp>2026-04-29T04:25:52Z</timestamp>
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        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <comment>/* 2020 */</comment>
      <origin>2806951</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7955" sha1="5vu5f9plf9lc0ouqb7tbduxys1kx80k" xml:space="preserve">See also: [[User:Jtneill/Research|Research]]
==[[../Research/Profiles|Profiles]]==
{{../Research/Profiles}}
&lt;!-- ==2026==
--&gt;
==2025==
{{Hanging indent|1=
Brichacek, A., Neill, J. T., Murray, K., Rieger, E., Watsford, C. (2025). Body Image Flexibility and Inflexibility Scale (BIFIS). In W. Ramseyer Winter, T. L. Tylka, &amp; A. M. Landor (Eds.), ''Handbook of body image-related measures''. Cambridge University Press (pp. 118–121). https://doi.org/10.1017/9781009398275.039

{{User:Jtneill/Publications/2025/Body}}&lt;!--
Neill, J. T., Herbert, S., Hartley, R., &amp; D'Cunha, N. (in preparation). ''Art for Wellbeing at the National Gallery of Australia: Thematic analysis of participant and staff perspectives''.

Lozancic Babic, V. &amp; Neill, J. T. ...  --&gt;
}}

==2024==
{{Hanging indent|1=

Black, H. M., &amp; Neill, J. T. (2024). Wellbeing through nature: A qualitative exploration of psychosocial aspects of a Landcare ACT nature-connection program. ''Journal of Outdoor and Environmental Education''. https://doi.org/10.1007/s42322-024-00184-2

Boerma, M., Beel, N., Neill, J. T., Jeffries, C., Krishnamoorthy, G., &amp; Guerri-Guttenberg, J. (2024). Male-friendly counselling for young men: a thematic analysis of client and caregiver experiences of Menslink counselling. ''Australian Psychologist'', 1–12. https://doi.org/10.1080/00050067.2024.2378119 ([https://www.researchgate.net/publication/385649063_Male-friendly_counselling_for_young_men_a_thematic_analysis_of_client_and_caregiver_experiences_of_Menslink_counselling#fullTextFileContent pdf])

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (accepted). The Body Image Flexibility and Inflexibility Scale (BIFIS). In V. Ramseyer Winter, T.  Tylka, &amp; A. Landor (Eds.), ''Handbook of body image-related measures'' (pp. *–*). Cambridge University Press.

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2024). The distinct affect regulation functions of body image flexibility and inflexibility: A prospective study in adolescents and emerging adults. ''Body Image'', ''50'', 101726. https://doi.org/10.1016/j.bodyim.2024.101726

{{User:Jtneill/Publications/2024/Collaborative}}

Neill, J. T. &amp; Black. H. (2024). ''[https://landcareact.org.au/wp-content/uploads/2026/01/Wellbeing-through-Nature-Final-Report.pdf Landcare ACT Wellbeing through Nature program evaluation: Final report'']. University of Canberra, Australia.

{{User:Jtneill/Publications/2024/Rich}}

}}

==2023==
{{Hanging indent|1=
Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2023). Ways of responding to body image threats: Development of the Body Image Flexibility and Inflexibility Scale for Youth. ''Journal of Contextual Behavioral Science'', ''30'', 31–40. https://doi.org/10.1016/j.jcbs.2023.08.007

{{/2023/WIL}}

Ross, B. M., &amp; Neill, J. T. (2023). Exploring the relationship between mental health, drug use, personality, and attitudes towards psilocybin-assisted therapy. ''[https://akjournals.com/view/journals/2054/2054-overview.xml Journal of Psychedelic Studies]'', ''7''(2), 114–118. https://doi.org/10.1556/2054.2023.00264}}

==2022==
{{Hanging indent|1=

Neill, J. T., Goch, I., Sullivan, A., &amp; Simons, M. (2022). The role of burn camp in the recovery of young people from burn injury: A qualitative study using long-term follow-up interviews with parents and participants. ''Burns'', ''48''(5), 1139–1148. https://doi.org/10.1016/j.burns.2021.09.020

Stevenson, D. J., Neill, J. T., Ball, K., Smith, R., &amp; Shores, M. C. (2022). How do preschool to year 6 educators prevent and cope with occupational violence from students? ''Australian Journal of Education'', ''66''(2), 154–170. https://doi.org/10.1177/00049441221092472. [https://www.teachermagazine.com/au_en/articles/the-research-files-episode-77-coping-with-violence-from-students Podcast].

}}

==2021==
{{Hanging indent|1=Brichacek, A. L., Murray, K., Neill, J. T., &amp; Rieger, E. (2021). Contextual behavioral approaches to understanding body image threats and coping in youth: A qualitative study. ''Journal of Adolescent Research'', ''39''(2), 328–360. https://doi.org/10.1177/07435584211007851}}

==2020==
{{Hanging indent|1=Boerma, M., &amp; Neill, D. J. (2020). The role of grit and self-control in university student academic achievement and satisfaction. ''College Student Journal''.

Boerma, M., Neill, J., &amp; Brown, P. (2020). Perseverance of effort moderates the relationship between psychological distress and life satisfaction. ''European Journal of Applied Positive Psychology'', ''4''(16), 1–11.}}

==2016==
{{Hanging indent|1=

Bowen, D. J., &amp; Neill, J. T. (2016). Effects of the PCYC Catalyst outdoor adventure intervention program on youths' life skills, mental health, and delinquent behaviour. ''International Journal of Adolescence and Youth'', ''21''(1), 34–55. https://doi.org/10.1080/02673843.2015.1027716

{{/2016/Internationalisation}}}}

==2013==
{{Hanging indent|1=
{{/2013/Promoting}}

{{/2013/Teaching}}
}}

==2011==
{{Hanging indent|1=

Gray, T. L. &amp; Neill, J. T. (2011). Program evaluation. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 164–182). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.010

Neill, J. T., &amp; Gray, T. L. (2011). Technology, risk and outdoor programming. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 132–149). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.008
}}

==2010==
Mackay, G. J., &amp; Neill, J. T. (2010). The effect of “green exercise” on state anxiety and the role of exercise duration, intensity, and greenness: A quasi-experimental study. ''Psychology of Sport and Exercise'', ''11''(3), 238–245. https://doi.org/10.1016/j.psychsport.2010.01.002

==2008==
{{Hanging indent|1=Neill, J. T. (2008). Enhancing life effectiveness: The impacts of outdoor education programs. [Unpublished doctoral dissertation]. University of Western Sydney. https://researchdirect.westernsydney.edu.au/islandora/object/uws:6441/}}

==2002==
{{/2002/Dramaturgy}}

==1997==
Hattie, J., Marsh, H. W., Neill, J. T., &amp; Richards, G. E. (1997). Adventure education and Outward Bound: Out-of-class experiences that make a lasting difference. ''Review of Educational Research'', ''67''(1), 43-87. https://doi.org/10.3102/00346543067001043

==Reports==
{{Hanging indent|1=
Neill, J. T. &amp; Bowen, D. J. (2014). ''[https://drive.google.com/file/d/0B2N4zSp4hmN9WUF3bzhuZ3JoNGM/view?usp=sharing&amp;resourcekey=0-y0ZTjcdhHXqQKNtz50BW0A Research evaluation of PCYC Bornhoffen Catalyst intervention programs for youth-at-risk &lt;nowiki&gt;[&lt;/nowiki&gt;2012-2013&lt;nowiki&gt;]&lt;/nowiki&gt;]''. University of Canberra.
}}

==Theses==
* [[User:Jtneill/PhD|PhD]]
&lt;!-- 
==Published==
* [http://www.wilderdom.com/JamesNeill/JamesNeillpublications.htm Articles &amp; presentations by James Neill]
--&gt;

==Ideas / In progress==
* [[User:Jtneill/4 pillars of free and open teaching|4 pillars of free and open teaching]]
* Some international trends in outdoor education - Past, present, and future
* Ingando camp (life effectiveness)
* Life Effectiveness Questionnaire psychometrics
* OE outcomes (longitudinal study)
* Adolescent Coping Scale psychometrics
* Resilience Scale psychometrics
* Overview of Outdoor Education Theory and/or Research
* Overview of Outdoor Education in Australia
* Overview of Adventure Therapy Theory and/or Education
* Past Trends and Future Directions for Outdoor Education
* Psychological Aspects of Outdoor Education
* Outdoor Education and Modern Technology
* Outdoor Education and Environmental Sustainability

==See also==
* [[User:Jtneill/Presentations]]</text>
      <sha1>5vu5f9plf9lc0ouqb7tbduxys1kx80k</sha1>
    </revision>
    <revision>
      <id>2806953</id>
      <parentid>2806951</parentid>
      <timestamp>2026-04-29T04:28:52Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <comment>/* 2020 */</comment>
      <origin>2806953</origin>
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      <format>text/x-wiki</format>
      <text bytes="8039" sha1="rhw0e8nk4od49ryi55o2mf5f96anx7t" xml:space="preserve">See also: [[User:Jtneill/Research|Research]]
==[[../Research/Profiles|Profiles]]==
{{../Research/Profiles}}
&lt;!-- ==2026==
--&gt;
==2025==
{{Hanging indent|1=
Brichacek, A., Neill, J. T., Murray, K., Rieger, E., Watsford, C. (2025). Body Image Flexibility and Inflexibility Scale (BIFIS). In W. Ramseyer Winter, T. L. Tylka, &amp; A. M. Landor (Eds.), ''Handbook of body image-related measures''. Cambridge University Press (pp. 118–121). https://doi.org/10.1017/9781009398275.039

{{User:Jtneill/Publications/2025/Body}}&lt;!--
Neill, J. T., Herbert, S., Hartley, R., &amp; D'Cunha, N. (in preparation). ''Art for Wellbeing at the National Gallery of Australia: Thematic analysis of participant and staff perspectives''.

Lozancic Babic, V. &amp; Neill, J. T. ...  --&gt;
}}

==2024==
{{Hanging indent|1=

Black, H. M., &amp; Neill, J. T. (2024). Wellbeing through nature: A qualitative exploration of psychosocial aspects of a Landcare ACT nature-connection program. ''Journal of Outdoor and Environmental Education''. https://doi.org/10.1007/s42322-024-00184-2

Boerma, M., Beel, N., Neill, J. T., Jeffries, C., Krishnamoorthy, G., &amp; Guerri-Guttenberg, J. (2024). Male-friendly counselling for young men: a thematic analysis of client and caregiver experiences of Menslink counselling. ''Australian Psychologist'', 1–12. https://doi.org/10.1080/00050067.2024.2378119 ([https://www.researchgate.net/publication/385649063_Male-friendly_counselling_for_young_men_a_thematic_analysis_of_client_and_caregiver_experiences_of_Menslink_counselling#fullTextFileContent pdf])

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (accepted). The Body Image Flexibility and Inflexibility Scale (BIFIS). In V. Ramseyer Winter, T.  Tylka, &amp; A. Landor (Eds.), ''Handbook of body image-related measures'' (pp. *–*). Cambridge University Press.

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2024). The distinct affect regulation functions of body image flexibility and inflexibility: A prospective study in adolescents and emerging adults. ''Body Image'', ''50'', 101726. https://doi.org/10.1016/j.bodyim.2024.101726

{{User:Jtneill/Publications/2024/Collaborative}}

Neill, J. T. &amp; Black. H. (2024). ''[https://landcareact.org.au/wp-content/uploads/2026/01/Wellbeing-through-Nature-Final-Report.pdf Landcare ACT Wellbeing through Nature program evaluation: Final report'']. University of Canberra, Australia.

{{User:Jtneill/Publications/2024/Rich}}

}}

==2023==
{{Hanging indent|1=
Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2023). Ways of responding to body image threats: Development of the Body Image Flexibility and Inflexibility Scale for Youth. ''Journal of Contextual Behavioral Science'', ''30'', 31–40. https://doi.org/10.1016/j.jcbs.2023.08.007

{{/2023/WIL}}

Ross, B. M., &amp; Neill, J. T. (2023). Exploring the relationship between mental health, drug use, personality, and attitudes towards psilocybin-assisted therapy. ''[https://akjournals.com/view/journals/2054/2054-overview.xml Journal of Psychedelic Studies]'', ''7''(2), 114–118. https://doi.org/10.1556/2054.2023.00264}}

==2022==
{{Hanging indent|1=

Neill, J. T., Goch, I., Sullivan, A., &amp; Simons, M. (2022). The role of burn camp in the recovery of young people from burn injury: A qualitative study using long-term follow-up interviews with parents and participants. ''Burns'', ''48''(5), 1139–1148. https://doi.org/10.1016/j.burns.2021.09.020

Stevenson, D. J., Neill, J. T., Ball, K., Smith, R., &amp; Shores, M. C. (2022). How do preschool to year 6 educators prevent and cope with occupational violence from students? ''Australian Journal of Education'', ''66''(2), 154–170. https://doi.org/10.1177/00049441221092472. [https://www.teachermagazine.com/au_en/articles/the-research-files-episode-77-coping-with-violence-from-students Podcast].

}}

==2021==
{{Hanging indent|1=Brichacek, A. L., Murray, K., Neill, J. T., &amp; Rieger, E. (2021). Contextual behavioral approaches to understanding body image threats and coping in youth: A qualitative study. ''Journal of Adolescent Research'', ''39''(2), 328–360. https://doi.org/10.1177/07435584211007851}}

==2020==
{{Hanging indent|1=Boerma, M., &amp; Neill, D. J. (2020). The role of grit and self-control in university student academic achievement and satisfaction. ''College Student Journal''.

Boerma, M., Neill, J., &amp; Brown, P. (2020). Perseverance of effort moderates the relationship between psychological distress and life satisfaction. ''European Journal of Applied Positive Psychology'', ''4''(16), 1–11. https://www.nationalwellbeingservice.org/volumes/volume-4-2020/volume-4-article-16/}}

==2016==
{{Hanging indent|1=

Bowen, D. J., &amp; Neill, J. T. (2016). Effects of the PCYC Catalyst outdoor adventure intervention program on youths' life skills, mental health, and delinquent behaviour. ''International Journal of Adolescence and Youth'', ''21''(1), 34–55. https://doi.org/10.1080/02673843.2015.1027716

{{/2016/Internationalisation}}}}

==2013==
{{Hanging indent|1=
{{/2013/Promoting}}

{{/2013/Teaching}}
}}

==2011==
{{Hanging indent|1=

Gray, T. L. &amp; Neill, J. T. (2011). Program evaluation. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 164–182). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.010

Neill, J. T., &amp; Gray, T. L. (2011). Technology, risk and outdoor programming. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 132–149). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.008
}}

==2010==
Mackay, G. J., &amp; Neill, J. T. (2010). The effect of “green exercise” on state anxiety and the role of exercise duration, intensity, and greenness: A quasi-experimental study. ''Psychology of Sport and Exercise'', ''11''(3), 238–245. https://doi.org/10.1016/j.psychsport.2010.01.002

==2008==
{{Hanging indent|1=Neill, J. T. (2008). Enhancing life effectiveness: The impacts of outdoor education programs. [Unpublished doctoral dissertation]. University of Western Sydney. https://researchdirect.westernsydney.edu.au/islandora/object/uws:6441/}}

==2002==
{{/2002/Dramaturgy}}

==1997==
Hattie, J., Marsh, H. W., Neill, J. T., &amp; Richards, G. E. (1997). Adventure education and Outward Bound: Out-of-class experiences that make a lasting difference. ''Review of Educational Research'', ''67''(1), 43-87. https://doi.org/10.3102/00346543067001043

==Reports==
{{Hanging indent|1=
Neill, J. T. &amp; Bowen, D. J. (2014). ''[https://drive.google.com/file/d/0B2N4zSp4hmN9WUF3bzhuZ3JoNGM/view?usp=sharing&amp;resourcekey=0-y0ZTjcdhHXqQKNtz50BW0A Research evaluation of PCYC Bornhoffen Catalyst intervention programs for youth-at-risk &lt;nowiki&gt;[&lt;/nowiki&gt;2012-2013&lt;nowiki&gt;]&lt;/nowiki&gt;]''. University of Canberra.
}}

==Theses==
* [[User:Jtneill/PhD|PhD]]
&lt;!-- 
==Published==
* [http://www.wilderdom.com/JamesNeill/JamesNeillpublications.htm Articles &amp; presentations by James Neill]
--&gt;

==Ideas / In progress==
* [[User:Jtneill/4 pillars of free and open teaching|4 pillars of free and open teaching]]
* Some international trends in outdoor education - Past, present, and future
* Ingando camp (life effectiveness)
* Life Effectiveness Questionnaire psychometrics
* OE outcomes (longitudinal study)
* Adolescent Coping Scale psychometrics
* Resilience Scale psychometrics
* Overview of Outdoor Education Theory and/or Research
* Overview of Outdoor Education in Australia
* Overview of Adventure Therapy Theory and/or Education
* Past Trends and Future Directions for Outdoor Education
* Psychological Aspects of Outdoor Education
* Outdoor Education and Modern Technology
* Outdoor Education and Environmental Sustainability

==See also==
* [[User:Jtneill/Presentations]]</text>
      <sha1>rhw0e8nk4od49ryi55o2mf5f96anx7t</sha1>
    </revision>
    <revision>
      <id>2806954</id>
      <parentid>2806953</parentid>
      <timestamp>2026-04-29T04:34:56Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <comment>/* 2020 */</comment>
      <origin>2806954</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8058" sha1="bci49ke8c5zk5hbewg4qx714vxjdx2o" xml:space="preserve">See also: [[User:Jtneill/Research|Research]]
==[[../Research/Profiles|Profiles]]==
{{../Research/Profiles}}
&lt;!-- ==2026==
--&gt;
==2025==
{{Hanging indent|1=
Brichacek, A., Neill, J. T., Murray, K., Rieger, E., Watsford, C. (2025). Body Image Flexibility and Inflexibility Scale (BIFIS). In W. Ramseyer Winter, T. L. Tylka, &amp; A. M. Landor (Eds.), ''Handbook of body image-related measures''. Cambridge University Press (pp. 118–121). https://doi.org/10.1017/9781009398275.039

{{User:Jtneill/Publications/2025/Body}}&lt;!--
Neill, J. T., Herbert, S., Hartley, R., &amp; D'Cunha, N. (in preparation). ''Art for Wellbeing at the National Gallery of Australia: Thematic analysis of participant and staff perspectives''.

Lozancic Babic, V. &amp; Neill, J. T. ...  --&gt;
}}

==2024==
{{Hanging indent|1=

Black, H. M., &amp; Neill, J. T. (2024). Wellbeing through nature: A qualitative exploration of psychosocial aspects of a Landcare ACT nature-connection program. ''Journal of Outdoor and Environmental Education''. https://doi.org/10.1007/s42322-024-00184-2

Boerma, M., Beel, N., Neill, J. T., Jeffries, C., Krishnamoorthy, G., &amp; Guerri-Guttenberg, J. (2024). Male-friendly counselling for young men: a thematic analysis of client and caregiver experiences of Menslink counselling. ''Australian Psychologist'', 1–12. https://doi.org/10.1080/00050067.2024.2378119 ([https://www.researchgate.net/publication/385649063_Male-friendly_counselling_for_young_men_a_thematic_analysis_of_client_and_caregiver_experiences_of_Menslink_counselling#fullTextFileContent pdf])

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (accepted). The Body Image Flexibility and Inflexibility Scale (BIFIS). In V. Ramseyer Winter, T.  Tylka, &amp; A. Landor (Eds.), ''Handbook of body image-related measures'' (pp. *–*). Cambridge University Press.

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2024). The distinct affect regulation functions of body image flexibility and inflexibility: A prospective study in adolescents and emerging adults. ''Body Image'', ''50'', 101726. https://doi.org/10.1016/j.bodyim.2024.101726

{{User:Jtneill/Publications/2024/Collaborative}}

Neill, J. T. &amp; Black. H. (2024). ''[https://landcareact.org.au/wp-content/uploads/2026/01/Wellbeing-through-Nature-Final-Report.pdf Landcare ACT Wellbeing through Nature program evaluation: Final report'']. University of Canberra, Australia.

{{User:Jtneill/Publications/2024/Rich}}

}}

==2023==
{{Hanging indent|1=
Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2023). Ways of responding to body image threats: Development of the Body Image Flexibility and Inflexibility Scale for Youth. ''Journal of Contextual Behavioral Science'', ''30'', 31–40. https://doi.org/10.1016/j.jcbs.2023.08.007

{{/2023/WIL}}

Ross, B. M., &amp; Neill, J. T. (2023). Exploring the relationship between mental health, drug use, personality, and attitudes towards psilocybin-assisted therapy. ''[https://akjournals.com/view/journals/2054/2054-overview.xml Journal of Psychedelic Studies]'', ''7''(2), 114–118. https://doi.org/10.1556/2054.2023.00264}}

==2022==
{{Hanging indent|1=

Neill, J. T., Goch, I., Sullivan, A., &amp; Simons, M. (2022). The role of burn camp in the recovery of young people from burn injury: A qualitative study using long-term follow-up interviews with parents and participants. ''Burns'', ''48''(5), 1139–1148. https://doi.org/10.1016/j.burns.2021.09.020

Stevenson, D. J., Neill, J. T., Ball, K., Smith, R., &amp; Shores, M. C. (2022). How do preschool to year 6 educators prevent and cope with occupational violence from students? ''Australian Journal of Education'', ''66''(2), 154–170. https://doi.org/10.1177/00049441221092472. [https://www.teachermagazine.com/au_en/articles/the-research-files-episode-77-coping-with-violence-from-students Podcast].

}}

==2021==
{{Hanging indent|1=Brichacek, A. L., Murray, K., Neill, J. T., &amp; Rieger, E. (2021). Contextual behavioral approaches to understanding body image threats and coping in youth: A qualitative study. ''Journal of Adolescent Research'', ''39''(2), 328–360. https://doi.org/10.1177/07435584211007851}}

==2020==
{{Hanging indent|1=Boerma, M., &amp; Neill, J. (2020). The role of grit and self-control in university student academic achievement and satisfaction. ''College Student Journal'', ''54''(4), 431–442.

Boerma, M., Neill, J., &amp; Brown, P. (2020). Perseverance of effort moderates the relationship between psychological distress and life satisfaction. ''European Journal of Applied Positive Psychology'', ''4''(16), 1–11. https://www.nationalwellbeingservice.org/volumes/volume-4-2020/volume-4-article-16/}}

==2016==
{{Hanging indent|1=

Bowen, D. J., &amp; Neill, J. T. (2016). Effects of the PCYC Catalyst outdoor adventure intervention program on youths' life skills, mental health, and delinquent behaviour. ''International Journal of Adolescence and Youth'', ''21''(1), 34–55. https://doi.org/10.1080/02673843.2015.1027716

{{/2016/Internationalisation}}}}

==2013==
{{Hanging indent|1=
{{/2013/Promoting}}

{{/2013/Teaching}}
}}

==2011==
{{Hanging indent|1=

Gray, T. L. &amp; Neill, J. T. (2011). Program evaluation. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 164–182). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.010

Neill, J. T., &amp; Gray, T. L. (2011). Technology, risk and outdoor programming. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 132–149). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.008
}}

==2010==
Mackay, G. J., &amp; Neill, J. T. (2010). The effect of “green exercise” on state anxiety and the role of exercise duration, intensity, and greenness: A quasi-experimental study. ''Psychology of Sport and Exercise'', ''11''(3), 238–245. https://doi.org/10.1016/j.psychsport.2010.01.002

==2008==
{{Hanging indent|1=Neill, J. T. (2008). Enhancing life effectiveness: The impacts of outdoor education programs. [Unpublished doctoral dissertation]. University of Western Sydney. https://researchdirect.westernsydney.edu.au/islandora/object/uws:6441/}}

==2002==
{{/2002/Dramaturgy}}

==1997==
Hattie, J., Marsh, H. W., Neill, J. T., &amp; Richards, G. E. (1997). Adventure education and Outward Bound: Out-of-class experiences that make a lasting difference. ''Review of Educational Research'', ''67''(1), 43-87. https://doi.org/10.3102/00346543067001043

==Reports==
{{Hanging indent|1=
Neill, J. T. &amp; Bowen, D. J. (2014). ''[https://drive.google.com/file/d/0B2N4zSp4hmN9WUF3bzhuZ3JoNGM/view?usp=sharing&amp;resourcekey=0-y0ZTjcdhHXqQKNtz50BW0A Research evaluation of PCYC Bornhoffen Catalyst intervention programs for youth-at-risk &lt;nowiki&gt;[&lt;/nowiki&gt;2012-2013&lt;nowiki&gt;]&lt;/nowiki&gt;]''. University of Canberra.
}}

==Theses==
* [[User:Jtneill/PhD|PhD]]
&lt;!-- 
==Published==
* [http://www.wilderdom.com/JamesNeill/JamesNeillpublications.htm Articles &amp; presentations by James Neill]
--&gt;

==Ideas / In progress==
* [[User:Jtneill/4 pillars of free and open teaching|4 pillars of free and open teaching]]
* Some international trends in outdoor education - Past, present, and future
* Ingando camp (life effectiveness)
* Life Effectiveness Questionnaire psychometrics
* OE outcomes (longitudinal study)
* Adolescent Coping Scale psychometrics
* Resilience Scale psychometrics
* Overview of Outdoor Education Theory and/or Research
* Overview of Outdoor Education in Australia
* Overview of Adventure Therapy Theory and/or Education
* Past Trends and Future Directions for Outdoor Education
* Psychological Aspects of Outdoor Education
* Outdoor Education and Modern Technology
* Outdoor Education and Environmental Sustainability

==See also==
* [[User:Jtneill/Presentations]]</text>
      <sha1>bci49ke8c5zk5hbewg4qx714vxjdx2o</sha1>
    </revision>
    <revision>
      <id>2806956</id>
      <parentid>2806954</parentid>
      <timestamp>2026-04-29T04:38:55Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>+ 2018</comment>
      <origin>2806956</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8311" sha1="bl5eu4kcdixatals2b3wjb7it6j8vs8" xml:space="preserve">See also: [[User:Jtneill/Research|Research]]
==[[../Research/Profiles|Profiles]]==
{{../Research/Profiles}}
&lt;!-- ==2026==
--&gt;
==2025==
{{Hanging indent|1=
Brichacek, A., Neill, J. T., Murray, K., Rieger, E., Watsford, C. (2025). Body Image Flexibility and Inflexibility Scale (BIFIS). In W. Ramseyer Winter, T. L. Tylka, &amp; A. M. Landor (Eds.), ''Handbook of body image-related measures''. Cambridge University Press (pp. 118–121). https://doi.org/10.1017/9781009398275.039

{{User:Jtneill/Publications/2025/Body}}&lt;!--
Neill, J. T., Herbert, S., Hartley, R., &amp; D'Cunha, N. (in preparation). ''Art for Wellbeing at the National Gallery of Australia: Thematic analysis of participant and staff perspectives''.

Lozancic Babic, V. &amp; Neill, J. T. ...  --&gt;
}}

==2024==
{{Hanging indent|1=

Black, H. M., &amp; Neill, J. T. (2024). Wellbeing through nature: A qualitative exploration of psychosocial aspects of a Landcare ACT nature-connection program. ''Journal of Outdoor and Environmental Education''. https://doi.org/10.1007/s42322-024-00184-2

Boerma, M., Beel, N., Neill, J. T., Jeffries, C., Krishnamoorthy, G., &amp; Guerri-Guttenberg, J. (2024). Male-friendly counselling for young men: a thematic analysis of client and caregiver experiences of Menslink counselling. ''Australian Psychologist'', 1–12. https://doi.org/10.1080/00050067.2024.2378119 ([https://www.researchgate.net/publication/385649063_Male-friendly_counselling_for_young_men_a_thematic_analysis_of_client_and_caregiver_experiences_of_Menslink_counselling#fullTextFileContent pdf])

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (accepted). The Body Image Flexibility and Inflexibility Scale (BIFIS). In V. Ramseyer Winter, T.  Tylka, &amp; A. Landor (Eds.), ''Handbook of body image-related measures'' (pp. *–*). Cambridge University Press.

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2024). The distinct affect regulation functions of body image flexibility and inflexibility: A prospective study in adolescents and emerging adults. ''Body Image'', ''50'', 101726. https://doi.org/10.1016/j.bodyim.2024.101726

{{User:Jtneill/Publications/2024/Collaborative}}

Neill, J. T. &amp; Black. H. (2024). ''[https://landcareact.org.au/wp-content/uploads/2026/01/Wellbeing-through-Nature-Final-Report.pdf Landcare ACT Wellbeing through Nature program evaluation: Final report'']. University of Canberra, Australia.

{{User:Jtneill/Publications/2024/Rich}}

}}

==2023==
{{Hanging indent|1=
Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2023). Ways of responding to body image threats: Development of the Body Image Flexibility and Inflexibility Scale for Youth. ''Journal of Contextual Behavioral Science'', ''30'', 31–40. https://doi.org/10.1016/j.jcbs.2023.08.007

{{/2023/WIL}}

Ross, B. M., &amp; Neill, J. T. (2023). Exploring the relationship between mental health, drug use, personality, and attitudes towards psilocybin-assisted therapy. ''[https://akjournals.com/view/journals/2054/2054-overview.xml Journal of Psychedelic Studies]'', ''7''(2), 114–118. https://doi.org/10.1556/2054.2023.00264}}

==2022==
{{Hanging indent|1=

Neill, J. T., Goch, I., Sullivan, A., &amp; Simons, M. (2022). The role of burn camp in the recovery of young people from burn injury: A qualitative study using long-term follow-up interviews with parents and participants. ''Burns'', ''48''(5), 1139–1148. https://doi.org/10.1016/j.burns.2021.09.020

Stevenson, D. J., Neill, J. T., Ball, K., Smith, R., &amp; Shores, M. C. (2022). How do preschool to year 6 educators prevent and cope with occupational violence from students? ''Australian Journal of Education'', ''66''(2), 154–170. https://doi.org/10.1177/00049441221092472. [https://www.teachermagazine.com/au_en/articles/the-research-files-episode-77-coping-with-violence-from-students Podcast].

}}

==2021==
{{Hanging indent|1=Brichacek, A. L., Murray, K., Neill, J. T., &amp; Rieger, E. (2021). Contextual behavioral approaches to understanding body image threats and coping in youth: A qualitative study. ''Journal of Adolescent Research'', ''39''(2), 328–360. https://doi.org/10.1177/07435584211007851}}

==2020==
{{Hanging indent|1=Boerma, M., &amp; Neill, J. (2020). The role of grit and self-control in university student academic achievement and satisfaction. ''College Student Journal'', ''54''(4), 431–442.

Boerma, M., Neill, J., &amp; Brown, P. (2020). Perseverance of effort moderates the relationship between psychological distress and life satisfaction. ''European Journal of Applied Positive Psychology'', ''4''(16), 1–11. https://www.nationalwellbeingservice.org/volumes/volume-4-2020/volume-4-article-16/}}

==2018==
Neill, J. T. (2018). ''[https://menslink.org.au/wp-content/uploads/2018/10/UC-Report-into-Long-term-Impacts-of-Menslink-Counselling-and-Mentoring-Oct-2018.pdf Long-term impacts of Menslink counselling and mentoring]''. University of Canberra.

==2016==
{{Hanging indent|1=

Bowen, D. J., &amp; Neill, J. T. (2016). Effects of the PCYC Catalyst outdoor adventure intervention program on youths' life skills, mental health, and delinquent behaviour. ''International Journal of Adolescence and Youth'', ''21''(1), 34–55. https://doi.org/10.1080/02673843.2015.1027716

{{/2016/Internationalisation}}}}

==2013==
{{Hanging indent|1=
{{/2013/Promoting}}

{{/2013/Teaching}}
}}

==2011==
{{Hanging indent|1=

Gray, T. L. &amp; Neill, J. T. (2011). Program evaluation. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 164–182). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.010

Neill, J. T., &amp; Gray, T. L. (2011). Technology, risk and outdoor programming. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 132–149). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.008
}}

==2010==
Mackay, G. J., &amp; Neill, J. T. (2010). The effect of “green exercise” on state anxiety and the role of exercise duration, intensity, and greenness: A quasi-experimental study. ''Psychology of Sport and Exercise'', ''11''(3), 238–245. https://doi.org/10.1016/j.psychsport.2010.01.002

==2008==
{{Hanging indent|1=Neill, J. T. (2008). Enhancing life effectiveness: The impacts of outdoor education programs. [Unpublished doctoral dissertation]. University of Western Sydney. https://researchdirect.westernsydney.edu.au/islandora/object/uws:6441/}}

==2002==
{{/2002/Dramaturgy}}

==1997==
Hattie, J., Marsh, H. W., Neill, J. T., &amp; Richards, G. E. (1997). Adventure education and Outward Bound: Out-of-class experiences that make a lasting difference. ''Review of Educational Research'', ''67''(1), 43-87. https://doi.org/10.3102/00346543067001043

==Reports==
{{Hanging indent|1=
Neill, J. T. &amp; Bowen, D. J. (2014). ''[https://drive.google.com/file/d/0B2N4zSp4hmN9WUF3bzhuZ3JoNGM/view?usp=sharing&amp;resourcekey=0-y0ZTjcdhHXqQKNtz50BW0A Research evaluation of PCYC Bornhoffen Catalyst intervention programs for youth-at-risk &lt;nowiki&gt;[&lt;/nowiki&gt;2012-2013&lt;nowiki&gt;]&lt;/nowiki&gt;]''. University of Canberra.
}}

==Theses==
* [[User:Jtneill/PhD|PhD]]
&lt;!-- 
==Published==
* [http://www.wilderdom.com/JamesNeill/JamesNeillpublications.htm Articles &amp; presentations by James Neill]
--&gt;

==Ideas / In progress==
* [[User:Jtneill/4 pillars of free and open teaching|4 pillars of free and open teaching]]
* Some international trends in outdoor education - Past, present, and future
* Ingando camp (life effectiveness)
* Life Effectiveness Questionnaire psychometrics
* OE outcomes (longitudinal study)
* Adolescent Coping Scale psychometrics
* Resilience Scale psychometrics
* Overview of Outdoor Education Theory and/or Research
* Overview of Outdoor Education in Australia
* Overview of Adventure Therapy Theory and/or Education
* Past Trends and Future Directions for Outdoor Education
* Psychological Aspects of Outdoor Education
* Outdoor Education and Modern Technology
* Outdoor Education and Environmental Sustainability

==See also==
* [[User:Jtneill/Presentations]]</text>
      <sha1>bl5eu4kcdixatals2b3wjb7it6j8vs8</sha1>
    </revision>
    <revision>
      <id>2806958</id>
      <parentid>2806956</parentid>
      <timestamp>2026-04-29T04:41:55Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* 2016 */ + 1</comment>
      <origin>2806958</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8549" sha1="kwu4hhhyfhyy68h10ym4g1odow0c2az" xml:space="preserve">See also: [[User:Jtneill/Research|Research]]
==[[../Research/Profiles|Profiles]]==
{{../Research/Profiles}}
&lt;!-- ==2026==
--&gt;
==2025==
{{Hanging indent|1=
Brichacek, A., Neill, J. T., Murray, K., Rieger, E., Watsford, C. (2025). Body Image Flexibility and Inflexibility Scale (BIFIS). In W. Ramseyer Winter, T. L. Tylka, &amp; A. M. Landor (Eds.), ''Handbook of body image-related measures''. Cambridge University Press (pp. 118–121). https://doi.org/10.1017/9781009398275.039

{{User:Jtneill/Publications/2025/Body}}&lt;!--
Neill, J. T., Herbert, S., Hartley, R., &amp; D'Cunha, N. (in preparation). ''Art for Wellbeing at the National Gallery of Australia: Thematic analysis of participant and staff perspectives''.

Lozancic Babic, V. &amp; Neill, J. T. ...  --&gt;
}}

==2024==
{{Hanging indent|1=

Black, H. M., &amp; Neill, J. T. (2024). Wellbeing through nature: A qualitative exploration of psychosocial aspects of a Landcare ACT nature-connection program. ''Journal of Outdoor and Environmental Education''. https://doi.org/10.1007/s42322-024-00184-2

Boerma, M., Beel, N., Neill, J. T., Jeffries, C., Krishnamoorthy, G., &amp; Guerri-Guttenberg, J. (2024). Male-friendly counselling for young men: a thematic analysis of client and caregiver experiences of Menslink counselling. ''Australian Psychologist'', 1–12. https://doi.org/10.1080/00050067.2024.2378119 ([https://www.researchgate.net/publication/385649063_Male-friendly_counselling_for_young_men_a_thematic_analysis_of_client_and_caregiver_experiences_of_Menslink_counselling#fullTextFileContent pdf])

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (accepted). The Body Image Flexibility and Inflexibility Scale (BIFIS). In V. Ramseyer Winter, T.  Tylka, &amp; A. Landor (Eds.), ''Handbook of body image-related measures'' (pp. *–*). Cambridge University Press.

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2024). The distinct affect regulation functions of body image flexibility and inflexibility: A prospective study in adolescents and emerging adults. ''Body Image'', ''50'', 101726. https://doi.org/10.1016/j.bodyim.2024.101726

{{User:Jtneill/Publications/2024/Collaborative}}

Neill, J. T. &amp; Black. H. (2024). ''[https://landcareact.org.au/wp-content/uploads/2026/01/Wellbeing-through-Nature-Final-Report.pdf Landcare ACT Wellbeing through Nature program evaluation: Final report'']. University of Canberra, Australia.

{{User:Jtneill/Publications/2024/Rich}}

}}

==2023==
{{Hanging indent|1=
Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2023). Ways of responding to body image threats: Development of the Body Image Flexibility and Inflexibility Scale for Youth. ''Journal of Contextual Behavioral Science'', ''30'', 31–40. https://doi.org/10.1016/j.jcbs.2023.08.007

{{/2023/WIL}}

Ross, B. M., &amp; Neill, J. T. (2023). Exploring the relationship between mental health, drug use, personality, and attitudes towards psilocybin-assisted therapy. ''[https://akjournals.com/view/journals/2054/2054-overview.xml Journal of Psychedelic Studies]'', ''7''(2), 114–118. https://doi.org/10.1556/2054.2023.00264}}

==2022==
{{Hanging indent|1=

Neill, J. T., Goch, I., Sullivan, A., &amp; Simons, M. (2022). The role of burn camp in the recovery of young people from burn injury: A qualitative study using long-term follow-up interviews with parents and participants. ''Burns'', ''48''(5), 1139–1148. https://doi.org/10.1016/j.burns.2021.09.020

Stevenson, D. J., Neill, J. T., Ball, K., Smith, R., &amp; Shores, M. C. (2022). How do preschool to year 6 educators prevent and cope with occupational violence from students? ''Australian Journal of Education'', ''66''(2), 154–170. https://doi.org/10.1177/00049441221092472. [https://www.teachermagazine.com/au_en/articles/the-research-files-episode-77-coping-with-violence-from-students Podcast].

}}

==2021==
{{Hanging indent|1=Brichacek, A. L., Murray, K., Neill, J. T., &amp; Rieger, E. (2021). Contextual behavioral approaches to understanding body image threats and coping in youth: A qualitative study. ''Journal of Adolescent Research'', ''39''(2), 328–360. https://doi.org/10.1177/07435584211007851}}

==2020==
{{Hanging indent|1=Boerma, M., &amp; Neill, J. (2020). The role of grit and self-control in university student academic achievement and satisfaction. ''College Student Journal'', ''54''(4), 431–442.

Boerma, M., Neill, J., &amp; Brown, P. (2020). Perseverance of effort moderates the relationship between psychological distress and life satisfaction. ''European Journal of Applied Positive Psychology'', ''4''(16), 1–11. https://www.nationalwellbeingservice.org/volumes/volume-4-2020/volume-4-article-16/}}

==2018==
Neill, J. T. (2018). ''[https://menslink.org.au/wp-content/uploads/2018/10/UC-Report-into-Long-term-Impacts-of-Menslink-Counselling-and-Mentoring-Oct-2018.pdf Long-term impacts of Menslink counselling and mentoring]''. University of Canberra.

==2016==
{{Hanging indent|1=

Bowen, D. J., &amp; Neill, J. T. (2016). Effects of the PCYC Catalyst outdoor adventure intervention program on youths' life skills, mental health, and delinquent behaviour. ''International Journal of Adolescence and Youth'', ''21''(1), 34–55. https://doi.org/10.1080/02673843.2015.1027716

Bowen, D. J., Neill, J. T., &amp; Crisp, S. J. (2016). Wilderness adventure therapy effects on the mental health of youth participants. ''Evaluation and Program Planning'', ''58'', 49–59. https://doi.org/10.1016/j.evalprogplan.2016.05.005

{{/2016/Internationalisation}}}}

==2013==
{{Hanging indent|1=
{{/2013/Promoting}}

{{/2013/Teaching}}
}}

==2011==
{{Hanging indent|1=

Gray, T. L. &amp; Neill, J. T. (2011). Program evaluation. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 164–182). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.010

Neill, J. T., &amp; Gray, T. L. (2011). Technology, risk and outdoor programming. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 132–149). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.008
}}

==2010==
Mackay, G. J., &amp; Neill, J. T. (2010). The effect of “green exercise” on state anxiety and the role of exercise duration, intensity, and greenness: A quasi-experimental study. ''Psychology of Sport and Exercise'', ''11''(3), 238–245. https://doi.org/10.1016/j.psychsport.2010.01.002

==2008==
{{Hanging indent|1=Neill, J. T. (2008). Enhancing life effectiveness: The impacts of outdoor education programs. [Unpublished doctoral dissertation]. University of Western Sydney. https://researchdirect.westernsydney.edu.au/islandora/object/uws:6441/}}

==2002==
{{/2002/Dramaturgy}}

==1997==
Hattie, J., Marsh, H. W., Neill, J. T., &amp; Richards, G. E. (1997). Adventure education and Outward Bound: Out-of-class experiences that make a lasting difference. ''Review of Educational Research'', ''67''(1), 43-87. https://doi.org/10.3102/00346543067001043

==Reports==
{{Hanging indent|1=
Neill, J. T. &amp; Bowen, D. J. (2014). ''[https://drive.google.com/file/d/0B2N4zSp4hmN9WUF3bzhuZ3JoNGM/view?usp=sharing&amp;resourcekey=0-y0ZTjcdhHXqQKNtz50BW0A Research evaluation of PCYC Bornhoffen Catalyst intervention programs for youth-at-risk &lt;nowiki&gt;[&lt;/nowiki&gt;2012-2013&lt;nowiki&gt;]&lt;/nowiki&gt;]''. University of Canberra.
}}

==Theses==
* [[User:Jtneill/PhD|PhD]]
&lt;!-- 
==Published==
* [http://www.wilderdom.com/JamesNeill/JamesNeillpublications.htm Articles &amp; presentations by James Neill]
--&gt;

==Ideas / In progress==
* [[User:Jtneill/4 pillars of free and open teaching|4 pillars of free and open teaching]]
* Some international trends in outdoor education - Past, present, and future
* Ingando camp (life effectiveness)
* Life Effectiveness Questionnaire psychometrics
* OE outcomes (longitudinal study)
* Adolescent Coping Scale psychometrics
* Resilience Scale psychometrics
* Overview of Outdoor Education Theory and/or Research
* Overview of Outdoor Education in Australia
* Overview of Adventure Therapy Theory and/or Education
* Past Trends and Future Directions for Outdoor Education
* Psychological Aspects of Outdoor Education
* Outdoor Education and Modern Technology
* Outdoor Education and Environmental Sustainability

==See also==
* [[User:Jtneill/Presentations]]</text>
      <sha1>kwu4hhhyfhyy68h10ym4g1odow0c2az</sha1>
    </revision>
    <revision>
      <id>2806959</id>
      <parentid>2806958</parentid>
      <timestamp>2026-04-29T04:44:37Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>+ 2017</comment>
      <origin>2806959</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8767" sha1="tjmk5buzdql3e4a24gx7ajc0mex9hsk" xml:space="preserve">See also: [[User:Jtneill/Research|Research]]
==[[../Research/Profiles|Profiles]]==
{{../Research/Profiles}}
&lt;!-- ==2026==
--&gt;
==2025==
{{Hanging indent|1=
Brichacek, A., Neill, J. T., Murray, K., Rieger, E., Watsford, C. (2025). Body Image Flexibility and Inflexibility Scale (BIFIS). In W. Ramseyer Winter, T. L. Tylka, &amp; A. M. Landor (Eds.), ''Handbook of body image-related measures''. Cambridge University Press (pp. 118–121). https://doi.org/10.1017/9781009398275.039

{{User:Jtneill/Publications/2025/Body}}&lt;!--
Neill, J. T., Herbert, S., Hartley, R., &amp; D'Cunha, N. (in preparation). ''Art for Wellbeing at the National Gallery of Australia: Thematic analysis of participant and staff perspectives''.

Lozancic Babic, V. &amp; Neill, J. T. ...  --&gt;
}}

==2024==
{{Hanging indent|1=

Black, H. M., &amp; Neill, J. T. (2024). Wellbeing through nature: A qualitative exploration of psychosocial aspects of a Landcare ACT nature-connection program. ''Journal of Outdoor and Environmental Education''. https://doi.org/10.1007/s42322-024-00184-2

Boerma, M., Beel, N., Neill, J. T., Jeffries, C., Krishnamoorthy, G., &amp; Guerri-Guttenberg, J. (2024). Male-friendly counselling for young men: a thematic analysis of client and caregiver experiences of Menslink counselling. ''Australian Psychologist'', 1–12. https://doi.org/10.1080/00050067.2024.2378119 ([https://www.researchgate.net/publication/385649063_Male-friendly_counselling_for_young_men_a_thematic_analysis_of_client_and_caregiver_experiences_of_Menslink_counselling#fullTextFileContent pdf])

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (accepted). The Body Image Flexibility and Inflexibility Scale (BIFIS). In V. Ramseyer Winter, T.  Tylka, &amp; A. Landor (Eds.), ''Handbook of body image-related measures'' (pp. *–*). Cambridge University Press.

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2024). The distinct affect regulation functions of body image flexibility and inflexibility: A prospective study in adolescents and emerging adults. ''Body Image'', ''50'', 101726. https://doi.org/10.1016/j.bodyim.2024.101726

{{User:Jtneill/Publications/2024/Collaborative}}

Neill, J. T. &amp; Black. H. (2024). ''[https://landcareact.org.au/wp-content/uploads/2026/01/Wellbeing-through-Nature-Final-Report.pdf Landcare ACT Wellbeing through Nature program evaluation: Final report'']. University of Canberra, Australia.

{{User:Jtneill/Publications/2024/Rich}}

}}

==2023==
{{Hanging indent|1=
Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2023). Ways of responding to body image threats: Development of the Body Image Flexibility and Inflexibility Scale for Youth. ''Journal of Contextual Behavioral Science'', ''30'', 31–40. https://doi.org/10.1016/j.jcbs.2023.08.007

{{/2023/WIL}}

Ross, B. M., &amp; Neill, J. T. (2023). Exploring the relationship between mental health, drug use, personality, and attitudes towards psilocybin-assisted therapy. ''[https://akjournals.com/view/journals/2054/2054-overview.xml Journal of Psychedelic Studies]'', ''7''(2), 114–118. https://doi.org/10.1556/2054.2023.00264}}

==2022==
{{Hanging indent|1=

Neill, J. T., Goch, I., Sullivan, A., &amp; Simons, M. (2022). The role of burn camp in the recovery of young people from burn injury: A qualitative study using long-term follow-up interviews with parents and participants. ''Burns'', ''48''(5), 1139–1148. https://doi.org/10.1016/j.burns.2021.09.020

Stevenson, D. J., Neill, J. T., Ball, K., Smith, R., &amp; Shores, M. C. (2022). How do preschool to year 6 educators prevent and cope with occupational violence from students? ''Australian Journal of Education'', ''66''(2), 154–170. https://doi.org/10.1177/00049441221092472. [https://www.teachermagazine.com/au_en/articles/the-research-files-episode-77-coping-with-violence-from-students Podcast].

}}

==2021==
{{Hanging indent|1=Brichacek, A. L., Murray, K., Neill, J. T., &amp; Rieger, E. (2021). Contextual behavioral approaches to understanding body image threats and coping in youth: A qualitative study. ''Journal of Adolescent Research'', ''39''(2), 328–360. https://doi.org/10.1177/07435584211007851}}

==2020==
{{Hanging indent|1=Boerma, M., &amp; Neill, J. (2020). The role of grit and self-control in university student academic achievement and satisfaction. ''College Student Journal'', ''54''(4), 431–442.

Boerma, M., Neill, J., &amp; Brown, P. (2020). Perseverance of effort moderates the relationship between psychological distress and life satisfaction. ''European Journal of Applied Positive Psychology'', ''4''(16), 1–11. https://www.nationalwellbeingservice.org/volumes/volume-4-2020/volume-4-article-16/}}

==2018==
{{Hanging indent|1=Neill, J. T. (2018). ''[https://menslink.org.au/wp-content/uploads/2018/10/UC-Report-into-Long-term-Impacts-of-Menslink-Counselling-and-Mentoring-Oct-2018.pdf Long-term impacts of Menslink counselling and mentoring]''. University of Canberra.}}

==2017==
{{Hanging indent|1=Booth, J. W., &amp; Neill, J. T. (2017). Coping strategies and the development of psychological resilience. Journal of Outdoor and Environmental Education, 20(1), 47-54.}}

==2016==
{{Hanging indent|1=

Bowen, D. J., &amp; Neill, J. T. (2016). Effects of the PCYC Catalyst outdoor adventure intervention program on youths' life skills, mental health, and delinquent behaviour. ''International Journal of Adolescence and Youth'', ''21''(1), 34–55. https://doi.org/10.1080/02673843.2015.1027716

Bowen, D. J., Neill, J. T., &amp; Crisp, S. J. (2016). Wilderness adventure therapy effects on the mental health of youth participants. ''Evaluation and Program Planning'', ''58'', 49–59. https://doi.org/10.1016/j.evalprogplan.2016.05.005

{{/2016/Internationalisation}}}}

==2013==
{{Hanging indent|1=
{{/2013/Promoting}}

{{/2013/Teaching}}
}}

==2011==
{{Hanging indent|1=

Gray, T. L. &amp; Neill, J. T. (2011). Program evaluation. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 164–182). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.010

Neill, J. T., &amp; Gray, T. L. (2011). Technology, risk and outdoor programming. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 132–149). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.008
}}

==2010==
Mackay, G. J., &amp; Neill, J. T. (2010). The effect of “green exercise” on state anxiety and the role of exercise duration, intensity, and greenness: A quasi-experimental study. ''Psychology of Sport and Exercise'', ''11''(3), 238–245. https://doi.org/10.1016/j.psychsport.2010.01.002

==2008==
{{Hanging indent|1=Neill, J. T. (2008). Enhancing life effectiveness: The impacts of outdoor education programs. [Unpublished doctoral dissertation]. University of Western Sydney. https://researchdirect.westernsydney.edu.au/islandora/object/uws:6441/}}

==2002==
{{/2002/Dramaturgy}}

==1997==
Hattie, J., Marsh, H. W., Neill, J. T., &amp; Richards, G. E. (1997). Adventure education and Outward Bound: Out-of-class experiences that make a lasting difference. ''Review of Educational Research'', ''67''(1), 43-87. https://doi.org/10.3102/00346543067001043

==Reports==
{{Hanging indent|1=
Neill, J. T. &amp; Bowen, D. J. (2014). ''[https://drive.google.com/file/d/0B2N4zSp4hmN9WUF3bzhuZ3JoNGM/view?usp=sharing&amp;resourcekey=0-y0ZTjcdhHXqQKNtz50BW0A Research evaluation of PCYC Bornhoffen Catalyst intervention programs for youth-at-risk &lt;nowiki&gt;[&lt;/nowiki&gt;2012-2013&lt;nowiki&gt;]&lt;/nowiki&gt;]''. University of Canberra.
}}

==Theses==
* [[User:Jtneill/PhD|PhD]]
&lt;!-- 
==Published==
* [http://www.wilderdom.com/JamesNeill/JamesNeillpublications.htm Articles &amp; presentations by James Neill]
--&gt;

==Ideas / In progress==
* [[User:Jtneill/4 pillars of free and open teaching|4 pillars of free and open teaching]]
* Some international trends in outdoor education - Past, present, and future
* Ingando camp (life effectiveness)
* Life Effectiveness Questionnaire psychometrics
* OE outcomes (longitudinal study)
* Adolescent Coping Scale psychometrics
* Resilience Scale psychometrics
* Overview of Outdoor Education Theory and/or Research
* Overview of Outdoor Education in Australia
* Overview of Adventure Therapy Theory and/or Education
* Past Trends and Future Directions for Outdoor Education
* Psychological Aspects of Outdoor Education
* Outdoor Education and Modern Technology
* Outdoor Education and Environmental Sustainability

==See also==
* [[User:Jtneill/Presentations]]</text>
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    </revision>
    <revision>
      <id>2806960</id>
      <parentid>2806959</parentid>
      <timestamp>2026-04-29T04:46:35Z</timestamp>
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        <username>Jtneill</username>
        <id>10242</id>
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      <text bytes="8812" sha1="m4q4iyctott3py2fi99shzhntg73mhh" xml:space="preserve">See also: [[User:Jtneill/Research|Research]]
==[[../Research/Profiles|Profiles]]==
{{../Research/Profiles}}
&lt;!-- ==2026==
--&gt;
==2025==
{{Hanging indent|1=
Brichacek, A., Neill, J. T., Murray, K., Rieger, E., Watsford, C. (2025). Body Image Flexibility and Inflexibility Scale (BIFIS). In W. Ramseyer Winter, T. L. Tylka, &amp; A. M. Landor (Eds.), ''Handbook of body image-related measures''. Cambridge University Press (pp. 118–121). https://doi.org/10.1017/9781009398275.039

{{User:Jtneill/Publications/2025/Body}}&lt;!--
Neill, J. T., Herbert, S., Hartley, R., &amp; D'Cunha, N. (in preparation). ''Art for Wellbeing at the National Gallery of Australia: Thematic analysis of participant and staff perspectives''.

Lozancic Babic, V. &amp; Neill, J. T. ...  --&gt;
}}

==2024==
{{Hanging indent|1=

Black, H. M., &amp; Neill, J. T. (2024). Wellbeing through nature: A qualitative exploration of psychosocial aspects of a Landcare ACT nature-connection program. ''Journal of Outdoor and Environmental Education''. https://doi.org/10.1007/s42322-024-00184-2

Boerma, M., Beel, N., Neill, J. T., Jeffries, C., Krishnamoorthy, G., &amp; Guerri-Guttenberg, J. (2024). Male-friendly counselling for young men: a thematic analysis of client and caregiver experiences of Menslink counselling. ''Australian Psychologist'', 1–12. https://doi.org/10.1080/00050067.2024.2378119 ([https://www.researchgate.net/publication/385649063_Male-friendly_counselling_for_young_men_a_thematic_analysis_of_client_and_caregiver_experiences_of_Menslink_counselling#fullTextFileContent pdf])

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (accepted). The Body Image Flexibility and Inflexibility Scale (BIFIS). In V. Ramseyer Winter, T.  Tylka, &amp; A. Landor (Eds.), ''Handbook of body image-related measures'' (pp. *–*). Cambridge University Press.

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2024). The distinct affect regulation functions of body image flexibility and inflexibility: A prospective study in adolescents and emerging adults. ''Body Image'', ''50'', 101726. https://doi.org/10.1016/j.bodyim.2024.101726

{{User:Jtneill/Publications/2024/Collaborative}}

Neill, J. T. &amp; Black. H. (2024). ''[https://landcareact.org.au/wp-content/uploads/2026/01/Wellbeing-through-Nature-Final-Report.pdf Landcare ACT Wellbeing through Nature program evaluation: Final report'']. University of Canberra, Australia.

{{User:Jtneill/Publications/2024/Rich}}

}}

==2023==
{{Hanging indent|1=
Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2023). Ways of responding to body image threats: Development of the Body Image Flexibility and Inflexibility Scale for Youth. ''Journal of Contextual Behavioral Science'', ''30'', 31–40. https://doi.org/10.1016/j.jcbs.2023.08.007

{{/2023/WIL}}

Ross, B. M., &amp; Neill, J. T. (2023). Exploring the relationship between mental health, drug use, personality, and attitudes towards psilocybin-assisted therapy. ''[https://akjournals.com/view/journals/2054/2054-overview.xml Journal of Psychedelic Studies]'', ''7''(2), 114–118. https://doi.org/10.1556/2054.2023.00264}}

==2022==
{{Hanging indent|1=

Neill, J. T., Goch, I., Sullivan, A., &amp; Simons, M. (2022). The role of burn camp in the recovery of young people from burn injury: A qualitative study using long-term follow-up interviews with parents and participants. ''Burns'', ''48''(5), 1139–1148. https://doi.org/10.1016/j.burns.2021.09.020

Stevenson, D. J., Neill, J. T., Ball, K., Smith, R., &amp; Shores, M. C. (2022). How do preschool to year 6 educators prevent and cope with occupational violence from students? ''Australian Journal of Education'', ''66''(2), 154–170. https://doi.org/10.1177/00049441221092472. [https://www.teachermagazine.com/au_en/articles/the-research-files-episode-77-coping-with-violence-from-students Podcast].

}}

==2021==
{{Hanging indent|1=Brichacek, A. L., Murray, K., Neill, J. T., &amp; Rieger, E. (2021). Contextual behavioral approaches to understanding body image threats and coping in youth: A qualitative study. ''Journal of Adolescent Research'', ''39''(2), 328–360. https://doi.org/10.1177/07435584211007851}}

==2020==
{{Hanging indent|1=Boerma, M., &amp; Neill, J. (2020). The role of grit and self-control in university student academic achievement and satisfaction. ''College Student Journal'', ''54''(4), 431–442.

Boerma, M., Neill, J., &amp; Brown, P. (2020). Perseverance of effort moderates the relationship between psychological distress and life satisfaction. ''European Journal of Applied Positive Psychology'', ''4''(16), 1–11. https://www.nationalwellbeingservice.org/volumes/volume-4-2020/volume-4-article-16/}}

==2018==
{{Hanging indent|1=Neill, J. T. (2018). ''[https://menslink.org.au/wp-content/uploads/2018/10/UC-Report-into-Long-term-Impacts-of-Menslink-Counselling-and-Mentoring-Oct-2018.pdf Long-term impacts of Menslink counselling and mentoring]''. University of Canberra.}}

==2017==
{{Hanging indent|1=Booth, J. W., &amp; Neill, J. T. (2017). Coping strategies and the development of psychological resilience. ''Journal of Outdoor and Environmental Education'', ''20''(1), 47–54. https://doi.org/10.1007/BF03401002}}

==2016==
{{Hanging indent|1=

Bowen, D. J., &amp; Neill, J. T. (2016). Effects of the PCYC Catalyst outdoor adventure intervention program on youths' life skills, mental health, and delinquent behaviour. ''International Journal of Adolescence and Youth'', ''21''(1), 34–55. https://doi.org/10.1080/02673843.2015.1027716

Bowen, D. J., Neill, J. T., &amp; Crisp, S. J. (2016). Wilderness adventure therapy effects on the mental health of youth participants. ''Evaluation and Program Planning'', ''58'', 49–59. https://doi.org/10.1016/j.evalprogplan.2016.05.005

{{/2016/Internationalisation}}}}

==2013==
{{Hanging indent|1=
{{/2013/Promoting}}

{{/2013/Teaching}}
}}

==2011==
{{Hanging indent|1=

Gray, T. L. &amp; Neill, J. T. (2011). Program evaluation. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 164–182). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.010

Neill, J. T., &amp; Gray, T. L. (2011). Technology, risk and outdoor programming. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 132–149). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.008
}}

==2010==
Mackay, G. J., &amp; Neill, J. T. (2010). The effect of “green exercise” on state anxiety and the role of exercise duration, intensity, and greenness: A quasi-experimental study. ''Psychology of Sport and Exercise'', ''11''(3), 238–245. https://doi.org/10.1016/j.psychsport.2010.01.002

==2008==
{{Hanging indent|1=Neill, J. T. (2008). Enhancing life effectiveness: The impacts of outdoor education programs. [Unpublished doctoral dissertation]. University of Western Sydney. https://researchdirect.westernsydney.edu.au/islandora/object/uws:6441/}}

==2002==
{{/2002/Dramaturgy}}

==1997==
Hattie, J., Marsh, H. W., Neill, J. T., &amp; Richards, G. E. (1997). Adventure education and Outward Bound: Out-of-class experiences that make a lasting difference. ''Review of Educational Research'', ''67''(1), 43-87. https://doi.org/10.3102/00346543067001043

==Reports==
{{Hanging indent|1=
Neill, J. T. &amp; Bowen, D. J. (2014). ''[https://drive.google.com/file/d/0B2N4zSp4hmN9WUF3bzhuZ3JoNGM/view?usp=sharing&amp;resourcekey=0-y0ZTjcdhHXqQKNtz50BW0A Research evaluation of PCYC Bornhoffen Catalyst intervention programs for youth-at-risk &lt;nowiki&gt;[&lt;/nowiki&gt;2012-2013&lt;nowiki&gt;]&lt;/nowiki&gt;]''. University of Canberra.
}}

==Theses==
* [[User:Jtneill/PhD|PhD]]
&lt;!-- 
==Published==
* [http://www.wilderdom.com/JamesNeill/JamesNeillpublications.htm Articles &amp; presentations by James Neill]
--&gt;

==Ideas / In progress==
* [[User:Jtneill/4 pillars of free and open teaching|4 pillars of free and open teaching]]
* Some international trends in outdoor education - Past, present, and future
* Ingando camp (life effectiveness)
* Life Effectiveness Questionnaire psychometrics
* OE outcomes (longitudinal study)
* Adolescent Coping Scale psychometrics
* Resilience Scale psychometrics
* Overview of Outdoor Education Theory and/or Research
* Overview of Outdoor Education in Australia
* Overview of Adventure Therapy Theory and/or Education
* Past Trends and Future Directions for Outdoor Education
* Psychological Aspects of Outdoor Education
* Outdoor Education and Modern Technology
* Outdoor Education and Environmental Sustainability

==See also==
* [[User:Jtneill/Presentations]]</text>
      <sha1>m4q4iyctott3py2fi99shzhntg73mhh</sha1>
    </revision>
  </page>
  <page>
    <title>Wikiversity:Request custodian action</title>
    <ns>4</ns>
    <id>75745</id>
    <revision>
      <id>2806871</id>
      <parentid>2806846</parentid>
      <timestamp>2026-04-28T14:19:34Z</timestamp>
      <contributor>
        <username>Codename Noreste</username>
        <id>2969951</id>
      </contributor>
      <comment>/* Request for Page Creation Approval - Research on Dr. Ian Stevenson */ reply ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
      <origin>2806871</origin>
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      <format>text/x-wiki</format>
      <text bytes="38023" sha1="iznr194pn6oadmxiqj0s1jr1z7ygcu0" xml:space="preserve">{{/Header}}
== Dan Polansky ==

I would like to ask you to assess the behavior of Dan Polansky, who in my opinion continues to violate [[Wikiversity:Etiquette|Etiquette]], calls users who disagree with him trolls, [https://en.wikiversity.org/w/index.php?title=User_talk:Harold_Foppele&amp;oldid=2760143#Your_qualification questions their expertise], tests them, etc. This is most evident in [[Wikiversity:Community Review/Dan Polansky]], where he has already indicated that two discussion opponents are trolls. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:05, 18 November 2025 (UTC)
: The coddling of overt disruptor Harold Foppele (substantiation is in RCA above) and proven provocateur and disruptor Juandev (substantiation in CR above) must stop. The English Wikiversity must start to properly curate its content and discipline disruptive editors. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:10, 18 November 2025 (UTC)
:[[Wikiversity:Community Review/Dan Polansky]] is underway; outcome pending. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:03, 27 November 2025 (UTC)
::It has been closed with consensus to ban him indefinitely from this project, I believe there is nothing else to do here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:06, 11 March 2026 (UTC)

== Sidewide count.js ==

i would like something like: [[Template:User contrib count/count.js]]. i created [[Template:User contrib count]] and a user/common.js. {{User contrib count}}.&lt;br&gt;&lt;br&gt; so a &quot;count.js&quot; would complete it. See [[User:Harold Foppele/common.js]].
If an Administrator could help please. Cheers [[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 19:22, 18 January 2026 (UTC)

== need to add my profile ==

im trying to add new profile content  [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 18:03, 4 February 2026 (UTC)

:You can edit it now. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:05, 4 February 2026 (UTC)
::where can create a new one [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 18:51, 4 February 2026 (UTC)
:::i have created and its in sandbox. i would like to know when it will be approved  [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 19:38, 4 February 2026 (UTC)
::::Please don’t create [[wv:spam|spam]] pages as it will be deleted. Please also read [[WV:Scope]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 04:01, 5 February 2026 (UTC)

== Im trying to add new profile while add content its shows not alowed ==

This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please [[Wikiversity:Request custodian action|inform an administrator]] of what you were trying to do. A brief description of the abuse rule which your action matched is: New User Exceeded New Page Limit

This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please [[Wikiversity:Request custodian action|inform an administrator]] of what you were trying to do. A brief description of the abuse rule which your action matched is: New User Created Page with External Link [[User:PAGURUMURTHY|PAGURUMURTHY]] ([[User talk:PAGURUMURTHY|discuss]] • [[Special:Contributions/PAGURUMURTHY|contribs]]) 18:51, 4 February 2026 (UTC)

== New User: cannot create talk page ==

Hi, I am a new user of Wikiversity and I wanted to create a talk page for the article [[ChatGPT's Essay on Kohlberg's Theory: AI's Use in Academic Writing]]. As a new user, I was barred from performing this action. The text that I wanted to add to the talk page is:

&lt;blockquote&gt;
I have doubts as to to the reliability of this essay. Take for rexample the sentence:

&lt;blockquote&gt;
Due to its efficiency, AI has saved 380,000-403,000 lives per year in European healthcare as reported in a recent Deloitte and MedTech Europe report&lt;ref&gt;Dantas, C., Mackiewicz, K., Tageo, V., Jacucci, G., Guardado, D., Ortet, S., Varlamis, I., Maniadakis, M., De Lera, E., Quintas, J., Kocsis, O., &amp; Vassiliou, C. (2021). Benefits and hurdles of AI in the workplace – what comes next? ''International Journal of Artificial Intelligence and Expert Systems, 10'', 9-17. https://www.researchgate.net/publication/351993615_Benefits_and_Hurdles_of_AI_In_The_Workplace_-What_Comes_Next&lt;/ref&gt;.
&lt;/blockquote&gt;

Reading the reference (freely available on ResearchGate), one notes:

# that the reference is from 2021 (predating the widespread use of LLMs such as ChatGPT and the associated 'AI' boom), and
# that the reference factually contradicts this essay.

Quoting from the reference:

&lt;blockquote&gt;
There are enormous benefits of applying AI-based solutions to monitor workers’ health and prevent accidents or, currently, COVID-19 infections, and those benefits are reported with enormous potential. According to the recent Deloitte and MedTech Europe report [11], implementing AI in European healthcare systems could save up 380,000 to 403,000 lives annually or €170.9 to 212.4 billion per year.
&lt;/blockquote&gt;

Not that the reference says ''could save'', not ''saves'' as in the essay.

This calls into question the reliability of the essay.

&lt;/blockquote&gt;

Could an administrator make this addition for me? Thank you! 
{{reflist}}
[[User:Æolus|Æolus]] ([[User talk:Æolus|discuss]] • [[Special:Contributions/Æolus|contribs]]) 06:53, 5 February 2026 (UTC)

:@[[User:Æolus|Æolus]] I have added it for you, you can change the header and sign it now. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 08:05, 5 February 2026 (UTC)
::Thank you! [[User:Æolus|Æolus]] ([[User talk:Æolus|discuss]] • [[Special:Contributions/Æolus|contribs]]) 12:43, 5 February 2026 (UTC)

== Disallowed to add a page on a course ==

I'm trying to populate a newly created course on Wikiversity, but it blocks me from creating more pages with &quot;New User Exceeded New Page Limit&quot;. Could this be lifted please? [[User:Berkeleywho|Berkeleywho]] ([[User talk:Berkeleywho|discuss]] • [[Special:Contributions/Berkeleywho|contribs]]) 13:21, 15 February 2026 (UTC)

:Sorry! Never mind. I was trying to create a new article instead of a new page. All good now. [[User:Berkeleywho|Berkeleywho]] ([[User talk:Berkeleywho|discuss]] • [[Special:Contributions/Berkeleywho|contribs]]) 14:03, 15 February 2026 (UTC)

== Harold Foppele adding LLM-generated nonsense and personal fiction ==

I became aware of [[User:Harold Foppele]]'s editing after I deleted some of his uploads on Commons. He appears to be adding a large amount of text and images that are some combination of personal fiction and LLM-generated nonsense. This includes:
*[[Quantum Ultra fast lasers#Future thought experiment|Personal speculative fiction]] in an otherwise &quot;nonfiction&quot; article
*Uploading nonsense LLM-created [[:File:Rontosecond pulse laser (Schematic).jpg|diagrams]] and [[:File:Rontosecond pulse laser (Futuristic).jpg|renders]] for nonexistent lab equipment, with fake source (on Commons, he indicated these files as having been created by him using an LLM)
*Uploading nonsense LLM-created images of equations with obvious artifacts. These images, such as [[:File:Redfield equation (non-Markovian).png]] and [[:File:Lindblad equation (Markovian).png]], don't even match the text he puts them with.
Much of his writing is also of extremely poor quality, to the point where it's not clear whether it's written by him or an LLM. I'm not an active editor on this project, so I'm not as familiar with the standards here, but I believe this is worth custodian attention. [[User:Pi.1415926535|Pi.1415926535]] ([[User talk:Pi.1415926535|discuss]] • [[Special:Contributions/Pi.1415926535|contribs]]) 03:06, 23 February 2026 (UTC)

:Fake source ''and'' contradictory copyright info, claiming both public domain and CC license. Moreover, if they are indeed nearly-direct LLM output, depending on jurisdiction they may not even be eligible for copyright.
:I've put speedy deletion marks for the equations, because they're obviously not coherent mathematical equations (the parentheses don't match, the symbols merge into each other the way text in image models often do, etc) [[User:Sesquilinear|Sesquilinear]] ([[User talk:Sesquilinear|discuss]] • [[Special:Contributions/Sesquilinear|contribs]]) 21:50, 7 March 2026 (UTC)

== Repeated removal of RFD notices by Harold Foppele ==

{{User|Harold Foppele }}

This editor is appearing in multiple noticeboards for behaviour which is contentious. Ther latest adventure is the repeated removal of tye RFD notice at [[Quantum/Henry C. Kapteyn]]. You will see from their contributions record the number of times. I have warned Tham on their user tag page that this is tantaomunt to volunteering to be blocked here. They have a track record of achieving blocks on enWiki and Commons already.

They have all the appearance of shooting not to understand when given direct information about their behaviour, whichever project they are editing, and are fast becoming a time sink. Their behaviour across multiple WMF sites may well lead to a Global Lock, but I do not believe they have quite reached the threshold for that.

I believe that what is required is a preventative block to seek to ensure thatchy understand the seriousness of their behaviour, and the need to be collegial. 🇵🇸&amp;zwj;🇺🇦&amp;nbsp;[[User:Timtrent|Timtrent]]&amp;nbsp;🇺🇦&amp;nbsp;[[User talk:Timtrent|talk to me]]&amp;nbsp;🇺🇦&amp;zwj;🇵🇸 23:03, 4 March 2026 (UTC)
: {{Done}} [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 11:45, 8 March 2026 (UTC)

== Blocks for sockpuppet ==

Please block [[User:Harold Foppele]] and [[User:Johnwilliamsiii]] for sockpuppetry based on [https://en.wikipedia.org/wiki/Wikipedia:Sockpuppet_investigations/Harold_Foppele en wiki] CU and [https://commons.wikimedia.org/w/index.php?diff=1177465640 commons] CU investigations. The user has also violated copyright, see the above discussion. A block is necessary to prevent further abuse. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:30, 8 March 2026 (UTC)

:&lt;small&gt;@[[User:MathXplore|MathXplore]]&lt;/small&gt; [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:31, 8 March 2026 (UTC)
:: {{Done}} [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 11:44, 8 March 2026 (UTC)
:CC. @[[User:Timtrent|Timtrent]], @[[User:Sesquilinear|Sesquilinear]], @[[User:Pi.1415926535|Pi.1415926535]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:33, 8 March 2026 (UTC)
::Thank you for the ping. I concur based on [[w:en:WP:DUCK|behaviour]]. CUs appear divided. 🇵🇸&amp;zwj;🇺🇦&amp;nbsp;[[User:Timtrent|Timtrent]]&amp;nbsp;🇺🇦&amp;nbsp;[[User talk:Timtrent|talk to me]]&amp;nbsp;🇺🇦&amp;zwj;🇵🇸 11:41, 8 March 2026 (UTC)

== Problem when trying to start a discussion with authors of the Plurilingual education portal ==

The authors I wanted to discuss with are called &quot;Project PEP&quot; and my name is Franch Chandler. How can I be allowed to do so ? [[User:French Chandler|French Chandler]] ([[User talk:French Chandler|discuss]] • [[Special:Contributions/French Chandler|contribs]]) 18:25, 16 March 2026 (UTC)

:@[[User:French Chandler|French Chandler]] place your qestion [https://en.wikiversity.org/w/index.php?title=User_talk:Projet_PEP&amp;action=edit into the dialog box] on this link and hit Publish page. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:22, 16 March 2026 (UTC)

== Please publish my post ==

My post is about &quot;Every child grows and develops at their own pace, but some may experience challenges that affect their ability to perform everyday tasks. These challenges can include difficulties with fine motor skills, sensory processing, handwriting, feeding, and self-regulation. When these issues are not addressed early, they can impact a child’s confidence, academic performance, and independence.

With the rise of digital healthcare services, '''online physical therapy''' has emerged as a powerful and accessible solution for parents seeking support for their children. This modern approach provides structured, personalized therapy programs that can be accessed from the comfort of home, making it easier for families to ensure consistent care.&quot; [[User:Skyabovetherapy|Skyabovetherapy]] ([[User talk:Skyabovetherapy|discuss]] • [[Special:Contributions/Skyabovetherapy|contribs]]) 12:28, 28 March 2026 (UTC)

:@[[User:Skyabovetherapy|Skyabovetherapy]] Well, you can publish it yourself, Wikiversity is a free environement, where everybody can create educational resources. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 14:11, 29 March 2026 (UTC)
::They actually triggered some abuse filters. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:24, 29 March 2026 (UTC)
:I looked at your attempts to add this text and I see a link to one website repeated many times, which reminds me of the misuse of Wikiversity for self-promotion or to increase the importance of the website. It is necessary to remind you here that Wikiversity is not a place for promotion, but a place for education. So if you want to educate, it will not be a problem to create the page without external links with a clearly defined procedure for how people should use it and what to expect from it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:07, 1 April 2026 (UTC)

== New user limit ==

Hi, I am creating an AIPA Method learning resource page. 

I am the author of the linked research, and I hit the “new user limit” and “new page with external link” filters while publishing. 

Here is the link to the page in creation: [https://en.wikiversity.org/w/index.php?title=AIPA_Method&amp;veaction=edit] 

Thank you for your help. 

Best regards, 

Senad Dizdarević [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:19, 30 March 2026 (UTC)

:@[[User:Senad Dizdarević|Senad Dizdarević]] I should admit I dont know, what is &quot;new user limit&quot;, but if filter blocks your page because of certain external link, you may force to save anyway and sometimes it works. It should not work, when the website is blacklisted. As of now, I am not seeing you to save page in main namespace, so try to save it without external links first. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:30, 30 March 2026 (UTC)
::Thank you, you are very kind. 
::I will wait a day, and try again (without links, too). 
::Today, I already created About Me info page, and maybe that is enough for the filters for one day.  [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 07:53, 30 March 2026 (UTC)
:::Well, I have analyzed your contribution to Wikiversity and I should point out here, that this project is not a place for advertising, so there is no way of promoting your books and authority this way. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:56, 1 April 2026 (UTC)
::::Hi, my About Me page is just an info page with the neutral as possible presentation of my work. 
::::There is a big difference between informing and advertising. Informing is neutrally stating that something exists and requiring no action, while advertising is a special communication form with intent to cause certain action from readers. For example, click here, click there, order this, buy that. 
::::There is no such intention, form, or terms on my info page. Just neutral information. I don't hide and I am not ashamed that I am write and author, and that is a part of the usual bio, including works. I checked your user page: &quot;I graduated from the Czech University of Life Sciences in Prague and studied information science at the Faculty of Arts of Charles University.&quot; I think that if you had written a book on Life Science, you would have mentioned that as well.
::::Most of the Info page is about my research and AIPA Method which is a valid contribution to psychology, consciousness studies, identity theory, and personality development. Actually, my paper '''AIPA Method: A Cognitive-Phenomenological Model for Identity Reconstruction and Stabilization in Pure Awareness''' is now in the peer review procedure at Journal of Consciousness Studies. 
::::Here is a part from the Wikiversity AIPA Method page in creation (waiting for the end of the time limit for new users):    [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 06:47, 2 April 2026 (UTC)
:::::For the unknown reasons, the form didn't publish my second part of the message: 
:::::I believe this is a valid contribution to Wikiversity. 
:::::Best Regards, 
:::::Senad [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 06:52, 2 April 2026 (UTC)
::::::And the third try: 
:::::: == Introduction ==
::::::The AIPA Method addresses a gap in contemporary personal development and consciousness science: most evidence‑based approaches (CBT, MBSR, MBCT, standard meditation) operate at the level of mental content—reframing thoughts, observing them, or reducing their impact—rather than at the level of identity structure. In contrast, AIPA targets the structural relationship between the self and the mind, aiming at durable identity reconstruction rooted in Pure Awareness rather than symptom management.
::::::The central research question of the primary AIPA preprint is whether a structured, sequentially staged method can produce permanent identity reconstruction rooted in Pure Awareness, and how such a method compares to established approaches in scope, mechanism, and outcome.
:::::: == Theoretical foundations ==
::::::The AIPA framework is grounded in the cognitive‑phenomenological tradition (e.g., McAdams, Varela, Metzinger, Erikson), contemporary consciousness science on minimal phenomenal experience, and qualitative methods advocacy in psychology. It builds directly on:
::::::* Empirical work on pure awareness and Minimal Phenomenal Experience (MPE), especially Gamma &amp; Metzinger’s large‑scale study of content‑reduced awareness states.
::::::* Metzinger’s proposal of minimal phenomenal experience as an entry point for a minimal unifying model of consciousness.
::::::* Narrative identity and partial‑self models within personality and identity theory.
::::::Within this backdrop, AIPA proposes Pure Awareness as a distinct, operationally specified state that can become a structural ground of identity rather than a transient meditative experience.
:::::: == Experiential empiricism ==
::::::The empirical foundation of the AIPA Method is explicitly first‑person and experiential, combining:
::::::* A 22‑year longitudinal autoethnographic self‑study (2003–2025) documenting partial personality episodes, protocol use, and outcomes.
::::::* A 13‑year prospective verification period with zero self‑reported recurrence of targeted harmful behaviors after a dated stabilization point (1 January 2006).
::::::* A high‑ecological‑validity “stress test” during acute bereavement, used to examine whether non‑reactive awareness remains stable under maximal provocation.
::::::* Two independent practitioner cases (an Amazon‑verified report and a structured questionnaire case) providing preliminary convergent signals across cognitive, emotional, behavioral, and identity dimensions.
::::::All central constructs (Pure Awareness, partial personalities, the Switch, identity stabilization) are operationalized with explicit phenomenological and behavioral criteria intended to enable replication and future third‑person measurement.
::::::I believe this is a valid contribution to Wikiversity. 
::::::Best regards, 
::::::Senad [[User:Senad Dizdarević|Senad Dizdarević]] ([[User talk:Senad Dizdarević|discuss]] • [[Special:Contributions/Senad Dizdarević|contribs]]) 06:54, 2 April 2026 (UTC)

== Unable to publish pages  ==

Whenever I try to publish a page with linked sources it gets flagged and says I'm a new user attempting to publish content with outside links. Those outside links are my sources. [[User:Soboyed|Soboyed]] ([[User talk:Soboyed|discuss]] • [[Special:Contributions/Soboyed|contribs]]) 04:52, 2 April 2026 (UTC)

:This restriction is automatically lifted after you have edited for a certain time (I don't recall that time off-hand, but it is not long). This is designed to stop spam. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:53, 2 April 2026 (UTC)

== Showing error to publish a Post ==

My  action was constructive, not destructive, please allow to publish it. [[Special:Contributions/&amp;#126;2026-20906-18|&amp;#126;2026-20906-18]] ([[User talk:&amp;#126;2026-20906-18|talk]]) 08:06, 4 April 2026 (UTC)

:Maybe you got caught in a filter. Consider [[Special:CreateAccount|creating an account]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:06, 4 April 2026 (UTC)
:Your edits, [https://en.wikiversity.org/w/index.php?title=Special:AbuseLog&amp;wpSearchUser=%7E2026-20906-18 these ones], seems to have tripped a filter when you tried to create a page on [[Create]] which external links. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:58, 4 April 2026 (UTC)
:Have you read my [https://en.wikiversity.org/w/index.php?title=Wikiversity:Request_custodian_action&amp;diff=prev&amp;oldid=2802219 previous reply] to you @[[User:~2026-20906-18|~2026-20906-18]]? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:02, 6 April 2026 (UTC)

== Abuse filters which should be deleted ==

Hi, there are some abuse filters which should probably be deleted.
* [[Special:AbuseFilter/1]] (not needed anymore)
* [[Special:AbuseFilter/2]] (no hits since 2018)
* [[Special:AbuseFilter/3]] (not needed since there are global filters that disallow this specific type of spam filter 3 would have catched)
* [[Special:AbuseFilter/4]] (looking at the logs, there are too many false positives)
* [[Special:AbuseFilter/5]] (no hits since 2023)
* Abuse filters 7, 8, 9, 10, 11, 12 (these filters are not needed anymore)
* [[Special:AbuseFilter/17]] (no hits since 2022)
* [[Special:AbuseFilter/19]] (no hits since 2019)
* [[Special:AbuseFilter/21]] (false positives, vandal currently inactive)
Thanks. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 03:51, 5 April 2026 (UTC)

:Why do these need to be deleted rather than inactivated? Do inactive abuse filters cause a server strain? —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:39, 5 April 2026 (UTC)
:: Deleted filters do not cause strain to the servers. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:28, 5 April 2026 (UTC)
:These sounds like sensible suggestions but, yes, would inactivation perhaps make more sense than deletion for at least some  filters? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:35, 5 April 2026 (UTC)
:I would keep them @[[User:Codename Noreste|Codename Noreste]]. Alternatively, I would turn off the ones that haven't caught anything for a long time, but I would leave them enabled in case they need to be turned on or improved. If someone has already written the code and we don't have hundreds of free man-hours of programmers on Wikiversity, the server load seems secondary to me, and is negligible compared to other things. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:11, 6 April 2026 (UTC)
:: I know how to write abuse filter code and regex, but I would recommend disabling filters that have never caught anything in a long time ''and'' those who made lots of false positives. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:09, 6 April 2026 (UTC)
:::Of course @[[User:Codename Noreste|Codename Noreste]], there are people here today who are capable of changing the code. But the question is what it will be like in a few years, the question is what will happen if those two are busy for a long time, etc. That's why I would leave it so that those who don't know much about code can be inspired by it and will need to do something with it someday - plus, more code for different types of filtering is actually great educational material on how those filters work. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 11:41, 17 April 2026 (UTC)

Here's the updated list of abuse filters under review with actions I've taken (several disabled, one basic code improvement, and some actions changed) - none have been deleted so they can all be edited and reactivayed - please suggest any further changes:
* [[Special:AbuseFilter/1]] (not needed anymore) - One time account spam bot - 4 hits over 10 years ago - Disabled in 2024 - May be useful in future
* [[Special:AbuseFilter/2]] (no hits since 2018) - Userspace spamming - 778 hits; none since 2018 likely due to global filters - Now disabled
* [[Special:AbuseFilter/3]] (not needed since there are global filters that disallow this specific type of spam filter 3 would have catched) - Specific user page spam - 1,101 hits most recent 7 March 2026 - Still active - Kept enabled
* [[Special:AbuseFilter/4]] (looking at the logs, there are too many false positives) - Questionable Language (profanity) - 6,055 hits including very recently - However it was logging hits without taking any actions - Edited to reduce likelihood of false positives by only applying filter to users with low (&lt; 20) edit count and applied weak actions to tag and warn but not prevent publishing the content
* [[Special:AbuseFilter/5]] (no hits since 2023) - Blocked Solicitation Links - 95 hits; none since 2023 - blocks specific historical spam sites - Non-active - Now disabled
* Abuse filters 7, 8, 9, 10, 11, 12 (these filters are not needed anymore) - Not reviewed - They are currently disabled
* [[Special:AbuseFilter/17]] (no hits since 2022) - Fundamental Physics Edits - 347 hits; none since 2022 - Non-active and very specific for a historical issue - Now disabled 
* [[Special:AbuseFilter/19]] (no hits since 2019) - Page Creation - 20 hits; none since 2019 - Retained for historical reference and possible future updates - Now disabled
* [[Special:AbuseFilter/21]] (false positives, vandal currently inactive) - Globally Banned Editor (renamed to Low-edit Spam Monitor) - 2,829 hits including very recent - Only applies to users with less than 5 edits and takes no actions / monitoring only - Reviewing the details of the hits I don't see many false positives and have strengthened its actions to add a tag and warning

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:57, 9 April 2026 (UTC)

== Block request ==

Please block ~2026-20985-80/~2026-21079-90/~2026-21223-88. Reason: Vandalism. [[User:Àncilu|Àncilu]] ([[User talk:Àncilu|discuss]] • [[Special:Contributions/Àncilu|contribs]]) 23:24, 5 April 2026 (UTC)

:All edits should be deleted and the first is blocked by Atcovi. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:33, 6 April 2026 (UTC)

== Antispam - Filter 12 ==
{{ping|Codename Noreste}} Thanks for contacting me with a suggested [[Special:AbuseFilter|abuse filter]] for the coupon spam we've been getting. A very much appreciated time saver. Per your suggestion, abuse filter 12 has been reactivated with your updated regex. It should tag and prevent page creation actions for coupon promo etc. Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:33, 15 April 2026 (UTC)

== Urgent! error message This action has been automatically identified as potentially harmful, and therefore disallowed ==

While submitting the post this error was coming &quot;This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please inform an administrator of what you were trying to do. A brief description of the abuse rule which your action matched is: New User Created Page with External Link&quot; How to resolve it?

Here is the content:

{{note|marketing material removed}}

[[User:EasyshikshaMarketing|EasyshikshaMarketing]] ([[User talk:EasyshikshaMarketing|discuss]] • [[Special:Contributions/EasyshikshaMarketing|contribs]]) 05:14, 17 April 2026 (UTC)
: That's because Wikiversity doesn't accept advertising. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:20, 17 April 2026 (UTC)
::So this is fine
::'''Online Internship and Digital Learning for Students'''
::Online learning has become an important part of modern education. With the help of the internet and digital tools, students can now study, practice, and gain experience without being physically present in a classroom. One key part of this system is the online internship, which helps students learn real-world skills along with their studies.
::An online internship allows students to work on tasks and projects through digital platforms. This makes learning more practical and useful, especially for those who want to understand how real work environments function.
::'''Background'''
::The concept of online learning developed from distance education, where students learned from remote locations. Over time, with the growth of digital technology, learning has become more interactive and flexible.
::The introduction of the online internship has added another important layer to digital education. It combines theoretical learning with practical experience, helping students prepare for future careers.
::During global events such as the COVID-19 pandemic, online education and online internship programs became essential. They helped students continue learning and gaining experience despite restrictions on physical movement.
::'''Importance of Online Internship'''
::An online internship plays an important role in student development. It helps bridge the gap between academic knowledge and practical skills.
::'''Some key points include:'''
::Students understand how real work is done
::They develop basic professional skills
::It supports career readiness
::It allows learning without location limits
::By participating in an online internship, students can improve their confidence and gain early exposure to different fields.
::'''Features of Online Internship-Based Learning'''
::Modern education platforms often include online internship opportunities as part of their learning system. These usually offer:
::Flexible schedules for students
::Access to learning from home
::Beginner-friendly tasks and projects
::A combination of theory and practice
::Such features make online internship programs suitable for a wide range of learners, including beginners.
::'''Learning Tasks'''
::Explore how online internship programs support student learning in digital environments.
::Identify how students can gain practical experience through an online internship
::Analyze the role of flexible learning in improving student engagement
::Understand how online education and internships work together
::'''References'''
::[https://ijpsl.in/wp-content/uploads/2020/09/E-Learning_Hanaaya-Navaneeth.pdf The Past, Present and Future of E-Learning: Hanaaya Varyani and Navaneeth M S]
::[https://ieeexplore.ieee.org/document/9788102 Current Trends and Future Perspectives of e-Learning in India]
::'''See Also'''
::[https://easyshiksha.com/online_courses/internship Online Internship]
::[https://easyshiksha.com/online_courses/ Online Courses]
::[https://easyshiksha.com/online_courses/kids-learning Kids Learning]
::[https://easyshiksha.com/career_helper/ Career Guidance] [[User:EasyshikshaMarketing|EasyshikshaMarketing]] ([[User talk:EasyshikshaMarketing|discuss]] • [[Special:Contributions/EasyshikshaMarketing|contribs]]) 05:27, 17 April 2026 (UTC)
:::@[[User:EasyshikshaMarketing|EasyshikshaMarketing]] Wikiversity is a resource for education, or a space for education. However, your intention to link to another website is obvious, and such content does not belong here, as it contradicts the purpose of Wikiversity. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 11:38, 17 April 2026 (UTC)

==AI-generated images==
Seeking your advice.

Myself and students use some AI-generated images in the [[Motivation and emotion]] project.

[[User:Dronebogus]] has been removing some of these images from Wikiversity pages and nominating them for deletion at Commons. As a result, some have been deleted and some have been kept.

Dronebogus has made some useful edits and image suggestions for [[Motivation and emotion]] which I've appreciated and  incorporated. However, there are other edits to remove an AI image by Dronebogus that I've reverted where I think the image is more educational than no image or an alternative image suggested by Dronebogus.

There are a couple of pages where Dronebogus has reverted my reversion, so we are at risk of edit warring. We have briefly discussed and warned each other on our user talk pages, but it seems to come down to a difference in perception about the educational usefulness of the AI images.

So, I'm asking here for others to please review the recent edit histories for these pages:
* [[Motivation and emotion/Lectures/Brain and physiological needs]]
* [[Motivation and emotion/Book/2025/Stockholm syndrome emotion]]

and let us know what you think about the AI image suitability vs. using no image or alternative images suggested by Dronebogus.

Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:45, 26 April 2026 (UTC)

:I think Jtneill needs to try harder to find non-AI alternatives both on Commons and the web. I’m not reiterating the well known problems with generative AI— you can read about those on Wikipedia and the broader Internet. Needless to say it’s kind of just inherently toxic. If you use it, it should be the last resort of last resorts. Just my stance, which I consider perfectly valid and reasonable. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 04:10, 27 April 2026 (UTC)

== Request for Page Creation Approval - Research on Dr. Ian Stevenson ==


'''Hello Administrators,'''

I am a digital archivist and researcher attempting to create an educational page regarding Dr. Ian Stevenson's scientific work, specifically titled &quot;Where Reincarnation and Biology Intersect&quot; and its official Vietnamese edition.

My edits are purely constructive, aimed at integrating local research context with global structured data (including '''Wikidata Q139548587'''). The automated filter is currently preventing publication due to &quot;new user external link limits.&quot;

I have ensured that all citations, references, and images (sourced from Wikimedia Commons) are strictly academic and non-promotional. Could you please review my contributions and consider whitelisting my account or approving this specific page creation?

Thank you for your assistance in preserving this research '''legacy''' within the open knowledge ecosystem.

'''Best regards,''' [[User:Ian Stevenson777|Ian Stevenson777]] ([[User talk:Ian Stevenson777|discuss]] • [[Special:Contributions/Ian Stevenson777|contribs]]) 07:26, 28 April 2026 (UTC)
: The page title should probably be in English, as this is English Wikiversity. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:19, 28 April 2026 (UTC)</text>
      <sha1>iznr194pn6oadmxiqj0s1jr1z7ygcu0</sha1>
    </revision>
  </page>
  <page>
    <title>VHDL programming in plain view</title>
    <ns>0</ns>
    <id>121359</id>
    <revision>
      <id>2806867</id>
      <parentid>2806818</parentid>
      <timestamp>2026-04-28T14:13:22Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Data */</comment>
      <origin>2806867</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2278" sha1="ft8hje0iu2fgbgkndvqfsotv6ix8s85" 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.20260428.pdf|A]], [[Media:Data.Object.1B.20260427.pdf|B]])
* Data Types ([[Media:Data.Type.2A.20260421.pdf|A]], [[Media:Data.Type.2B.20260421.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>ft8hje0iu2fgbgkndvqfsotv6ix8s85</sha1>
    </revision>
    <revision>
      <id>2806869</id>
      <parentid>2806867</parentid>
      <timestamp>2026-04-28T14:15:29Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Data */</comment>
      <origin>2806869</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2278" sha1="0c530g0ofvs7vageenvpbae74gw94ry" 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.20260428.pdf|A]], [[Media:Data.Object.1B.20260428.pdf|B]])
* Data Types ([[Media:Data.Type.2A.20260421.pdf|A]], [[Media:Data.Type.2B.20260421.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>0c530g0ofvs7vageenvpbae74gw94ry</sha1>
    </revision>
    <revision>
      <id>2806949</id>
      <parentid>2806869</parentid>
      <timestamp>2026-04-29T04:24:40Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Data */</comment>
      <origin>2806949</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2278" sha1="sryiqw6nvmfp97qyijv6vvgczjvrton" 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.20260428.pdf|A]], [[Media:Data.Object.1B.20260428.pdf|B]])
* Data Types ([[Media:Data.Type.2A.20260427.pdf|A]], [[Media:Data.Type.2B.20260427.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>sryiqw6nvmfp97qyijv6vvgczjvrton</sha1>
    </revision>
    <revision>
      <id>2806955</id>
      <parentid>2806949</parentid>
      <timestamp>2026-04-29T04:38:04Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Data */</comment>
      <origin>2806955</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2278" sha1="08n4pho74u7ooy9td0linuw91lr1bbi" 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.20260428.pdf|A]], [[Media:Data.Object.1B.20260428.pdf|B]])
* Data Types ([[Media:Data.Type.2A.20260428.pdf|A]], [[Media:Data.Type.2B.20260427.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>08n4pho74u7ooy9td0linuw91lr1bbi</sha1>
    </revision>
    <revision>
      <id>2806962</id>
      <parentid>2806955</parentid>
      <timestamp>2026-04-29T04:55:50Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Data */</comment>
      <origin>2806962</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2278" sha1="nittz8rh55qjamyxbzscetb5dwbib37" 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.20260428.pdf|A]], [[Media:Data.Object.1B.20260428.pdf|B]])
* Data Types ([[Media:Data.Type.2A.20260428.pdf|A]], [[Media:Data.Type.2B.20260428.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>nittz8rh55qjamyxbzscetb5dwbib37</sha1>
    </revision>
  </page>
  <page>
    <title>Wikipedia/Types of user accounts</title>
    <ns>0</ns>
    <id>122308</id>
    <revision>
      <id>2806930</id>
      <parentid>2359306</parentid>
      <timestamp>2026-04-29T00:27:31Z</timestamp>
      <contributor>
        <username>~2026-25992-66</username>
        <id>3069181</id>
      </contributor>
      <comment>I've changed &quot;auto patrolled&quot; to &quot;autopatrolled.&quot; I attempted to fix the grammar in this section.</comment>
      <origin>2806930</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5894" sha1="hw071l9qucc6r09vq5vticd35khr7rd" xml:space="preserve">{{Template:Writing an article on Wikipedia navigation|module=1|nextpage=The language of Wikipedia}}
In a sense, on Wikipedia all users are equal: all people contributing to the project are expected to stick to the same rules, and when it comes time to discuss how to manage issues, all contributors have an equal voice. That said, there is a difference in what different contributors can do, and that is where this lesson comes into play.

== IP editors ==
Anyone editing without first creating an account is identified by their [[w:IP address|IP address]]. An IP address is a series of numbers which identify the network location of your computer. As such, IP editors lose a certain degree of privacy, as depending on the circumstances that IP address can be used to discover the country, city and potentially the location from which the contributor is accessing Wikipedia. Along with the reduction in privacy, IP addresses often change, and as a result it is difficult for someone editing without an account to build a reputation as a valued contributor.

In terms of editing, IP editors can edit any [[w:Wikipedia:Protection policy|unprotected page]]. However, they are unable to create new pages or upload files, and their edits tend to be given special attention by those looking at [[w:Special:RecentChanges|recent changes]].

== New accounts ==
New users are able to make changes to unprotected resources (which is the default status for the vast majority of pages), but are not able to edit protected resources or to upload files. By having an account, they are no longer identified by their IP address, which provides a greater degree of privacy. The single account name also makes it easier for contributors to build a positive reputation and relationships with other contributors.

The [[m:Help:Unified login|unified login]] (SUL) process permits users with an account to have a common username across all the [[wmf:Our projects|Wikimedia Foundation projects]]. Thus the same username and password could be used to log into Wikiversity, Wikipedia, [[b:Wikibooks:What is Wikibooks?|Wikibooks]] and [[commons:Commons:Welcome|Wikimedia Commons]], to name but four of the projects.

== Autoconfirmed editors ==
After four days and 10 edits, most users with a user account is automatically autoconfirmed. This provides them with the ability to upload files, edit semi-protected pages and perform page moves.

== Established users ==
There is no formal definition of an &quot;established user&quot;, and they don't necessarily have any more rights than an autoconfirmed user. In practice, though, an established user is someone who has been around long enough, and who has done enough, to build up a bit of a reputation as a valued contributor. Established users may take on extra responsibilities, such as mentoring new editors, and they may gain extra user rights, such as [[w:Wikipedia:Rollback feature|rollback]] and [[wikipedia:Wikipedia:Autopatrolled|autopatrolled]], but the main advantage is that these are people who have been around for long enough to know how things work, understand policy, and have demonstrated that they are able to work within Wikipedia's framework.

== Administrators ==
Administrators are, (as their name would suggest), responsible for administrative actions, such as [[w:Wikipedia:Blocking policy|blocking and unblocking accounts]], [[w:Wikipedia:Deletion policy|deleting and restoring pages and files]], and [[w:Wikipedia:Protection policy|protecting and unprotecting]] content. As these tasks are both essential and open to misuse, custodians are carefully selected by the Wikipedia community. Administrators need to be trusted contributors who have demonstrated that they understand Wikiversity's policies and are able to interpret [[w:Wikipedia:Consensus|consensus]] accurately and consistently. As a result, the community has big expectations of administrators, such that it is expected that contributors have at least 6 months experience and over 2000 edits before applying for the role.

It is important to note that administrators may have extra tools that they can employ, but in terms of finding consensus they are no more able to direct debate than any other contributor. The administrator role is an important one, but determining policy, selecting administrators and engaging in debate are things that all contributors are encouraged to take part in as equal members of the community.

== Checkusers ==
CheckUsers have the ability to compare user accounts and IP addresses to determine if they are being used by the same people. Abusing accounts, in particular to get around a block or ban, or to lend the appearance of additional support in a debate, is not permitted on Wikipedia, and the CheckUsers have access to the tools required to investigate where such abuse is suspected. However, doing so requires that CheckUsers have access to potentially private information, such as the IP address from which a registered account is being used. Thus they are limited to highly trusted members of the community, who must be over 18 years of age, are required to provide identification to the Wikimedia Foundation, and who must work within the [[wmf:Privacy policy|privacy policy]].

== Bureaucrats ==
Bureaucrats are primarily responsible for three jobs: they determine consensus and decide whether or not to promote someone to administrator or bureaucrat; they can [[:Wikipedia:Changing username|rename user accounts]]; and they can permit [[w:Wikipedia:Bots|bots]] to operate. As with custodians, bureaucrats are expected to have a deep understanding of Wikipedia's processes, and are highly trusted members of the community.

== Further reading ==
* [[w:Wikipedia:Who writes Wikipedia|Wikipedia:Who writes Wikipedia]]
* [[w:Wikipedia:Wikipedians|Wikipedia:Wikipedians]]
* [[w:Wikipedia:User access levels|Wikipedia:User access levels]]

{{CourseCat}}</text>
      <sha1>hw071l9qucc6r09vq5vticd35khr7rd</sha1>
    </revision>
    <revision>
      <id>2806931</id>
      <parentid>2806930</parentid>
      <timestamp>2026-04-29T00:43:39Z</timestamp>
      <contributor>
        <username>~2026-25992-66</username>
        <id>3069181</id>
      </contributor>
      <comment>&quot;...Understand policy,&quot; is unclear. I've changed it to: &quot;understand the privacy policy,&quot; in attempt to clear up some confusion. Please revert or start a talk if needed.</comment>
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In a sense, on Wikipedia all users are equal: all people contributing to the project are expected to stick to the same rules, and when it comes time to discuss how to manage issues, all contributors have an equal voice. That said, there is a difference in what different contributors can do, and that is where this lesson comes into play.

== IP editors ==
Anyone editing without first creating an account is identified by their [[w:IP address|IP address]]. An IP address is a series of numbers which identify the network location of your computer. As such, IP editors lose a certain degree of privacy, as depending on the circumstances that IP address can be used to discover the country, city and potentially the location from which the contributor is accessing Wikipedia. Along with the reduction in privacy, IP addresses often change, and as a result it is difficult for someone editing without an account to build a reputation as a valued contributor.

In terms of editing, IP editors can edit any [[w:Wikipedia:Protection policy|unprotected page]]. However, they are unable to create new pages or upload files, and their edits tend to be given special attention by those looking at [[w:Special:RecentChanges|recent changes]].

== New accounts ==
New users are able to make changes to unprotected resources (which is the default status for the vast majority of pages), but are not able to edit protected resources or to upload files. By having an account, they are no longer identified by their IP address, which provides a greater degree of privacy. The single account name also makes it easier for contributors to build a positive reputation and relationships with other contributors.

The [[m:Help:Unified login|unified login]] (SUL) process permits users with an account to have a common username across all the [[wmf:Our projects|Wikimedia Foundation projects]]. Thus the same username and password could be used to log into Wikiversity, Wikipedia, [[b:Wikibooks:What is Wikibooks?|Wikibooks]] and [[commons:Commons:Welcome|Wikimedia Commons]], to name but four of the projects.

== Autoconfirmed editors ==
After four days and 10 edits, most users with a user account is automatically autoconfirmed. This provides them with the ability to upload files, edit semi-protected pages and perform page moves.

== Established users ==
There is no formal definition of an &quot;established user&quot;, and they don't necessarily have any more rights than an autoconfirmed user. In practice, though, an established user is someone who has been around long enough, and has done enough, to build up a bit of a reputation as a valued contributor. Established users may take on extra responsibilities, such as mentoring new editors, and they may gain extra user rights, such as [[w:Wikipedia:Rollback feature|rollback]] and [[wikipedia:Wikipedia:Autopatrolled|autopatrolled]], but the main advantage is that these are people who have been around for long enough to know how things work, understand the privacy policy, and have demonstrated that they are able to work within Wikipedia's framework.

== Administrators ==
Administrators are, (as their name would suggest), responsible for administrative actions, such as [[w:Wikipedia:Blocking policy|blocking and unblocking accounts]], [[w:Wikipedia:Deletion policy|deleting and restoring pages and files]], and [[w:Wikipedia:Protection policy|protecting and unprotecting]] content. As these tasks are both essential and open to misuse, custodians are carefully selected by the Wikipedia community. Administrators need to be trusted contributors who have demonstrated that they understand Wikiversity's policies and are able to interpret [[w:Wikipedia:Consensus|consensus]] accurately and consistently. As a result, the community has big expectations of administrators, such that it is expected that contributors have at least 6 months experience and over 2000 edits before applying for the role.

It is important to note that administrators may have extra tools that they can employ, but in terms of finding consensus they are no more able to direct debate than any other contributor. The administrator role is an important one, but determining policy, selecting administrators and engaging in debate are things that all contributors are encouraged to take part in as equal members of the community.

== Checkusers ==
CheckUsers have the ability to compare user accounts and IP addresses to determine if they are being used by the same people. Abusing accounts, in particular to get around a block or ban, or to lend the appearance of additional support in a debate, is not permitted on Wikipedia, and the CheckUsers have access to the tools required to investigate where such abuse is suspected. However, doing so requires that CheckUsers have access to potentially private information, such as the IP address from which a registered account is being used. Thus they are limited to highly trusted members of the community, who must be over 18 years of age, are required to provide identification to the Wikimedia Foundation, and who must work within the [[wmf:Privacy policy|privacy policy]].

== Bureaucrats ==
Bureaucrats are primarily responsible for three jobs: they determine consensus and decide whether or not to promote someone to administrator or bureaucrat; they can [[:Wikipedia:Changing username|rename user accounts]]; and they can permit [[w:Wikipedia:Bots|bots]] to operate. As with custodians, bureaucrats are expected to have a deep understanding of Wikipedia's processes, and are highly trusted members of the community.

== Further reading ==
* [[w:Wikipedia:Who writes Wikipedia|Wikipedia:Who writes Wikipedia]]
* [[w:Wikipedia:Wikipedians|Wikipedia:Wikipedians]]
* [[w:Wikipedia:User access levels|Wikipedia:User access levels]]

{{CourseCat}}</text>
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In a sense, on Wikipedia all users are equal: all people contributing to the project are expected to stick to the same rules, and when it comes time to discuss how to manage issues, all contributors have an equal voice. That said, there is a difference in what different contributors can do, and that is where this lesson comes into play.

== IP editors ==
Anyone editing without first creating an account is identified by their [[w:IP address|IP address]]. An IP address is a series of numbers which identify the network location of your computer. As such, IP editors lose a certain degree of privacy, as depending on the circumstances that IP address can be used to discover the country, city and potentially the location from which the contributor is accessing Wikipedia. Along with the reduction in privacy, IP addresses often change, and as a result it is difficult for someone editing without an account to build a reputation as a valued contributor.

In terms of editing, IP editors can edit any [[w:Wikipedia:Protection policy|unprotected page]]. However, they are unable to create new pages or upload files, and their edits tend to be given special attention by those looking at [[w:Special:RecentChanges|recent changes]].

== New accounts ==
New users are able to make changes to unprotected resources (which is the default status for the vast majority of pages), but are not able to edit protected resources or to upload files. By having an account, they are no longer identified by their IP address, which provides a greater degree of privacy. The single account name also makes it easier for contributors to build a positive reputation and relationships with other contributors.

The [[m:Help:Unified login|unified login]] (SUL) process permits users with an account to have a common username across all the [[wmf:Our projects|Wikimedia Foundation projects]]. Thus the same username and password could be used to log into Wikiversity, Wikipedia, [[b:Wikibooks:What is Wikibooks?|Wikibooks]] and [[commons:Commons:Welcome|Wikimedia Commons]], to name but four of the projects.

== Autoconfirmed editors ==
After four days and 10 edits, most users with a user account is automatically autoconfirmed. This provides them with the ability to upload files, edit semi-protected pages and perform page moves.

== Established users ==
There is no formal definition of an &quot;established user&quot;, and they don't necessarily have any more rights than an autoconfirmed user. In practice, though, an established user is someone who has been around long enough, and have done enough, to build up a bit of a reputation as a valued contributor. Established users may take on extra responsibilities, such as mentoring new editors, and they may gain extra user rights, such as [[w:Wikipedia:Rollback feature|rollback]] and [[wikipedia:Wikipedia:Autopatrolled|autopatrolled]], but the main advantage is that these are people who have been around for long enough to know how things work, understand the privacy policy, and have demonstrated that they are able to work within Wikipedia's framework.

== Administrators ==
Administrators are, (as their name would suggest), responsible for administrative actions, such as [[w:Wikipedia:Blocking policy|blocking and unblocking accounts]], [[w:Wikipedia:Deletion policy|deleting and restoring pages and files]], and [[w:Wikipedia:Protection policy|protecting and unprotecting]] content. As these tasks are both essential and open to misuse, custodians are carefully selected by the Wikipedia community. Administrators need to be trusted contributors who have demonstrated that they understand Wikiversity's policies and are able to interpret [[w:Wikipedia:Consensus|consensus]] accurately and consistently. As a result, the community has big expectations of administrators, such that it is expected that contributors have at least 6 months experience and over 2000 edits before applying for the role.

It is important to note that administrators may have extra tools that they can employ, but in terms of finding consensus they are no more able to direct debate than any other contributor. The administrator role is an important one, but determining policy, selecting administrators and engaging in debate are things that all contributors are encouraged to take part in as equal members of the community.

== Checkusers ==
CheckUsers have the ability to compare user accounts and IP addresses to determine if they are being used by the same people. Abusing accounts, in particular to get around a block or ban, or to lend the appearance of additional support in a debate, is not permitted on Wikipedia, and the CheckUsers have access to the tools required to investigate where such abuse is suspected. However, doing so requires that CheckUsers have access to potentially private information, such as the IP address from which a registered account is being used. Thus they are limited to highly trusted members of the community, who must be over 18 years of age, are required to provide identification to the Wikimedia Foundation, and who must work within the [[wmf:Privacy policy|privacy policy]].

== Bureaucrats ==
Bureaucrats are primarily responsible for three jobs: they determine consensus and decide whether or not to promote someone to administrator or bureaucrat; they can [[:Wikipedia:Changing username|rename user accounts]]; and they can permit [[w:Wikipedia:Bots|bots]] to operate. As with custodians, bureaucrats are expected to have a deep understanding of Wikipedia's processes, and are highly trusted members of the community.

== Further reading ==
* [[w:Wikipedia:Who writes Wikipedia|Wikipedia:Who writes Wikipedia]]
* [[w:Wikipedia:Wikipedians|Wikipedia:Wikipedians]]
* [[w:Wikipedia:User access levels|Wikipedia:User access levels]]

{{CourseCat}}</text>
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      <comment>I've changed, &quot;autoconfirmed editors,&quot; to: &quot;autoconfirmed users.&quot;</comment>
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In a sense, on Wikipedia all users are equal: all people contributing to the project are expected to stick to the same rules, and when it comes time to discuss how to manage issues, all contributors have an equal voice. That said, there is a difference in what different contributors can do, and that is where this lesson comes into play.

== IP editors ==
Anyone editing without first creating an account is identified by their [[w:IP address|IP address]]. An IP address is a series of numbers which identify the network location of your computer. As such, IP editors lose a certain degree of privacy, as depending on the circumstances that IP address can be used to discover the country, city and potentially the location from which the contributor is accessing Wikipedia. Along with the reduction in privacy, IP addresses often change, and as a result it is difficult for someone editing without an account to build a reputation as a valued contributor.

In terms of editing, IP editors can edit any [[w:Wikipedia:Protection policy|unprotected page]]. However, they are unable to create new pages or upload files, and their edits tend to be given special attention by those looking at [[w:Special:RecentChanges|recent changes]].

== New accounts ==
New users are able to make changes to unprotected resources (which is the default status for the vast majority of pages), but are not able to edit protected resources or to upload files. By having an account, they are no longer identified by their IP address, which provides a greater degree of privacy. The single account name also makes it easier for contributors to build a positive reputation and relationships with other contributors.

The [[m:Help:Unified login|unified login]] (SUL) process permits users with an account to have a common username across all the [[wmf:Our projects|Wikimedia Foundation projects]]. Thus the same username and password could be used to log into Wikiversity, Wikipedia, [[b:Wikibooks:What is Wikibooks?|Wikibooks]] and [[commons:Commons:Welcome|Wikimedia Commons]], to name but four of the projects.

== Autoconfirmed users ==
After four days and 10 edits, most users with a user account is automatically autoconfirmed. This provides them with the ability to upload files, edit semi-protected pages and perform page moves.

== Established users ==
There is no formal definition of an &quot;established user&quot;, and they don't necessarily have any more rights than an autoconfirmed user. In practice, though, an established user is someone who has been around long enough, and have done enough, to build up a bit of a reputation as a valued contributor. Established users may take on extra responsibilities, such as mentoring new editors, and they may gain extra user rights, such as [[w:Wikipedia:Rollback feature|rollback]] and [[wikipedia:Wikipedia:Autopatrolled|autopatrolled]], but the main advantage is that these are people who have been around for long enough to know how things work, understand the privacy policy, and have demonstrated that they are able to work within Wikipedia's framework.

== Administrators ==
Administrators are, (as their name would suggest), responsible for administrative actions, such as [[w:Wikipedia:Blocking policy|blocking and unblocking accounts]], [[w:Wikipedia:Deletion policy|deleting and restoring pages and files]], and [[w:Wikipedia:Protection policy|protecting and unprotecting]] content. As these tasks are both essential and open to misuse, custodians are carefully selected by the Wikipedia community. Administrators need to be trusted contributors who have demonstrated that they understand Wikiversity's policies and are able to interpret [[w:Wikipedia:Consensus|consensus]] accurately and consistently. As a result, the community has big expectations of administrators, such that it is expected that contributors have at least 6 months experience and over 2000 edits before applying for the role.

It is important to note that administrators may have extra tools that they can employ, but in terms of finding consensus they are no more able to direct debate than any other contributor. The administrator role is an important one, but determining policy, selecting administrators and engaging in debate are things that all contributors are encouraged to take part in as equal members of the community.

== Checkusers ==
CheckUsers have the ability to compare user accounts and IP addresses to determine if they are being used by the same people. Abusing accounts, in particular to get around a block or ban, or to lend the appearance of additional support in a debate, is not permitted on Wikipedia, and the CheckUsers have access to the tools required to investigate where such abuse is suspected. However, doing so requires that CheckUsers have access to potentially private information, such as the IP address from which a registered account is being used. Thus they are limited to highly trusted members of the community, who must be over 18 years of age, are required to provide identification to the Wikimedia Foundation, and who must work within the [[wmf:Privacy policy|privacy policy]].

== Bureaucrats ==
Bureaucrats are primarily responsible for three jobs: they determine consensus and decide whether or not to promote someone to administrator or bureaucrat; they can [[:Wikipedia:Changing username|rename user accounts]]; and they can permit [[w:Wikipedia:Bots|bots]] to operate. As with custodians, bureaucrats are expected to have a deep understanding of Wikipedia's processes, and are highly trusted members of the community.

== Further reading ==
* [[w:Wikipedia:Who writes Wikipedia|Wikipedia:Who writes Wikipedia]]
* [[w:Wikipedia:Wikipedians|Wikipedia:Wikipedians]]
* [[w:Wikipedia:User access levels|Wikipedia:User access levels]]

{{CourseCat}}</text>
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In a sense, on Wikipedia all users are equal: all people contributing to the project are expected to stick to the same rules, and when it comes time to discuss how to manage issues, all contributors have an equal voice. That said, there is a difference in what different contributors can do, and that is where this lesson comes into play.

== IP editors ==
Anyone editing without first creating an account is identified by their [[w:IP address|IP address]]. An IP address is a series of numbers which identify the network location of your computer. As such, IP editors lose a certain degree of privacy, as depending on the circumstances that IP address can be used to discover the country, city and potentially the location from which the contributor is accessing Wikipedia. Along with the reduction in privacy, IP addresses often change, and as a result it is difficult for someone editing without an account to build a reputation as a valued contributor.

In terms of editing, IP editors can edit any [[w:Wikipedia:Protection policy|unprotected page]]. However, they are unable to create new pages or upload files, and their edits tend to be given special attention by those looking at [[w:Special:RecentChanges|recent changes]].

== New accounts ==
New users are able to make changes to unprotected resources (which is the default status for the vast majority of pages), but are not able to edit protected resources or to upload files. By having an account, they are no longer identified by their IP address, which provides a greater degree of privacy. The single account name also makes it easier for contributors to build a positive reputation and relationships with other contributors.

The [[m:Help:Unified login|unified login]] (SUL) process permits users with an account to have a common username across all the [[wmf:Our projects|Wikimedia Foundation projects]]. Thus the same username and password could be used to log into Wikiversity, Wikipedia, [[b:Wikibooks:What is Wikibooks?|Wikibooks]] and [[commons:Commons:Welcome|Wikimedia Commons]], to name but four of the projects.

== Autoconfirmed users ==
After four days and 10 edits, most users with a user account are automatically autoconfirmed. This provides them with the ability to upload files, edit semi-protected pages and perform page moves.

== Established users ==
There is no formal definition of an &quot;established user&quot;, and they don't necessarily have any more rights than an autoconfirmed user. In practice, though, an established user is someone who has been around long enough, and have done enough, to build up a bit of a reputation as a valued contributor. Established users may take on extra responsibilities, such as mentoring new editors, and they may gain extra user rights, such as [[w:Wikipedia:Rollback feature|rollback]] and [[wikipedia:Wikipedia:Autopatrolled|autopatrolled]], but the main advantage is that these are people who have been around for long enough to know how things work, understand the privacy policy, and have demonstrated that they are able to work within Wikipedia's framework.

== Administrators ==
Administrators are, (as their name would suggest), responsible for administrative actions, such as [[w:Wikipedia:Blocking policy|blocking and unblocking accounts]], [[w:Wikipedia:Deletion policy|deleting and restoring pages and files]], and [[w:Wikipedia:Protection policy|protecting and unprotecting]] content. As these tasks are both essential and open to misuse, custodians are carefully selected by the Wikipedia community. Administrators need to be trusted contributors who have demonstrated that they understand Wikiversity's policies and are able to interpret [[w:Wikipedia:Consensus|consensus]] accurately and consistently. As a result, the community has big expectations of administrators, such that it is expected that contributors have at least 6 months experience and over 2000 edits before applying for the role.

It is important to note that administrators may have extra tools that they can employ, but in terms of finding consensus they are no more able to direct debate than any other contributor. The administrator role is an important one, but determining policy, selecting administrators and engaging in debate are things that all contributors are encouraged to take part in as equal members of the community.

== Checkusers ==
CheckUsers have the ability to compare user accounts and IP addresses to determine if they are being used by the same people. Abusing accounts, in particular to get around a block or ban, or to lend the appearance of additional support in a debate, is not permitted on Wikipedia, and the CheckUsers have access to the tools required to investigate where such abuse is suspected. However, doing so requires that CheckUsers have access to potentially private information, such as the IP address from which a registered account is being used. Thus they are limited to highly trusted members of the community, who must be over 18 years of age, are required to provide identification to the Wikimedia Foundation, and who must work within the [[wmf:Privacy policy|privacy policy]].

== Bureaucrats ==
Bureaucrats are primarily responsible for three jobs: they determine consensus and decide whether or not to promote someone to administrator or bureaucrat; they can [[:Wikipedia:Changing username|rename user accounts]]; and they can permit [[w:Wikipedia:Bots|bots]] to operate. As with custodians, bureaucrats are expected to have a deep understanding of Wikipedia's processes, and are highly trusted members of the community.

== Further reading ==
* [[w:Wikipedia:Who writes Wikipedia|Wikipedia:Who writes Wikipedia]]
* [[w:Wikipedia:Wikipedians|Wikipedia:Wikipedians]]
* [[w:Wikipedia:User access levels|Wikipedia:User access levels]]

{{CourseCat}}</text>
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In a sense, on Wikipedia, all users are equal: all people contributing to the project are expected to stick to the same rules, and when it comes time to discuss how to manage issues, all contributors have an equal voice. That said, there is a difference in what different contributors can do, and that is where this lesson comes into play.

== IP editors ==
Anyone editing without first creating an account is identified by their [[w:IP address|IP address]]. An IP address is a series of numbers which identify the network location of your computer. As such, IP editors lose a certain degree of privacy, as depending on the circumstances that IP address can be used to discover the country, city and potentially the location from which the contributor is accessing Wikipedia. Along with the reduction in privacy, IP addresses often change, and as a result it is difficult for someone editing without an account to build a reputation as a valued contributor.

In terms of editing, IP editors can edit any [[w:Wikipedia:Protection policy|unprotected page]]. However, they are unable to create new pages or upload files, and their edits tend to be given special attention by those looking at [[w:Special:RecentChanges|recent changes]].

== New accounts ==
New users are able to make changes to unprotected resources (which is the default status for the vast majority of pages), but are not able to edit protected resources or to upload files. By having an account, they are no longer identified by their IP address, which provides a greater degree of privacy. The single account name also makes it easier for contributors to build a positive reputation and relationships with other contributors.

The [[m:Help:Unified login|unified login]] (SUL) process permits users with an account to have a common username across all the [[wmf:Our projects|Wikimedia Foundation projects]]. Thus the same username and password could be used to log into Wikiversity, Wikipedia, [[b:Wikibooks:What is Wikibooks?|Wikibooks]] and [[commons:Commons:Welcome|Wikimedia Commons]], to name but four of the projects.

== Autoconfirmed users ==
After four days and 10 edits, most users with a user account are automatically autoconfirmed. This provides them with the ability to upload files, edit semi-protected pages and perform page moves.

== Established users ==
There is no formal definition of an &quot;established user&quot;, and they don't necessarily have any more rights than an autoconfirmed user. In practice, though, an established user is someone who has been around long enough, and have done enough, to build up a bit of a reputation as a valued contributor. Established users may take on extra responsibilities, such as mentoring new editors, and they may gain extra user rights, such as [[w:Wikipedia:Rollback feature|rollback]] and [[wikipedia:Wikipedia:Autopatrolled|autopatrolled]], but the main advantage is that these are people who have been around for long enough to know how things work, understand the privacy policy, and have demonstrated that they are able to work within Wikipedia's framework.

== Administrators ==
Administrators are, (as their name would suggest), responsible for administrative actions, such as [[w:Wikipedia:Blocking policy|blocking and unblocking accounts]], [[w:Wikipedia:Deletion policy|deleting and restoring pages and files]], and [[w:Wikipedia:Protection policy|protecting and unprotecting]] content. As these tasks are both essential and open to misuse, custodians are carefully selected by the Wikipedia community. Administrators need to be trusted contributors who have demonstrated that they understand Wikiversity's policies and are able to interpret [[w:Wikipedia:Consensus|consensus]] accurately and consistently. As a result, the community has big expectations of administrators, such that it is expected that contributors have at least 6 months experience and over 2000 edits before applying for the role.

It is important to note that administrators may have extra tools that they can employ, but in terms of finding consensus they are no more able to direct debate than any other contributor. The administrator role is an important one, but determining policy, selecting administrators and engaging in debate are things that all contributors are encouraged to take part in as equal members of the community.

== Checkusers ==
CheckUsers have the ability to compare user accounts and IP addresses to determine if they are being used by the same people. Abusing accounts, in particular to get around a block or ban, or to lend the appearance of additional support in a debate, is not permitted on Wikipedia, and the CheckUsers have access to the tools required to investigate where such abuse is suspected. However, doing so requires that CheckUsers have access to potentially private information, such as the IP address from which a registered account is being used. Thus they are limited to highly trusted members of the community, who must be over 18 years of age, are required to provide identification to the Wikimedia Foundation, and who must work within the [[wmf:Privacy policy|privacy policy]].

== Bureaucrats ==
Bureaucrats are primarily responsible for three jobs: they determine consensus and decide whether or not to promote someone to administrator or bureaucrat; they can [[:Wikipedia:Changing username|rename user accounts]]; and they can permit [[w:Wikipedia:Bots|bots]] to operate. As with custodians, bureaucrats are expected to have a deep understanding of Wikipedia's processes, and are highly trusted members of the community.

== Further reading ==
* [[w:Wikipedia:Who writes Wikipedia|Wikipedia:Who writes Wikipedia]]
* [[w:Wikipedia:Wikipedians|Wikipedia:Wikipedians]]
* [[w:Wikipedia:User access levels|Wikipedia:User access levels]]

{{CourseCat}}</text>
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      <comment>I added three new hyperlinks and replaced the WikiBooks hyperlink. I also corrected the grammar of this section.</comment>
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In a sense, on Wikipedia, all users are equal: all people contributing to the project are expected to stick to the same rules, and when it comes time to discuss how to manage issues, all contributors have an equal voice. That said, there is a difference in what different contributors can do, and that is where this lesson comes into play.

== IP editors ==
Anyone editing without first creating an account is identified by their [[w:IP address|IP address]]. An IP address is a series of numbers which identify the network location of your computer. As such, IP editors lose a certain degree of privacy, as depending on the circumstances that IP address can be used to discover the country, city and potentially the location from which the contributor is accessing Wikipedia. Along with the reduction in privacy, IP addresses often change, and as a result it is difficult for someone editing without an account to build a reputation as a valued contributor.

In terms of editing, IP editors can edit any [[w:Wikipedia:Protection policy|unprotected page]]. However, they are unable to create new pages or upload files, and their edits tend to be given special attention by those looking at [[w:Special:RecentChanges|recent changes]].

== New accounts ==
New users are able to make changes to unprotected resources (which is the default status for the vast majority of pages), but are not able to edit protected resources or to upload files. By having an account, they are no longer identified by their IP address, which provides a greater degree of privacy. The single account name also makes it easier for contributors to build a positive reputation and relationships with other contributors.

The [[m:Help:Unified login|unified login]] (SUL) process permits users with an account to have a common username across all the [[wmf:Our projects|Wikimedia Foundation projects]]. Thus, the same username and password could be used to log into [[Wikiversity:Welcome|Wikiversity]], [[Wikipedia]], [[wikibooks:Wikibooks:Welcome|Wikibooks]] and [[commons:Commons:Welcome|Wikimedia Commons]], to name four of the projects.

== Autoconfirmed users ==
After four days and 10 edits, most users with a user account are automatically autoconfirmed. This provides them with the ability to upload files, edit semi-protected pages and perform page moves.

== Established users ==
There is no formal definition of an &quot;established user&quot;, and they don't necessarily have any more rights than an autoconfirmed user. In practice, though, an established user is someone who has been around long enough, and have done enough, to build up a bit of a reputation as a valued contributor. Established users may take on extra responsibilities, such as mentoring new editors, and they may gain extra user rights, such as [[w:Wikipedia:Rollback feature|rollback]] and [[wikipedia:Wikipedia:Autopatrolled|autopatrolled]], but the main advantage is that these are people who have been around for long enough to know how things work, understand the privacy policy, and have demonstrated that they are able to work within Wikipedia's framework.

== Administrators ==
Administrators are, (as their name would suggest), responsible for administrative actions, such as [[w:Wikipedia:Blocking policy|blocking and unblocking accounts]], [[w:Wikipedia:Deletion policy|deleting and restoring pages and files]], and [[w:Wikipedia:Protection policy|protecting and unprotecting]] content. As these tasks are both essential and open to misuse, custodians are carefully selected by the Wikipedia community. Administrators need to be trusted contributors who have demonstrated that they understand Wikiversity's policies and are able to interpret [[w:Wikipedia:Consensus|consensus]] accurately and consistently. As a result, the community has big expectations of administrators, such that it is expected that contributors have at least 6 months experience and over 2000 edits before applying for the role.

It is important to note that administrators may have extra tools that they can employ, but in terms of finding consensus they are no more able to direct debate than any other contributor. The administrator role is an important one, but determining policy, selecting administrators and engaging in debate are things that all contributors are encouraged to take part in as equal members of the community.

== Checkusers ==
CheckUsers have the ability to compare user accounts and IP addresses to determine if they are being used by the same people. Abusing accounts, in particular to get around a block or ban, or to lend the appearance of additional support in a debate, is not permitted on Wikipedia, and the CheckUsers have access to the tools required to investigate where such abuse is suspected. However, doing so requires that CheckUsers have access to potentially private information, such as the IP address from which a registered account is being used. Thus they are limited to highly trusted members of the community, who must be over 18 years of age, are required to provide identification to the Wikimedia Foundation, and who must work within the [[wmf:Privacy policy|privacy policy]].

== Bureaucrats ==
Bureaucrats are primarily responsible for three jobs: they determine consensus and decide whether or not to promote someone to administrator or bureaucrat; they can [[:Wikipedia:Changing username|rename user accounts]]; and they can permit [[w:Wikipedia:Bots|bots]] to operate. As with custodians, bureaucrats are expected to have a deep understanding of Wikipedia's processes, and are highly trusted members of the community.

== Further reading ==
* [[w:Wikipedia:Who writes Wikipedia|Wikipedia:Who writes Wikipedia]]
* [[w:Wikipedia:Wikipedians|Wikipedia:Wikipedians]]
* [[w:Wikipedia:User access levels|Wikipedia:User access levels]]

{{CourseCat}}</text>
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== Group 1 JDR ==
[[User:User|Klaas Kelchtermans]]

[[User:s111778|Jose Bosca Moya]]

[[User:s111785|Diego Panadero Roser]]

[[User:User|Juan Ricardo Rios Rivas]]

== Group 2 ==
[[User:s111312|Jinghua Wang]]

[[User:s110906|Elakkiya Ellavarason]]

[[User:s042478|Nis Schmidt]]

== Group 3 ==
[[User:s114581|Graeme Best]]

[[User:s114516|Oscar Rahnama]]

[[User:s091820|Wojciech Pawlak]]

[[User:s114580|Matt Birman]]

=== Network on a Chip ===

[https://www.campusnet.dtu.dk/cnnet/FileSharing/SADownload.aspx?ElementId=371901&amp;FolderId=622947&amp;FileId=3000951 Project Proposal]

[https://github.com/mattbirman/Network-on-Chip-in-VHDL Open-source project]

== Group 4 ==
[[User:Andreas B. Svendsen|Andreas B. Svendsen]]

[[User:Attila Sükösd|Attila Sükösd]]

[[User:Rasmus T. Karstensen|Rasmus T. Karstensen]]
{{CourseCat}}</text>
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  <page>
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      <contributor>
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      <text bytes="3077" sha1="kc5rbazejzt6b8n2qcremcx8hioj5c5" xml:space="preserve">This course belongs to the [[Electrical &amp; Computer Engineering Studies]]

== Introduction ==
* Introduction ([[Media:SysP.Intro.20161128.pdf|pdf]])

== File System ==
* File System ([[Media:SysP.FileSystem.20251023.pdf|pdf]])
* File Pointer ([[Media:SysP..FilePointer.20161103.pdf|pdf]])
* System Calls ([[Media:SysP.File.SysCall.20161128.pdf|pdf]])
* File IO ([[Media:SysP.FileIO.20251023.pdf|pdf]])


* Copilot: File System ([[Media:glibcFileSystem.20251029-2.pdf|pdf]])
* Copilot: File Buffer ([[Media:glibcFileBuffer.20251025-2.pdf|pdf]])
* Copilot: File IO ([[Media:glibcFileIO.20251025-2.pdf|pdf]])
* Copilot: File Permission ([[Media:glibcFilePerm.20260121.pdf|pdf]])
* Copilot: File Control ([[Media:CP.FileCntl.20260427.pdf|pdf]])

&lt;br&gt;
&lt;br&gt;

== Process ==
* Process  ([[Media:SysP.Process.20251120.pdf|pdf]])
* Fork  ([[Media:SysP.Fork.20251126.pdf|pdf]])


* Copilot: Process Information ([[Media:glibc.Process.1Info.20251101.pdf|pdf]])
* Copilot: Process Control ([[Media:glibc.Process.2Control.20251103.pdf|pdf]])
* Copilot: Process Execution ([[Media:glibc.Proc.3Exec.20251105.pdf|pdf]])
* Copilot: Process Fork ([[Media:glibc.Proc.4Fork.20251106.pdf|pdf]])
* Copilot: Process Context Switching ([[Media:glibc.Proc.5Context.20251107.pdf|pdf]])
* Copilot: Process Exec family of functions ([[Media:glibc.Proc.6ExecCall.20251112.pdf|pdf]])
* Copilot: Process Wait family of functions ([[Media:glibc.Proc.7WaitCall.20251112.pdf|pdf]])
* Copilot: Process Exit ([[Media:glibc.Proc.8Exit.20251113.pdf|pdf]])

&lt;/br&gt;

== Inter Process Communication==

=== Signal ===
* Signal  ([[Media:SysP.7.A.Signal.20121206.pdf|pdf]])


* Copilot: Signal 1. Alarm ([[Media:glibc.Signal.Alarm.20251201.pdf|pdf]])
* Copilot: Signal 2. Other Functions ([[Media:glibc.Signal.2Other.20251205.pdf|pdf]])

&lt;/br&gt;

=== Pipe ===

* Pipe ([[Media:SysP.3.A.IPC.20121115.pdf|pdf]])


* Copilot: Pipe 1. A Special File ([[Media:glibc.Pipe.File.20260307.pdf|pdf]])

&lt;/br&gt;

=== System V IPC ===

* Message Queue  ([[Media:SysP.5.A.MessageQ.20121213.pdf|pdf]])

* Shared Memory  ([[Media:SysP.8.A.SharedMem.20121227.pdf|pdf]])

* Semaphore  ([[Media:SysP.6.A.Semaphore.20251215.pdf|pdf]])

&lt;/br&gt;

* Copilot: Message Queue ([[Media:glibc.MessageQ.20251202.pdf|pdf]])
* Copilot: Shared Memory ([[Media:glibc.SharedMem.20251203.pdf|pdf]])
* Copilot: Semaphore ([[Media:glibc.Semaphore.20251215.pdf|pdf]])

&lt;/br&gt;

=== Socket ===

* Socket  ([[Media:SysP.4.A.Socket.20121122.pdf|pdf]])

&lt;/br&gt;

== Thread ==
* POSIX thread (pthread)  ([[Media:SysP.9.A.Pthread.20130225.pdf|pdf]])

==External links==
* [http://www.tldp.org/LDP/tlk/tlk.html The Linux Kernel]
* [http://www.tldp.org/LDP/lpg/lpg.html The Linux Programmer's Guide]
* [http://www.cs.cf.ac.uk/Dave/C/ Programming in C - UNIX System Calls and Subroutines using C.]
* [http://www.cs.cmu.edu/afs/cs/academic/class/15492-f07/www/pthreads.html POSIX thread (pthread) libraries]
* [https://computing.llnl.gov/tutorials/pthreads/#Thread POSIX Threads Programming]

[[Category:Linux]]
[[Category:Computer programming]]
[[Category:C programming language]]</text>
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      <text bytes="3077" sha1="f2zpkpvo1jumf81gztzv4v7asbbwxje" xml:space="preserve">This course belongs to the [[Electrical &amp; Computer Engineering Studies]]

== Introduction ==
* Introduction ([[Media:SysP.Intro.20161128.pdf|pdf]])

== File System ==
* File System ([[Media:SysP.FileSystem.20251023.pdf|pdf]])
* File Pointer ([[Media:SysP..FilePointer.20161103.pdf|pdf]])
* System Calls ([[Media:SysP.File.SysCall.20161128.pdf|pdf]])
* File IO ([[Media:SysP.FileIO.20251023.pdf|pdf]])


* Copilot: File System ([[Media:glibcFileSystem.20251029-2.pdf|pdf]])
* Copilot: File Buffer ([[Media:glibcFileBuffer.20251025-2.pdf|pdf]])
* Copilot: File IO ([[Media:glibcFileIO.20251025-2.pdf|pdf]])
* Copilot: File Permission ([[Media:glibcFilePerm.20260121.pdf|pdf]])
* Copilot: File Control ([[Media:CP.FileCntl.20260428.pdf|pdf]])

&lt;br&gt;
&lt;br&gt;

== Process ==
* Process  ([[Media:SysP.Process.20251120.pdf|pdf]])
* Fork  ([[Media:SysP.Fork.20251126.pdf|pdf]])


* Copilot: Process Information ([[Media:glibc.Process.1Info.20251101.pdf|pdf]])
* Copilot: Process Control ([[Media:glibc.Process.2Control.20251103.pdf|pdf]])
* Copilot: Process Execution ([[Media:glibc.Proc.3Exec.20251105.pdf|pdf]])
* Copilot: Process Fork ([[Media:glibc.Proc.4Fork.20251106.pdf|pdf]])
* Copilot: Process Context Switching ([[Media:glibc.Proc.5Context.20251107.pdf|pdf]])
* Copilot: Process Exec family of functions ([[Media:glibc.Proc.6ExecCall.20251112.pdf|pdf]])
* Copilot: Process Wait family of functions ([[Media:glibc.Proc.7WaitCall.20251112.pdf|pdf]])
* Copilot: Process Exit ([[Media:glibc.Proc.8Exit.20251113.pdf|pdf]])

&lt;/br&gt;

== Inter Process Communication==

=== Signal ===
* Signal  ([[Media:SysP.7.A.Signal.20121206.pdf|pdf]])


* Copilot: Signal 1. Alarm ([[Media:glibc.Signal.Alarm.20251201.pdf|pdf]])
* Copilot: Signal 2. Other Functions ([[Media:glibc.Signal.2Other.20251205.pdf|pdf]])

&lt;/br&gt;

=== Pipe ===

* Pipe ([[Media:SysP.3.A.IPC.20121115.pdf|pdf]])


* Copilot: Pipe 1. A Special File ([[Media:glibc.Pipe.File.20260307.pdf|pdf]])

&lt;/br&gt;

=== System V IPC ===

* Message Queue  ([[Media:SysP.5.A.MessageQ.20121213.pdf|pdf]])

* Shared Memory  ([[Media:SysP.8.A.SharedMem.20121227.pdf|pdf]])

* Semaphore  ([[Media:SysP.6.A.Semaphore.20251215.pdf|pdf]])

&lt;/br&gt;

* Copilot: Message Queue ([[Media:glibc.MessageQ.20251202.pdf|pdf]])
* Copilot: Shared Memory ([[Media:glibc.SharedMem.20251203.pdf|pdf]])
* Copilot: Semaphore ([[Media:glibc.Semaphore.20251215.pdf|pdf]])

&lt;/br&gt;

=== Socket ===

* Socket  ([[Media:SysP.4.A.Socket.20121122.pdf|pdf]])

&lt;/br&gt;

== Thread ==
* POSIX thread (pthread)  ([[Media:SysP.9.A.Pthread.20130225.pdf|pdf]])

==External links==
* [http://www.tldp.org/LDP/tlk/tlk.html The Linux Kernel]
* [http://www.tldp.org/LDP/lpg/lpg.html The Linux Programmer's Guide]
* [http://www.cs.cf.ac.uk/Dave/C/ Programming in C - UNIX System Calls and Subroutines using C.]
* [http://www.cs.cmu.edu/afs/cs/academic/class/15492-f07/www/pthreads.html POSIX thread (pthread) libraries]
* [https://computing.llnl.gov/tutorials/pthreads/#Thread POSIX Threads Programming]

[[Category:Linux]]
[[Category:Computer programming]]
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      <text bytes="11783" sha1="ix0m9ftu8wkdmwvgg5p2ftmyq44mrbc" xml:space="preserve">{{WikiJMed_top_menu}}
{{WikiJMed_right_menu}}
__NOTOC____NOEDITSECTION__

{{WikiJournal/Publishing/Intro}} 

The journal publishes both review articles and original research in [[WikiJournal of Medicine/Publishing#Publication_formats|various formats]]. WikiJournals enable academic and medical professionals to contribute expert knowledge to the Wikimedia movement in the [[Wikipedia:Academic publishing|academic publishing]] format that directly rewards them with citable publications. Included works are assigned [[Wikipedia:Digital object identifier|DOI codes]] (permanent links to each work via [[W:Crossref|Crossref]]) and are indexed by [https://scholar.google.co.uk/scholar?hl=en&amp;as_sdt=0%2C5&amp;q=source%3AWikiJournal+source%3AMedicine Google Scholar], [https://doaj.org/toc/2002-4436 DOAJ] and [[{{ROOTPAGENAME}}/About#Indexing_and_impact|others]].

The journal targets a broad population spanning from advanced researchers and clinicians to students and laypersons, wherein the latter can get quick explanations of advanced terms by [[Wikipedia:Hyperlink#Hyperlinks_in_wikis|in-line links to Wikipedia]].

{{anchor|PDF}}

{{WikiJMed_h2|Unique publication features}}
&lt;div style=&quot;display:table&quot;&gt;

{{WikiJournal about box
|[[File:Open_Access_logo_PLoS_transparent.svg|40px|link=]]
|Open Access
|All of our published articles are [[w:Open access journal|openly accessible]] under a free [https://creativecommons.org/licenses/by-sa/4.0/ Creative Commons] [[File:Cc.logo.circle.svg|16px|class=skin-invert link=]] or [[Wikipedia:Wikipedia:FAQ/Copyright#Can_I_add_something_to_Wikipedia_that_I_got_from_somewhere_else?|similar license]]|#ff6600}}

{{WikiJournal about box
|[[File:Fxemoji u1F4B2 narrow.svg|40px|link=]]
|Free to publish
|We are a fully non-profit journal with a volunteer board of editors, and we therefore have no publication charges of any kind|#0b550b}}

{{WikiJournal about box
|[[File:Journal Icon.svg|50px|link=]]
|Public peer-review
|All peer reviews are published and publicly accessible|#222}}

{{WikiJournal about box
|[[File:Wikipedia-W-bold-in-square-Clean.svg|50px|class=skin-invert|link=]]
|Wikipedia-integrated
|Appropriate material is integrated into [[w:Main Page|Wikipedia]] for added reach and exposure|#1111aa}}
&lt;/div&gt;

Further information:
* [https://drive.google.com/file/d/1ZQ3AsK1H_kAYuJQ8082xCuPgQyYmJDje/view?usp=sharing Presentation of WikiJournal at Wikimania 2019]
* [[WikiJournal User Group/Comparison to other journals|Comparison to other journals]]

{{WikiJMed_h2|Structure}}

=== Publication ===
All accepted articles are published in two forms:
* An editable &quot;wiki&quot; version, hosted using [[Wikipedia:MediaWiki|MediaWiki software]] by the [https://wikimediafoundation.org/wiki/Home Wikimedia Foundation]
* A stable [[Wikipedia:PDF|PDF]] version, which is the cite-able &quot;version of record&quot;
Before inclusion in the journal, article submissions are [[{{ROOTPAGENAME}}/Peer reviewers|peer reviewed]] by at least two medical experts. The quality of the peer reviews is weighted in when the [[{{ROOTPAGENAME}}/Editorial board|editorial board]] makes final decision on including the article in the journal. The journal emphasises transparency throughout the process, from peer reviews to final version and all versions are freely accessible online.

=== Contributors ===
The journal offers its services for free, by donations to the [https://wikimediafoundation.org/wiki/Home Wikimedia Foundation], and fully volunteer [[{{ROOTPAGENAME}}/Peer reviewers|peer reviewers]], [[{{ROOTPAGENAME}}/Editors|editors]] and [[{{ROOTPAGENAME}}/Editors|editorial board]] members. {{ROOTPAGENAME}} is part of the [[meta:WikiJournal User Group|WikiJournal Publishing Group]], which is also the formal ''publisher'' of the journal.

Anyone may edit journal pages, but only minor edits to accepted journal articles (e.g. spelling) are permitted. Any edits that change the content of a published article require an additional round of peer-review, and if accepted, are published in an updated version of the page. Suggestions for updates of the main text of published articles may be created as separate drafts that are re-submitted to undergo peer review before being used to update the article.

All articles are open to post-publication peer review, and edit suggestions can be added at the &quot;Discuss&quot; tab at the top of each page.

=== Guidelines ===
The [[{{ROOTPAGENAME}}/Ethics statement|publication ethics statement]] of the {{ROOTPAGENAME}} abides by several international guidelines:
*[[wikipedia:ICMJE Recommendations (Uniform Requirements for Manuscripts)|ICMJE]] Recommendations for the Conduct, Reporting, Editing, and Publication: [http://www.icmje.org/icmje-recommendations.pdf]
*[[Wikipedia:Committee on Publication Ethics|COPE]] code of conduct for journal editors: [http://publicationethics.org/files/Code_of_conduct_for_journal_editors_Mar11.pdf]
*[[Wikipedia:Budapest Open Access Initiative|Budapest Open Access Initiative]] recommendations: [http://www.budapestopenaccessinitiative.org/boai-10-recommendations]

{{WikiJMed_h2|Indexing and impact}}
{{fig||Scopus logo.svg|float=right|width=120px|caption=[[Wikipedia:Scopus|Scopus]]}}
{{fig||COPE logo for WikiJMed.png|float=right|width=120px|caption=[[Wikipedia:Committee on Publication Ethics|Committee on Publication Ethics]]}}
{{fig||OPM-vertical.png|float=right|width=120px|caption=[https://openpublishingawards.org/recipients/wikijournals/index.html Open Publishing Award winner]&lt;br&gt;(open publishing model category)}}

The journal is currently indexed in the following databasesː
* [https://scholar.google.co.uk/scholar?hl=en&amp;as_sdt=0%2C5&amp;q=source%3AWikiJournal+source%3AMedicine Google Scholar]
* [https://doaj.org/toc/2002-4436 Directory of Open Access Journals (DOAJ)]
* [https://www.informit.org/product-details/623/WIKIJMED Informit], including archiving of published articles
* [https://www.worldcat.org/title/WikiJournal-of-medicine/oclc/961872227 Worldcat]
* [https://www.altmetric.com/explorer/report/ab3b956e-bf4a-4277-8084-6605d397fcb7 AltMetric]
* [https://dx.doi.org/10.15347 Crossref]
* [https://portal.issn.org/resource/ISSN/2002-4436 ROAD]
* [https://www.scienceopen.com/search#collection/5d82d3f1-9597-475c-88c8-c7f0b6cebd27 ScienceOpen]
* [https://app.dimensions.ai/discover/publication?and_facet_source_title=jour.1155558 Dimensions]
* [https://tools.wmflabs.org/scholia/topic/Q24657325 Scholia] (via [[wikidata:Q24657325|WikiData]])
* [https://suggestor.step.scopus.com/progressTracker/?trackingID=B5576DA669EC06F0 Scopus]
* [https://www.semanticscholar.org/paper/WikiJournal-of-Medicine%2C-the-first-academic-journal-Shafee-Das/f6c1d041f71e6127bbace3ea072e01a0836272dd Semantic Scholar]

The journal is preparing to be indexed in the following locations: [https://www.ncbi.nlm.nih.gov/pmc/ PubMed Central], [https://clarivate.com/products/web-of-science/ Web of Science]

The journal is also listed in:
* [https://publicationethics.org/members/wikijournal-medicine Committee on Publication Ethics]
* [https://transpose-publishing.github.io/#/ Transpose]
* [https://reimaginereview.asapbio.org/listing/wikijournal-user-group/ ReimagineReview]

The journal prioritises diverse impact across a range of metrics, such as [https://www.altmetric.com/explorer/report/ab3b956e-bf4a-4277-8084-6605d397fcb7 its AltMetric score], in addition to traditional Impact Factor (Not yet assigned by Web of Science; unofficial estimation [[{{ROOTPAGENAME}}/Citation metrics|here]]). Additionally high reach is achieved for those articles integrated into Wikipedia [https://tools.wmflabs.org/massviews/?platform=all-access&amp;agent=user&amp;source=category&amp;target=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FCategory%3AWikipedia_articles_published_in_WikiJournal_of_Medicine&amp;range=latest-356&amp;subjectpage=0&amp;subcategories=0&amp;sort=views in full] or [https://tools.wmflabs.org/massviews/?platform=all-access&amp;agent=user&amp;source=category&amp;target=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FCategory%3AWikipedia_articles_with_sections_published_in_WikiJournal_of_Medicine&amp;range=latest-356&amp;subjectpage=0&amp;subcategories=0&amp;sort=views&amp;direction=1&amp;view=list in part].  

{{WikiJMed_h2|History}}
:''More detailed updates are located at the [[Talk:{{ROOTPAGENAME}}|discussion page]]''

{{fig||WikiJournal logo.svg| Part of the [[:m:WikiJournal User Group|WikiJournal Publishing Group]] |float=right|width=100px}}

*2004-07-05: [https://meta.wikimedia.org/w/index.php?title=Wiki_Scholarly_Journals&amp;oldid=52153 Proposal of &quot;Wiki Scholarly Journals&quot;], where articles undergo peer review
*2009-09-12: [https://strategy.wikimedia.org/wiki/Proposal:Journal_(A_peer-review_journal_to_allow/encourage_academics_to_write_Wikipedia_articles) Proposal:Journal], a structured proposal for creating a journal as a [[Wikipedia:Wikimedia project|Wikimedia project]]
*2010-09-28: [[Wikiversity:Journal of the future|Core criteria]] for a journal are described in Wikiversity
*2011-01-22: A [[:File:Hormones estradiol, progesterone, LH and FSH during menstrual cycle.svg|diagram of hormone levels throughout the menstrual cycle]], that would later become the first WikiJournal article, is uploaded
*2012-09-11: [[:File:Peer review certification of 2012 Uppsala teaspoon study.jpg|First official external peer review]] performed on a pilot study
*2013-12-11: The [[:File:Hormones estradiol, progesterone, LH and FSH during menstrual cycle.svg|diagram of hormone levels]] is peer reviewed
*2014-[[Wikipedia:Waffle Day|03-25]]: ''Wikiversity Journal'' is officially created by [[User:Mikael Häggström|Mikael Häggström]], with ''Wikiversity Journal of Medicine'' becoming the first member journal
*2014-03-28: The journal is registered by the [[Wikipedia:National Library of Sweden|National Library of Sweden]] and is assigned an [[Wikipedia:International Standard Serial Number|ISSN]]
*2014-04-12: Articles are assigned [[wikipedia:digital object identifier|DOI codes]] by Crossref
*2015-01-12: The first [[{{ROOTPAGENAME}}/Editorial board|editorial board]] is formed, with members Dr [[User:Doc James|James Heilman]] (former president of [[meta:Wiki_Project_Med|WikiProject Medicine Foundation]] and ex-member of the Wikimedia Foundation Board of Trustees), Gwinyai Masukume, Lisa Kipersztok and Mikael Häggström
*2016-02-08: Inspired by the newly created [[WikiJournal of Science]], the journal expands accepted forms of submissions to include any medical Wikipedia content
*2016-06-01: The project is accepted as a [[Meta:Wikimedia_user_groups|Wikimedia user group]]
*2016-07: Major layout and formatting update by [[User:Evolution and evolvability|Thomas Shafee]]
*2016-09-07: ''Wikiversity Journal'' is renamed to become ''WikiJournal''
*2016-09-29: [https://doaj.org/toc/2002-4436 Listing in the Directory of Open Access Journals]
*2016-11-30: Registration as a non-profit organization in Sweden
*2017-01-04: Requirement of peer review of at least 2 independent medical experts for all publications
*2017-03-17: The [http://form.wikijmed.org/ authorship declaration form] was implemented, replacing the previous [[{{ROOTPAGENAME}}/Submission letter for single author|submission letter]]
*2017-12-11: [https://www.scienceopen.com/search#collection/5d82d3f1-9597-475c-88c8-c7f0b6cebd27 Indexed in ScienceOpen]
*2018-02-21: Formal [[{{ROOTPAGENAME}}/Ethics statement|ethics statement]] ratified
*2018-07-06: The journal is accepted as a member of the [[Wikipedia:Committee on Publication Ethics|Committee on Publication Ethics]] (COPE).
*2019: Winner of the [https://openpublishingawards.org/index.php/the-wikijournal-user-group/ Open Publishing Awards (open publishing model category)]
*2020-06-19: The journal is [[{{ROOTPAGENAME}}/Applications/SCOPUS#Acceptance|accepted]] for indexing in SCOPUS

[[Category:{{ROOTPAGENAME}}]]</text>
      <sha1>ix0m9ftu8wkdmwvgg5p2ftmyq44mrbc</sha1>
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    <revision>
      <id>2806909</id>
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      <timestamp>2026-04-28T19:51:05Z</timestamp>
      <contributor>
        <username>Nintendofan885</username>
        <id>2887676</id>
      </contributor>
      <comment>invert on dark mode for light green</comment>
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      <text bytes="11801" sha1="8px0d3ionhqqs44850q7mlewpjw4z9d" xml:space="preserve">{{WikiJMed_top_menu}}
{{WikiJMed_right_menu}}
__NOTOC____NOEDITSECTION__

{{WikiJournal/Publishing/Intro}} 

The journal publishes both review articles and original research in [[WikiJournal of Medicine/Publishing#Publication_formats|various formats]]. WikiJournals enable academic and medical professionals to contribute expert knowledge to the Wikimedia movement in the [[Wikipedia:Academic publishing|academic publishing]] format that directly rewards them with citable publications. Included works are assigned [[Wikipedia:Digital object identifier|DOI codes]] (permanent links to each work via [[W:Crossref|Crossref]]) and are indexed by [https://scholar.google.co.uk/scholar?hl=en&amp;as_sdt=0%2C5&amp;q=source%3AWikiJournal+source%3AMedicine Google Scholar], [https://doaj.org/toc/2002-4436 DOAJ] and [[{{ROOTPAGENAME}}/About#Indexing_and_impact|others]].

The journal targets a broad population spanning from advanced researchers and clinicians to students and laypersons, wherein the latter can get quick explanations of advanced terms by [[Wikipedia:Hyperlink#Hyperlinks_in_wikis|in-line links to Wikipedia]].

{{anchor|PDF}}

{{WikiJMed_h2|Unique publication features}}
&lt;div style=&quot;display:table&quot;&gt;

{{WikiJournal about box
|[[File:Open_Access_logo_PLoS_transparent.svg|40px|link=]]
|Open Access
|All of our published articles are [[w:Open access journal|openly accessible]] under a free [https://creativecommons.org/licenses/by-sa/4.0/ Creative Commons] [[File:Cc.logo.circle.svg|16px|class=skin-invert link=]] or [[Wikipedia:Wikipedia:FAQ/Copyright#Can_I_add_something_to_Wikipedia_that_I_got_from_somewhere_else?|similar license]]|#ff6600}}

{{WikiJournal about box
|[[File:Fxemoji u1F4B2 narrow.svg|40px|class=skin-invert|link=]]
|Free to publish
|We are a fully non-profit journal with a volunteer board of editors, and we therefore have no publication charges of any kind|#0b550b}}

{{WikiJournal about box
|[[File:Journal Icon.svg|50px|link=]]
|Public peer-review
|All peer reviews are published and publicly accessible|#222}}

{{WikiJournal about box
|[[File:Wikipedia-W-bold-in-square-Clean.svg|50px|class=skin-invert|link=]]
|Wikipedia-integrated
|Appropriate material is integrated into [[w:Main Page|Wikipedia]] for added reach and exposure|#1111aa}}
&lt;/div&gt;

Further information:
* [https://drive.google.com/file/d/1ZQ3AsK1H_kAYuJQ8082xCuPgQyYmJDje/view?usp=sharing Presentation of WikiJournal at Wikimania 2019]
* [[WikiJournal User Group/Comparison to other journals|Comparison to other journals]]

{{WikiJMed_h2|Structure}}

=== Publication ===
All accepted articles are published in two forms:
* An editable &quot;wiki&quot; version, hosted using [[Wikipedia:MediaWiki|MediaWiki software]] by the [https://wikimediafoundation.org/wiki/Home Wikimedia Foundation]
* A stable [[Wikipedia:PDF|PDF]] version, which is the cite-able &quot;version of record&quot;
Before inclusion in the journal, article submissions are [[{{ROOTPAGENAME}}/Peer reviewers|peer reviewed]] by at least two medical experts. The quality of the peer reviews is weighted in when the [[{{ROOTPAGENAME}}/Editorial board|editorial board]] makes final decision on including the article in the journal. The journal emphasises transparency throughout the process, from peer reviews to final version and all versions are freely accessible online.

=== Contributors ===
The journal offers its services for free, by donations to the [https://wikimediafoundation.org/wiki/Home Wikimedia Foundation], and fully volunteer [[{{ROOTPAGENAME}}/Peer reviewers|peer reviewers]], [[{{ROOTPAGENAME}}/Editors|editors]] and [[{{ROOTPAGENAME}}/Editors|editorial board]] members. {{ROOTPAGENAME}} is part of the [[meta:WikiJournal User Group|WikiJournal Publishing Group]], which is also the formal ''publisher'' of the journal.

Anyone may edit journal pages, but only minor edits to accepted journal articles (e.g. spelling) are permitted. Any edits that change the content of a published article require an additional round of peer-review, and if accepted, are published in an updated version of the page. Suggestions for updates of the main text of published articles may be created as separate drafts that are re-submitted to undergo peer review before being used to update the article.

All articles are open to post-publication peer review, and edit suggestions can be added at the &quot;Discuss&quot; tab at the top of each page.

=== Guidelines ===
The [[{{ROOTPAGENAME}}/Ethics statement|publication ethics statement]] of the {{ROOTPAGENAME}} abides by several international guidelines:
*[[wikipedia:ICMJE Recommendations (Uniform Requirements for Manuscripts)|ICMJE]] Recommendations for the Conduct, Reporting, Editing, and Publication: [http://www.icmje.org/icmje-recommendations.pdf]
*[[Wikipedia:Committee on Publication Ethics|COPE]] code of conduct for journal editors: [http://publicationethics.org/files/Code_of_conduct_for_journal_editors_Mar11.pdf]
*[[Wikipedia:Budapest Open Access Initiative|Budapest Open Access Initiative]] recommendations: [http://www.budapestopenaccessinitiative.org/boai-10-recommendations]

{{WikiJMed_h2|Indexing and impact}}
{{fig||Scopus logo.svg|float=right|width=120px|caption=[[Wikipedia:Scopus|Scopus]]}}
{{fig||COPE logo for WikiJMed.png|float=right|width=120px|caption=[[Wikipedia:Committee on Publication Ethics|Committee on Publication Ethics]]}}
{{fig||OPM-vertical.png|float=right|width=120px|caption=[https://openpublishingawards.org/recipients/wikijournals/index.html Open Publishing Award winner]&lt;br&gt;(open publishing model category)}}

The journal is currently indexed in the following databasesː
* [https://scholar.google.co.uk/scholar?hl=en&amp;as_sdt=0%2C5&amp;q=source%3AWikiJournal+source%3AMedicine Google Scholar]
* [https://doaj.org/toc/2002-4436 Directory of Open Access Journals (DOAJ)]
* [https://www.informit.org/product-details/623/WIKIJMED Informit], including archiving of published articles
* [https://www.worldcat.org/title/WikiJournal-of-medicine/oclc/961872227 Worldcat]
* [https://www.altmetric.com/explorer/report/ab3b956e-bf4a-4277-8084-6605d397fcb7 AltMetric]
* [https://dx.doi.org/10.15347 Crossref]
* [https://portal.issn.org/resource/ISSN/2002-4436 ROAD]
* [https://www.scienceopen.com/search#collection/5d82d3f1-9597-475c-88c8-c7f0b6cebd27 ScienceOpen]
* [https://app.dimensions.ai/discover/publication?and_facet_source_title=jour.1155558 Dimensions]
* [https://tools.wmflabs.org/scholia/topic/Q24657325 Scholia] (via [[wikidata:Q24657325|WikiData]])
* [https://suggestor.step.scopus.com/progressTracker/?trackingID=B5576DA669EC06F0 Scopus]
* [https://www.semanticscholar.org/paper/WikiJournal-of-Medicine%2C-the-first-academic-journal-Shafee-Das/f6c1d041f71e6127bbace3ea072e01a0836272dd Semantic Scholar]

The journal is preparing to be indexed in the following locations: [https://www.ncbi.nlm.nih.gov/pmc/ PubMed Central], [https://clarivate.com/products/web-of-science/ Web of Science]

The journal is also listed in:
* [https://publicationethics.org/members/wikijournal-medicine Committee on Publication Ethics]
* [https://transpose-publishing.github.io/#/ Transpose]
* [https://reimaginereview.asapbio.org/listing/wikijournal-user-group/ ReimagineReview]

The journal prioritises diverse impact across a range of metrics, such as [https://www.altmetric.com/explorer/report/ab3b956e-bf4a-4277-8084-6605d397fcb7 its AltMetric score], in addition to traditional Impact Factor (Not yet assigned by Web of Science; unofficial estimation [[{{ROOTPAGENAME}}/Citation metrics|here]]). Additionally high reach is achieved for those articles integrated into Wikipedia [https://tools.wmflabs.org/massviews/?platform=all-access&amp;agent=user&amp;source=category&amp;target=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FCategory%3AWikipedia_articles_published_in_WikiJournal_of_Medicine&amp;range=latest-356&amp;subjectpage=0&amp;subcategories=0&amp;sort=views in full] or [https://tools.wmflabs.org/massviews/?platform=all-access&amp;agent=user&amp;source=category&amp;target=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FCategory%3AWikipedia_articles_with_sections_published_in_WikiJournal_of_Medicine&amp;range=latest-356&amp;subjectpage=0&amp;subcategories=0&amp;sort=views&amp;direction=1&amp;view=list in part].  

{{WikiJMed_h2|History}}
:''More detailed updates are located at the [[Talk:{{ROOTPAGENAME}}|discussion page]]''

{{fig||WikiJournal logo.svg| Part of the [[:m:WikiJournal User Group|WikiJournal Publishing Group]] |float=right|width=100px}}

*2004-07-05: [https://meta.wikimedia.org/w/index.php?title=Wiki_Scholarly_Journals&amp;oldid=52153 Proposal of &quot;Wiki Scholarly Journals&quot;], where articles undergo peer review
*2009-09-12: [https://strategy.wikimedia.org/wiki/Proposal:Journal_(A_peer-review_journal_to_allow/encourage_academics_to_write_Wikipedia_articles) Proposal:Journal], a structured proposal for creating a journal as a [[Wikipedia:Wikimedia project|Wikimedia project]]
*2010-09-28: [[Wikiversity:Journal of the future|Core criteria]] for a journal are described in Wikiversity
*2011-01-22: A [[:File:Hormones estradiol, progesterone, LH and FSH during menstrual cycle.svg|diagram of hormone levels throughout the menstrual cycle]], that would later become the first WikiJournal article, is uploaded
*2012-09-11: [[:File:Peer review certification of 2012 Uppsala teaspoon study.jpg|First official external peer review]] performed on a pilot study
*2013-12-11: The [[:File:Hormones estradiol, progesterone, LH and FSH during menstrual cycle.svg|diagram of hormone levels]] is peer reviewed
*2014-[[Wikipedia:Waffle Day|03-25]]: ''Wikiversity Journal'' is officially created by [[User:Mikael Häggström|Mikael Häggström]], with ''Wikiversity Journal of Medicine'' becoming the first member journal
*2014-03-28: The journal is registered by the [[Wikipedia:National Library of Sweden|National Library of Sweden]] and is assigned an [[Wikipedia:International Standard Serial Number|ISSN]]
*2014-04-12: Articles are assigned [[wikipedia:digital object identifier|DOI codes]] by Crossref
*2015-01-12: The first [[{{ROOTPAGENAME}}/Editorial board|editorial board]] is formed, with members Dr [[User:Doc James|James Heilman]] (former president of [[meta:Wiki_Project_Med|WikiProject Medicine Foundation]] and ex-member of the Wikimedia Foundation Board of Trustees), Gwinyai Masukume, Lisa Kipersztok and Mikael Häggström
*2016-02-08: Inspired by the newly created [[WikiJournal of Science]], the journal expands accepted forms of submissions to include any medical Wikipedia content
*2016-06-01: The project is accepted as a [[Meta:Wikimedia_user_groups|Wikimedia user group]]
*2016-07: Major layout and formatting update by [[User:Evolution and evolvability|Thomas Shafee]]
*2016-09-07: ''Wikiversity Journal'' is renamed to become ''WikiJournal''
*2016-09-29: [https://doaj.org/toc/2002-4436 Listing in the Directory of Open Access Journals]
*2016-11-30: Registration as a non-profit organization in Sweden
*2017-01-04: Requirement of peer review of at least 2 independent medical experts for all publications
*2017-03-17: The [http://form.wikijmed.org/ authorship declaration form] was implemented, replacing the previous [[{{ROOTPAGENAME}}/Submission letter for single author|submission letter]]
*2017-12-11: [https://www.scienceopen.com/search#collection/5d82d3f1-9597-475c-88c8-c7f0b6cebd27 Indexed in ScienceOpen]
*2018-02-21: Formal [[{{ROOTPAGENAME}}/Ethics statement|ethics statement]] ratified
*2018-07-06: The journal is accepted as a member of the [[Wikipedia:Committee on Publication Ethics|Committee on Publication Ethics]] (COPE).
*2019: Winner of the [https://openpublishingawards.org/index.php/the-wikijournal-user-group/ Open Publishing Awards (open publishing model category)]
*2020-06-19: The journal is [[{{ROOTPAGENAME}}/Applications/SCOPUS#Acceptance|accepted]] for indexing in SCOPUS

[[Category:{{ROOTPAGENAME}}]]</text>
      <sha1>8px0d3ionhqqs44850q7mlewpjw4z9d</sha1>
    </revision>
  </page>
  <page>
    <title>The necessities in Filter Theory</title>
    <ns>0</ns>
    <id>199550</id>
    <revision>
      <id>2806979</id>
      <parentid>2806565</parentid>
      <timestamp>2026-04-29T10:43:37Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Sample Processing Methods */</comment>
      <origin>2806979</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1524" sha1="tkaciq0bse25gke0bxqwvi2b4h5p13v" xml:space="preserve">==''' Background '''==
=== Bode plot ===

See [http://lpsa.swarthmore.edu/Bode/Bode.html swarthmore]
&lt;/br&gt;

=== OP Amp ===
Overview ([[Media:OPAmp.A.1.20151203.pdf  |pdf]])
 
See [http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/opampcon.html#c1 Hyperphysics]
&lt;/br&gt;

==''' Analog Filter Analysis (Continuous Time) '''==
=== First Order Filters ===
&lt;/br&gt;

=== Second Order Filters ===
&lt;/br&gt;

==''' Digital Filter Analysis (Discrete Time) '''==

=== Sample Processing Methods  ===
* Tapped Delays ([[Media:Sample.TappedDelay.20260427.pdf  |A.pdf]])
* Programming Considerations
* Circular Buffers

=== FIR Filter Realizations ===
* Direct Form FIR Filter 
* Canonical Form FIR Filter
* Cascade Form FIR Filter

=== IIR Filter Realizations ===
* Direct Form IIR Filter ([[Media:IIR.DirectForm.20231209.pdf  |A.pdf]])
* Canonical Form IIR Filter
* Cascade Form IIR Filter 
&lt;/br&gt;

=== FIR (Finite Impulse Response) Filters ===
* Block Processing Methods
* Sample Processing Methods
* Window Method
* Kaiser Window
* Frequency Sampling Method
&lt;/br&gt;

=== IIR (Infinite Impulse Response) Filters ===
* Bilinear Transform
* 1st Order Lowpass and Highpass Filters
* 2nd Order Lowpass and Highpass Filters
* Parametric Equalizer Filters
* Comb Filters
* High Order Filters
&lt;/br&gt;


=== Example Octave Codes for Digital Filters === 

==== Octave Functions for Filters ====
* Octave Functions for Filters ([[Media:Octave.1.Function.1.A.20180219.pdf  |A.pdf]]) 


&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]</text>
      <sha1>tkaciq0bse25gke0bxqwvi2b4h5p13v</sha1>
    </revision>
    <revision>
      <id>2806981</id>
      <parentid>2806979</parentid>
      <timestamp>2026-04-29T10:44:48Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Sample Processing Methods */</comment>
      <origin>2806981</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1524" sha1="40n2f240lrncmo6bprn6ykyaojr3f1z" xml:space="preserve">==''' Background '''==
=== Bode plot ===

See [http://lpsa.swarthmore.edu/Bode/Bode.html swarthmore]
&lt;/br&gt;

=== OP Amp ===
Overview ([[Media:OPAmp.A.1.20151203.pdf  |pdf]])
 
See [http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/opampcon.html#c1 Hyperphysics]
&lt;/br&gt;

==''' Analog Filter Analysis (Continuous Time) '''==
=== First Order Filters ===
&lt;/br&gt;

=== Second Order Filters ===
&lt;/br&gt;

==''' Digital Filter Analysis (Discrete Time) '''==

=== Sample Processing Methods  ===
* Tapped Delays ([[Media:Sample.TappedDelay.20260428.pdf  |A.pdf]])
* Programming Considerations
* Circular Buffers

=== FIR Filter Realizations ===
* Direct Form FIR Filter 
* Canonical Form FIR Filter
* Cascade Form FIR Filter

=== IIR Filter Realizations ===
* Direct Form IIR Filter ([[Media:IIR.DirectForm.20231209.pdf  |A.pdf]])
* Canonical Form IIR Filter
* Cascade Form IIR Filter 
&lt;/br&gt;

=== FIR (Finite Impulse Response) Filters ===
* Block Processing Methods
* Sample Processing Methods
* Window Method
* Kaiser Window
* Frequency Sampling Method
&lt;/br&gt;

=== IIR (Infinite Impulse Response) Filters ===
* Bilinear Transform
* 1st Order Lowpass and Highpass Filters
* 2nd Order Lowpass and Highpass Filters
* Parametric Equalizer Filters
* Comb Filters
* High Order Filters
&lt;/br&gt;


=== Example Octave Codes for Digital Filters === 

==== Octave Functions for Filters ====
* Octave Functions for Filters ([[Media:Octave.1.Function.1.A.20180219.pdf  |A.pdf]]) 


&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]</text>
      <sha1>40n2f240lrncmo6bprn6ykyaojr3f1z</sha1>
    </revision>
  </page>
  <page>
    <title>Python programming in plain view</title>
    <ns>0</ns>
    <id>212733</id>
    <revision>
      <id>2806969</id>
      <parentid>2806545</parentid>
      <timestamp>2026-04-29T09:34:25Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Using Libraries */</comment>
      <origin>2806969</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2647" sha1="p2c0xex0nu508ghocfj1v4ws1qhnzhd" 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.20260427.pdf |pdf]])
* Namespaces ([[Media:Python.Work2.Scope.1A.20231021.pdf |pdf]])


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

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

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

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

=== Handling a Big Work ===

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

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

=== Handling Series of Data ===

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

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

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


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

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

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

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

==External links==
* [http://www.southampton.ac.uk/~fangohr/training/python/pdfs/Python-for-Computational-Science-and-Engineering.pdf Python and Computational Science and Engineering]</text>
      <sha1>p2c0xex0nu508ghocfj1v4ws1qhnzhd</sha1>
    </revision>
    <revision>
      <id>2806971</id>
      <parentid>2806969</parentid>
      <timestamp>2026-04-29T09:41:27Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Using Libraries */</comment>
      <origin>2806971</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2647" sha1="3slxbzvpmn0cmuaysq511oao8z0me86" 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.20260428.pdf |pdf]])
* Namespaces ([[Media:Python.Work2.Scope.1A.20231021.pdf |pdf]])


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

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

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

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

=== Handling a Big Work ===

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

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

=== Handling Series of Data ===

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

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

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


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

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

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

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

==External links==
* [http://www.southampton.ac.uk/~fangohr/training/python/pdfs/Python-for-Computational-Science-and-Engineering.pdf Python and Computational Science and Engineering]</text>
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      <comment>dark mode</comment>
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[[File:WikiJournal of Medicine logo.svg|80px|center|link=WikiJournal of Medicine]]
&lt;div style=&quot;font-size:  13pt;
            text-align: center;&quot;&gt;
WikiJournal of Medicine&lt;/div&gt;

&lt;div style=&quot;text-align:  center;&quot;&gt;
An [[Wikipedia:Open access journal|open access journal]] with&lt;br&gt; no publication costs – [[WikiJournal of Medicine/About|About]]&lt;/div&gt;
&lt;br&gt;
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[http://www.wikijmed.org www.WikiJMed.org]&lt;br&gt;
ISSN: 2002-4436&lt;br&gt;
Frequency: Continuous&lt;br&gt;
Since: March 2014&lt;br&gt;
Funding: [[W:Wikimedia Foundation|Wikimedia Foundation]]&lt;br&gt;
Publisher: [[meta:WikiJournal User Group|WikiJournal User Group]]

--------

'''On social media'''

{{Category news feed|cat=Articles included in WikiJournal of Medicine}}&lt;br&gt;
[[File:2023 Facebook icon.svg|16px|link=]] [https://www.facebook.com/pages/WikiJMed Facebook]&lt;br&gt;
[[File:X logo 2023.svg|16px|link=]] [https://twitter.com/WikiJMed Twitter]&lt;br&gt;
[[File:YouTube social white square (2017).svg|16px|link=]] [https://www.youtube.com/channel/UCcy1F3VGeIE5mP9SBwZSnBQ YouTube]&lt;br&gt;
[[File:Aiga_mail.svg|16px|link=]] [https://groups.google.com/forum/#!forum/{{WikiJXyz}}/join Mailing list]

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{{WikiJMed right menu/Member of}}

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{{WikiJMed right menu/In the news}}

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[[WikiJournal of Medicine/Contact|Contact]] &amp; [[WikiJournal_of_Medicine/Editorial_guidelines#How_to_contribute|Contribute]]
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&lt;noinclude&gt;[[Category:WikiJournal of Medicine formatting templates]] [[Category:WikiJournal headers, banners and infoboxes]]&lt;/noinclude&gt;</text>
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      <comment>also invert icons</comment>
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            text-align: center;&quot;&gt;
WikiJournal of Medicine&lt;/div&gt;

&lt;div style=&quot;text-align:  center;&quot;&gt;
An [[Wikipedia:Open access journal|open access journal]] with&lt;br&gt; no publication costs – [[WikiJournal of Medicine/About|About]]&lt;/div&gt;
&lt;br&gt;
--------

[http://www.wikijmed.org www.WikiJMed.org]&lt;br&gt;
ISSN: 2002-4436&lt;br&gt;
Frequency: Continuous&lt;br&gt;
Since: March 2014&lt;br&gt;
Funding: [[W:Wikimedia Foundation|Wikimedia Foundation]]&lt;br&gt;
Publisher: [[meta:WikiJournal User Group|WikiJournal User Group]]

--------

'''On social media'''

{{Category news feed|cat=Articles included in WikiJournal of Medicine}}&lt;br&gt;
[[File:2023 Facebook icon.svg|16px|link=]] [https://www.facebook.com/pages/WikiJMed Facebook]&lt;br&gt;
[[File:X logo 2023.svg|16px|class=skin-invert|link=]] [https://twitter.com/WikiJMed Twitter]&lt;br&gt;
[[File:YouTube social white square (2017).svg|16px|link=]] [https://www.youtube.com/channel/UCcy1F3VGeIE5mP9SBwZSnBQ YouTube]&lt;br&gt;
[[File:Aiga mail.svg|16px|class=skin-invert|link=]] [https://groups.google.com/forum/#!forum/{{WikiJXyz}}/join Mailing list]

--------

{{WikiJMed right menu/Member of}}

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{{WikiJMed right menu/In the news}}

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[[WikiJournal of Medicine/Contact|Contact]] &amp; [[WikiJournal_of_Medicine/Editorial_guidelines#How_to_contribute|Contribute]]
&lt;/div&gt;&lt;div style=&quot;color:transparent;font-size:0;&quot;&gt;&lt;references group=&quot;lower-alpha&quot;/&gt;{{reflist}}{{notelist}}&lt;/div&gt;

&lt;noinclude&gt;[[Category:WikiJournal of Medicine formatting templates]] [[Category:WikiJournal headers, banners and infoboxes]]&lt;/noinclude&gt;</text>
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      <contributor>
        <username>Nintendofan885</username>
        <id>2887676</id>
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      <comment>Codex colours for dark mode</comment>
      <origin>2806894</origin>
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             color:{{{4|{{{colour|#ff6600}}}}}}&quot;&gt;
{{{2|{{{title}}}}}}&lt;/span&gt;&lt;br&gt;
{{{3|{{{text}}}}}}&lt;/div&gt;&lt;noinclude&gt;{{documentation}}&lt;/noinclude&gt;</text>
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      <contributor>
        <username>Nintendofan885</username>
        <id>2887676</id>
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      <comment>HTTPS</comment>
      <origin>2806886</origin>
      <model>wikitext</model>
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      <text bytes="491" sha1="8elh7597z3keoa1stp8gehi2wkr7bto" xml:space="preserve">'''Share article'''&lt;small&gt;&lt;br&gt;
{{Share email          |icon=true|subject=Journal article: {{SUBPAGENAME}}|contents={{{email|https://en.wikiversity.org/wiki/{{FULLPAGENAMEE}}}}}}}
| {{Share facebook     |icon=true}}
| {{Share twitter      |icon=true|via={{WikiJXyz}}|related=WikiJMed,WikiJSci,WikiJHum,Wikipedia,OpenAccess}}
| {{Share linkedin     |icon=true}}
| {{Share mendeley     |icon=true}}
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&lt;/small&gt;&lt;noinclude&gt;[[Category:Share templates]]&lt;/noinclude&gt;</text>
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      <contributor>
        <username>Nintendofan885</username>
        <id>2887676</id>
      </contributor>
      <comment>HTTPS, invert icon on dark mode</comment>
      <origin>2806887</origin>
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      <format>text/x-wiki</format>
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|%0A}}]]&amp;nbsp;|}}[mailto:{{{target|}}}?subject={{#invoke:String|replace|{{{subject|{{SUBPAGENAME}}}}}| |%20}}&amp;body={{#invoke:String|replace|{{#invoke:String|replace|{{#invoke:String|replace|{{{contents|https://en.wikipedia.org/wiki/{{FULLPAGENAMEE}}}}}| |%20}}|&amp;nbsp;|%20}}|
|%0A}} {{{show|Email}}}]&lt;/span&gt;&lt;noinclude&gt;{{documentation}}&lt;/noinclude&gt;</text>
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      <comment>/* Temporary Accounts */ new section</comment>
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      <model>wikitext</model>
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      <text bytes="862" sha1="iv6ck8x2ae41rlke7gvfvu04oulhlxs" xml:space="preserve">==Illustrations/screenshots==
How am I going to upload useful images for my tutorials without being autoconfirmed? [[User:Query boss|Query boss]] ([[User talk:Query boss|discuss]] • [[Special:Contributions/Query boss|contribs]]) 21:06, 23 September 2016 (UTC)

:You aren't. You'll need to wait for auto-confirmation. That's a number of edits / number of days. Please continue participating until you are able to upload. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 01:19, 24 September 2016 (UTC)

== Temporary Accounts ==

Temporary accounts aren't described at all. Please add an informative section for temporary accounts. [[Special:Contributions/&amp;#126;2026-25992-66|&amp;#126;2026-25992-66]] ([[User talk:&amp;#126;2026-25992-66|talk]]) 02:19, 29 April 2026 (UTC)</text>
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      <contributor>
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        <id>3069181</id>
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      <comment>/* Temporary Accounts */</comment>
      <origin>2806940</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="904" sha1="s775fdu1my8v6yqhza8s3nmq155l48z" xml:space="preserve">==Illustrations/screenshots==
How am I going to upload useful images for my tutorials without being autoconfirmed? [[User:Query boss|Query boss]] ([[User talk:Query boss|discuss]] • [[Special:Contributions/Query boss|contribs]]) 21:06, 23 September 2016 (UTC)

:You aren't. You'll need to wait for auto-confirmation. That's a number of edits / number of days. Please continue participating until you are able to upload. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 01:19, 24 September 2016 (UTC)

== Temporary Accounts ==

Temporary accounts aren't described at all. Please add an informative section for temporary accounts. What should their status be described as? [[Special:Contributions/&amp;#126;2026-25992-66|&amp;#126;2026-25992-66]] ([[User talk:&amp;#126;2026-25992-66|talk]]) 02:19, 29 April 2026 (UTC)</text>
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==Information==
This is an alternate version of [[Template:Cc-by-3.0]], without the box, and describing the subject as the more general &quot;work&quot; rather than &quot;file&quot;.

==See also==
*{{tlx|PD work}}
*{{tlx|CC-BY work}}
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[[Category:Creative Commons templates|{{PAGENAME}}]]
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This is an open access article distributed under the&amp;nbsp;[http://creativecommons.org/licenses/by-sa/3.0/ Creative Commons Attribution ShareAlike License], which permits unrestricted use, distribution, and reproduction, provided the original author and source are credited.&lt;noinclude&gt;

==Information==
This is an alternate version of [[Template:Cc-by-sa-3.0]], without the box, and describing the subject as the more general &quot;work&quot; rather than &quot;file&quot;.

==See also==
*{{tlx|PD work}}
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[[Category:Creative Commons templates|{{PAGENAME}}]]
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[[hu:Wikiegyetem]]
[[File:Open Content Development Process.png|thumb|Staircase of workload (red) of OER building on existing development]]
'''Open educational resources''' ('''OER''') are freely accessible, openly licensed text, media, and other digital assets that are useful for teaching, learning, and assessing as well as for research purposes&lt;ref&gt;[[Wikipedia:Open educational resources]]&lt;/ref&gt;. Communities use and alter OERs according to their requirements and constraints. A scientific analysis of these alterations, forkings, or extensions of OERs provides insights in the drivers and needs of communities.

== Contents ==
The following resources can be found in these {{Subpages/Simple}}:
* [[/Introduction/]] - Introduction to OERs
* [[/Lesson/]] - Lesson on OERs
* [[/Sources/]] - Sources of OERs (table of resources w/ counts of data available from each)
* [[/Student Authoring/]] - Resources supporting student-authored content
* [[/Adoptions/]] - Schools and courses using OERs
* [[Classifying educational resources]]
* [[/Open Courses/]] - Open course lists
* [[/Open Textbooks/]] - Open textbook lists
* [[/Training/]] - For teachers
* [[/Localization/|Geographic Information Systems and tailored educational resources for a specific region]]
* [[/Tailored WikiBooks/]]
* [[Green Open Access]]
* [[Wikiversity:Edit-a-thon]] to create OERs

== Learning Activities ==
* Explore the recommendation of '''[[w:en:UNESCO|UNESCO]]''' on Open Educational Resources and derive an application to your institutional setting:
** as a '''teacher''' for designing classroom activities,
** as a '''learner/student''' to perform the learning tasks in the classroom (e.g. supporting material as OER for performing a learning task)
* '''(Digital Public Good)''' Explore and compare the concept of 
** digital public good, 
** [[w:common good|common goods]] and
** OERs
: and identify similarities and differences between these concepts. How can digital public goods can trigger as building blocks for an [[Open Innovation Ecosystem]]. What are the benefits, challenges and constraints?

== See Also ==
* [[Intellectual Honesty]]
* [[Open academia]]
* [[Open Badges]] as digital certificates and record of achievements for OER courses and learning activities
* [[Open Community Approach]] and OER as one main constituent
* [[PanDocElectron]] for using Wikiversity Resource as Slide Presentations with the option to annotate slides in a browser with a stylus
* [[Reproducible Science]]
* [[Wikipedia: Open educational resources|Open educational resources]] (Wikipedia)
* [[Open Innovation Ecosystem]]

== External Learning Resources for OER ==
* [https://courses.lumenlearning.com/pathways Lumen Learning: Adopting Open Educational Resources in the Classroom]
* [http://pressbooks.oer.hawaii.edu/oertraining2018 Hawaii.edu: OER Training]

== References ==
{{reflist}}

[[Category:Capacity Building]]
[[Category:Learning]]
[[Category:Resources]]
&lt;noinclude&gt;
[[de:Open Educational Resources]]
&lt;/noinclude&gt;
&lt;noinclude&gt;
[[fr:Aide:Ressources_éducatives_libres]]
&lt;/noinclude&gt;
&lt;noinclude&gt;
[[zh:開放教育資源]]
&lt;/noinclude&gt;</text>
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      <contributor>
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        <id>2887676</id>
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      <comment>HTTPS, invert icon on dark mode</comment>
      <origin>2806889</origin>
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      <text bytes="485" sha1="f9j2nsog630zlmlrw4ikyc2kzbkcar1" xml:space="preserve">{{#ifeq:{{{icon|true}}}|true|[[File:Researchgate logo.svg|13px|alt=|class=skin-invert|link=https://www.addthis.com/bookmark.php?v=300&amp;s=researchgate&amp;url=https://en.wikiversity.org/wiki/WikiJournal_of_Medicine/{{PAGENAMEE}}&amp;title={{SUBPAGENAMEE}}]]&amp;nbsp;}}&lt;span class = &quot;plainlinks&quot;&gt;[https://www.addthis.com/bookmark.php?v=300&amp;s=researchgate&amp;url=https://en.wikiversity.org/wiki/{{PAGENAMEE}}&amp;title={{SUBPAGENAMEE}} ResearchGate]&lt;/span&gt;&lt;noinclude&gt;[[Category:Share templates]]&lt;/noinclude&gt;</text>
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      <comment>dark mode?</comment>
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    <title>Portal:Complex Systems Digital Campus/CS-DC 2020 elections, manifestos and results</title>
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      <comment>updating 2025 elections to 2026 elections</comment>
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      <text bytes="57596" sha1="5g66ekgw7tpd85ru7shh0i7w3d3i868" xml:space="preserve">&lt;nowiki&gt;**&lt;/nowiki&gt;Please ''login'' in Wikiversity and then use the ''&lt;nowiki/&gt;'edit''' button: your edition mode will be 'WYSIWYG'. Each Candidature with its manifesto can take inspiration from those for previous years.  

''Decision of the 2025 General Assembly'' ''(as in 2024)'':

''(i) There will be an election for President starting in April.''

''(ii) To save time and effort - Instead of an election for 1/3 of the Executive Committee'' ''(EC) for 2025 we propose to keep the existing EC and invite members of the Council '''not on the EC''' to offer themselves as members of an enlarged Executive Committee. If less than six people propose themselves, they will be co-opted onto the Executive Committee. If more than six propose themselves we will hold an election for 1/3 EC.''

The deadline for your candidature is Saturday 16th May 2026 at 24:00 CET. 

== Candidature for the New Vice-President for [2026, 2027] ==
 
'''Manifesto – Candidature for New Vice-President of UNESCO UniTwin CS-DC'''


== New Candidatures to the Executive Committee for [2026] ==

=== Manifesto – Candidature for the Executive Committee of UNESCO UniTwin CS-DC ===


== Candidature for the New President for [2025, 2026] ==
 
'''Manifesto – Candidature for New President of UNESCO UniTwin CS-DC'''

'''Prof. Paul Bourgine''' : 

If elected, my main commitment is to create the conditions for a self-organized development of our UNESCO UniTwin CS-DC as autonomous communities of communities for our flagship TIMES and its Knowledge &amp; Knowhow Accelerator one-for-all &amp; all-for-one (KKA). We know now how to realize —for the two above commitments— the 3&lt;sup&gt;rd&lt;/sup&gt; UNESCO commitment, i.e., the ‘computational ecosystem’. It will use the mature part of Web 3.0, especially the InterPlanetary File System (IPFS). Thanks to our previous efforts especially of the two last years, the remaining work amount is ten times less than we were anticipating at the beginning of the 2&lt;sup&gt;nd&lt;/sup&gt; renewal of our UniTwin by UNESCO 2020-2026.  

If elected, my duty will be not only to fulfill entirely the commitments of our Cooperation Program with UNESCO but also starting an exponential increasing development wave for our UniTwin network (through their continent and country Councils) and of our e-Campus (through CS-DC’25 and e-Labs’26 Conferences especially for our Flagships for sustainable development). The Knowledge &amp; Knowhow Accelerator will directly benefit from 1) such conference series, 2) our past and new flagships for sustainable development and 3) a new decentralized strategy for collecting donations in our decentralized network of X-Legal Entities.

== New Candidatures to the Executive Committee for [2025] ==

=== Manifesto – Candidature for the Executive Committee of UNESCO UniTwin CS-DC ===
'''Prof. Silvius STANCIU, PhD in Economics, PhD in Engineering, Habil.'''

Full Professor, “Dunărea de Jos” University of Galați (UDJG), Romania

Editor-in-Chief, ''Journal of Agriculture and Rural Development Studies (JARDS)''

Former Vice-Rector, Former Director of DFCTT and CTT UGAL
----'''Dear Councillors,'''

It is a great honor for me to submit my candidacy for the Executive Committee of the UNESCO UniTwin Complex Systems Digital Campus (CS-DC). With more than '''30 years of experience in academia''', I am currently Full Professor and doctoral advisor at “Dunărea de Jos” University of Galați (UDJG), Romania — a public research university with a strong regional impact and a long-standing tradition in interdisciplinary education and innovation.

I hold two doctoral degrees — one in Economics and one in Engineering — and I am a habilitated professor. I have published '''163 ISI-indexed scientific articles''' and have a '''Clarivate H-index of 14'''. My research focuses on '''food security, circular economy, technological innovation, rural development''', and '''complex systems in agro-food value chains'''.

I am the founder and coordinator of Romania’s first doctoral program in ''Engineering and Management in Agriculture and Rural Development (IMADR)'', with '''9 PhD graduates''' and '''9 doctoral students''' currently under my supervision. I also serve as '''Editor-in-Chief''' of the ''Journal of Agriculture and Rural Development Studies (JARDS)'', dedicated to interdisciplinary research in sustainable food and rural systems.

Over the past decade, I have been involved as director or expert in '''more than 45 national and international research projects''', including Horizon-compatible initiatives and cross-border cooperation programs. I coordinated a '''Romania–Republic of Moldova cross-border project''' (2020–2021) and currently lead '''two new ROMD-funded projects''' entering implementation.

My former institutional leadership roles include:

* '''Vice-Rector for Research and Innovation'''
* '''Director of the Department for Institutional Development (DFCTT)''' and of the '''Technology Transfer Center (CTT UGAL)'''
* '''Member of national and international quality and research bodies''', including CNATDCU, ARACIS, and CMPTJ

----'''If elected, I am committed to:'''

* Expanding the CS-DC network in '''Eastern Europe and the Black Sea region''', enhancing scientific and territorial diversity;
* Supporting '''POEM''' and '''FOOD flagship programs''' through digital education, doctoral/postdoctoral collaboration, and innovation ecosystems;
* Promoting '''open science''', international e-seminars, and interdisciplinary MOOCs;
* Coordinating thematic initiatives in '''agro-complexity, food systems resilience''', and '''sustainable rural innovation'''.

As a representative of a '''UniTwin member institution''', I see this candidacy as a unique opportunity to strengthen UDJG’s role within the CS-DC ecosystem. I fully embrace the CS-DC mission to foster global collaboration, education, and research in complexity science.

I am ready to bring '''vision, experience, and energy''' to the Executive Committee and help shape the future of our UniTwin community.
----'''Sincerely,'''

'''Prof. Silvius STANCIU, PhD in Economics, PhD in Engineering, Habil.'''

Representative of “Dunărea de Jos” University of Galați (UDJG)

'''Professional Identifiers:'''

* Web of Science Author ID: R-8246-2017
* ORCID: 0000-0001-7697-0968
* Scopus ID: 36633317700
* Google Scholar: Silvius Stanciu
* ResearchGate: Silvius Stanciu

== New Candidatures to the Executive Committee for [2024] ==

'''Enver Oruro Puma, Ph.D., Principal Investigator of Neurocomputing, Social Simulation, and Complex Systems Laboratory at the Instituto Científico of Universidad Andina del Cusco, Peru''' 

Dear Councillor of the UNESCO UniTwin CS-DC. I am very honored to place my candidature for the UNESCO UniTwin CS-DC Executive Committee. I am Enver Miguel Oruro Puma, Ph.D., principal investigator of Neurocomputing, Social Simulation, and Complex Systems Laboratory at the Instituto Científico of Universidad Andina del Cusco, Peru (https://sites.google.com/view/orurolab/). Since 2009, I have promoted and organized conferences and academic events on Complex Systems in Latin America. Recently, I have promoted the area of computational neuroscience on infant attachment (https://sites.google.com/view/envermiguel/seminar-in-maternal-infant-relationship-studies).

It would be a great honor for me if given the opportunity to contribute to the Executive Committee of UNESCO UniTwin CS-DC in the integration of Complex System research groups in the Latin American Region. For this, I propose the creation of two periodical activities: 1) A Special Lectures Series on Complex Systems UNESCO UniTwin CS-DC oriented to experts on Complex Systems, and 2) A Invited Advanced Lectures UNESCO UniTwin CS-DC oriented to experts who do not identify explicitly with complex systems

'''Pierre Collet, full professor of Strasbourg University, on secondment to Universidad Andrés Bello, Instituto de Tecnología para la Innovación en Salud y Bienestar, Viña del Mar, Valparaiso, Chile'''.

Since 2012, I have contributed to the elaboration of the CS-DC Unesco UniTwin together with Paul Bourigne, Jeffrey Johnson and many others, and I have been co-coordinator of the CS-DC UniTwin with Cyrille Bertelle since its creation in 2014. Starting part of this great adventure has changed my academic and personal life: thanks to the UniTwin, I have changed my research from stochastic optimisation, artificial evolution and AI in general to complex systems and epistemology.

Participating in this UniTwin allowed me to make new contacts and start incredible projects that I could not have imagined before. It has even changed my life, as I am now living in Chile, having been recruited by ITISB, an institute founded by Carla Taramasco, the CS-DC representative for South America.

Together with Paul and others, we would like to revive UniTwin by preparing another world conference inspired by the great success of [https://cs-dc-15.org CS-DC'15] and also develop flagship projects such as POEM (Personalised Open Education for the Masses) and the [https://en.wikiversity.org/wiki/Portal:Complex_Systems_Digital_Campus/E-Laboratory_on_complex_computational_ecosystems ECCE e-lab], which this year has welcomed a new very active [[Figures of Play/Les figures du Jeu e-team|Figures of Play]] that has started the [https://ludocorpus.org/ Ludocorpus] in France.

As said before, this incredible UniTwin adventure always pays off for those who invest in it and in its great challenge: to develop the new science of complex systems through research and education. Through its projects, it contributes to making the world a better place to live in, despite the constant attacks on science coming from the most unlikely places.

Science is the solution, not the problem, to many of the world's plagues. We must put our energy into developing it and defend it against all its detractors.

That is why I am once again standing for election to the Executive Committee of this great CS-DC UniTwin. Modern science is Complex Systems science. It is important that its beacon continues to illuminate the world, and we must invest our time and energy in it.

== Candidature Deputy President for [2024, 2025] ==
'''Jeffrey JOHNSON, Professor of Complexity Science and Design, The Open University, UK'''

I offer myself as a candidate both to be President and to the Executive Committee of The UNESCO UniTwin Complex Systems Digital Campus (CS-DC) so that I can help to drive it forward to achieve it goals.

I am particularly committed to our educational efforts. I have made four MOOCs on the FutureLearn Platform for CS-DC ( &lt;nowiki&gt;https://www.futurelearn.com/partners/unesco-unitwin-complex-systems-digital-campus&lt;/nowiki&gt; ): Global Systems Science (2015-16); Systems Thinking and Complexity (2017-18); First Steps in Data Science with Google Analytics (2018-19) and COVID-19 - Pandemics, Modelling and Policy (2020). CS-DC has a great opportunity to become the global university providing interdisciplinary education for a better world.

I am also committed to our research mission with UNESCO towards the achieving the U.N. Sustainable Development Goals. My own research on representing the dynamics of complex multilevel systems is relevant to many of the research initiatives of CS-DC.

I have extensive experience working within the complex systems community. I have run various coordination actions supporting research programmes funded by the European Commission, I am a founder member and past president of the Complex Systems Society, and I am Deputy-President of the CS-DC. I believe this experience will enable me to make a significant contribution the CS-DC over the next three years.

== New Elected President for [2023, 2024] ==
Paul BOURGINE, present President of the UNESCO UniTwin CS-DC, Complex Systems Institute of Paris 
I offer myself as a candidate to be President of The UNESCO UniTwin Complex Systems Digital Campus (CS-DC).
My previous commitment two years ago is below. The bad news is that it was not achieved. The good new is that we know now how to create 'autonomous community of autonomous communities' as a social network with IPFS (the InterPlanetary File System) like the new development of Wikipedia. 
If elected, my first commitment is to finish this job as quickly as possible. My second commitment is simultaneously to visit each country of the UniTwin for creating its country.CS-DC and its roadmap with young eTeams shared by their Universities with a senior scientific committee. The eTeam projects will have the opportunity to be submitted to the EU calls or other ones. 

Enver Oruro PhD, Head of Neurocomputing, Social Simulation and Complex Systems Laboratory, Universidad Andina del Cusco, Peru.
'''I would like to nominate Professor Paul Bourgine.'''

== New Elected Members to the Executive Committee for [2022,2023, 2024,2025] ==
'''2Dr Mohamed Abdellahi (Ould BABAH) Ebbe, Mauritania,''' 

* Senior Advisor for the CILSS Executif Secretary for international Partnership and formal General Director of the     Institut du Sahel/CILSS www.insah.org;

·       Commissionaire General of CILSS for Horticulture Universal  Expo of DOHA 2023-2024   &lt;nowiki&gt;https://www.dohaexpo2023.gov.qa/en/&lt;/nowiki&gt; with central thems:

'''CENTRAL THEME: GREEN DESERT, BETTER ENVIRONMENT'''

* Executive Director of the     Orthopterist Society (400 researchers among the globe) &lt;nowiki&gt;https://orthsoc.org/&lt;/nowiki&gt;
* We have organized our last     congress during 16-20 0ctober in Merida Mexique

&lt;nowiki&gt;https://ico2023mexico.com/&lt;/nowiki&gt;

By obtaining the honor of having your hoped-for confidence for continuing this post of member of the executive council of the CS-DC, I will work, in priority and in the short term on two main subjects:

## '''The transboundary plague of the     Desert Locust (Schistocerca gregaria (Forska l , 1775))''' This plague of the Desert     Locust of more than 3000 years that cites all our holy books (the Tourah,     the Bible and the Koran) and which continues to be present to this day and     to wreak devastating devastation. In case of invasion, it can affect the     agriculture and pastures of about 25 countries including those of the     poorest countries of the world, from Mauritania to India, while its best     and most effective strategy of struggle is preventive struggle by     targeting its first centers of gregarization which are very small in space     and much better known today. In 2005, the costs of its struggle in the     Sahel and North Africa amounted to half a billion dollars, with 8 million     farmers and pastoralists affected in the Sahel. It also massively invaded     Asia and Africa. 'East Africa in 2020. On this subject, I have spent 30     years studying and fighting and developing a national strategy against     this scourge which has made it possible to establish a whole prevention     model and an institutional, technical, operational mechanism. and     scientific effective in my country that can be adapted and copied and in     all other affected countries: Biogeography of the desert locust     Schistocerca gregaria, Forskal, 1775: Identification, characterization and     originality of a gregarious focus in central Mauritania (HR.HORS COLLEC.)     (French Edition) - Babah Ebbe, Mohamed Abdallahi | 9782705670573 |     Amazon.com.au | Books &lt;nowiki&gt;http://www.worldbank.org/en/news/feature/2010/01/07/improved-ways-to-prevent-the-desert-locust-in-mauritania-and-the-sahel&lt;/nowiki&gt;,     http: //     whatsnext.blogs.cnn.com/2012/02/02/in-mauritania-sunny-with-a-chance-of-locusts/ I was invited last year by Royal Society 20-21 may 2024 to moderate one session on locust research management (La plasticité des criquets et des abeilles dans un monde en mutation | Société royale) and in a “International Conference on New Technology and Concepts for Sustainable Management of Locusts and Grasshoppers” held from 2 to 7 June 2024 in Jinan, Shandong, China.We are also preparing our Orthopterist congress in Argentina during the next mars 2026  &lt;nowiki&gt;https://ico2026.com.ar&lt;/nowiki&gt;

'''All this is in addition of more than 110 publications or joint publications on the locust, its environment and management''' 

# '''Senior Adviser to the CILSS Executive Secretary for International Partnerships'''] [Assistance to Mauritania (or 3 months) in the preparation of the organisation of the Nouakchott+10 High-Level Forum on pastoralism held in Nouakchott from 6 to 8 November 2024, various advising for the international partnership and the mobilization of resources including preparation of the organization of a round table planned in OPEC Vienna Austria for the mobilization of Arabic and Islamic funds for the financing of the CILSS 2050 strategic plan
# '''The Sahel Institute (INSAH)     www.insah.org of the Permanent Interstate Committee for Drought Control     (CILSS)''' that I     lead and which has been doing extraordinary work for almost half a century     in the field of research and development of animal and plant production     techniques and also in the field of support for demographic, population     and development policies, in favor of the populations of our 13 Sahelian,     coastal and island member countries. This work covered the majority of     good practice technologies in the field of plant and animal production,     natural resource management, land rstauration, cultivation techniques,     post-harvest, machining, dehulling operations technology. / ginning,     Conservation and storage, good resilience practices

Research on the demographic dividend, gender and the empowerment of women and the Population / Development interrelations ... etc The results of all this work are contained in a database. data, online &lt;nowiki&gt;http://publications.insah.org/&lt;/nowiki&gt;, containing more than 1,500 books, scientific and technical articles that will have to be modernized and connected to the CS Meta data.

As General Commissionaire  of CILSS for Horticulture Universal  Expo of DOHA 2023-2024   &lt;nowiki&gt;https://www.dohaexpo2023.gov.qa/en/&lt;/nowiki&gt; with central thems:

'''CENTRAL THEME: GREEN DESERT, BETTER ENVIRONMENT''' I am working  in introducing as detailed below:

'''CILSS ''contribution to the improvement of sustainable horticultural agricultural production in a context of drought''''' 

'''I. PRESENTATION OF THE EXPO'''  

Expo 2023 in Doha is part of the fight against desertification. The Expo will be held from 2 October 2023 to 28 March 2024 under the theme &quot;'''''Green Desert, Better Environment'''''&quot;. The aim is to encourage, inspire and inform people about innovative solutions to reduce desertification. The exhibition will provide an international platform for participants, stakeholders, decision-makers, nongovernmental organizations and experts to address the global challenge of &quot;desertification&quot;, while making a valuable contribution to achieving a sustainable future. During the 6 months of the Expo, nearly 3 million visitors from over 80 countries are expected

The objectives of this Expo are in line with those of the CILSS, which seeks to improve the living conditions of the people of the Sahel in a sustainable manner. This is why the participation of CILSS in this Expo is important for the region and its vulnerable populations. 

'''OBJECTIVES OF EXPO 2023 DOHA, QATAR'''

Expo 2023 Doha, Qatar is defined by the following objectives:

-   Encourage horticultural innovation by focusing on Qatar's climate, water and soil.

-   Promote Expo 2023 in Doha, Qatar, as a catalyst for international investment and business opportunities.

-   To propose innovative actions that would allow humanity to fight against desertification more quickly and decisively before it is too late.

-   To build up useful environmental outputs for future generations.  

'''II. ORGANISATION OF THE CILSS PARTICIPATION'''

'''II.1. GOALS OF CILSS EXHIBITION:''' 

1. Sharing experiences and best practices,

2. Building International Partnership,

3. Promoting technology and innovation

Finally, I will continue to work actively with my colleagues on the Executive Board on all aspects of other cross-border scourges but also all aspects of improving agro-sylvo-pastoral production


'''Dr. Xabier E. Barandiaran, Lecturer at the University of the Basque Country (UPV/EHU), Department of Philosophy, Donostia - San Sebastian, Spain'''

I would like to present [https://xabier.barandiaran.net myself] as a candidate for the Executive Committee. I have been the representative and coordinator between CS-DC and the [https://ehu.eus University of the Basque Country] since 2013. I develop my academic research at the [https://ias-research.net IAS-Research Centre for Life, Mind, and Society], with a focus on the understanding of autonomous and complex adaptive systems (from biology to cognition, from brains to societies). I am the author of over 50 indexed publications on topics related to complex systems, philosophy of mind, complex epistemology, simulation models of the origins of life, minimal agency, evolutionary robotics, complex social network analysis, etc. I recently received the “Award for Distinguished Early-Career Investigator” by the International Society for Artificial Life. Overal I have been awarded with 7 different grants and have actively participated on 15 different research projects. I have also supervised 2 PhD thesis (4 more still in development) and I hold an extensive record of scientific and innovative management experience in different academic and public institutions as founder of research networks ReteCog.Net and FLOK Society – Buen Conocer and head of RDI at Barcelona City Council (2016-2018). I have also organized several national workshops, summer schools and conferences, and 2 international summer schools, 4 international workshops and one international conference. I am currently the Principal Investigator of a founded research project (with more than 30 research-collaborators) on a complex systems' approach to the concept of autonomy beyond its classical conception as an individual bounded property.

As part of my university's goal of fostering international collaboration and opening up e-learning and research initiatives I would like to get more deeply involved on CS-DC with the following goals:

* To desing the infrastructure, learning-experience, research-experience and content for distributed, open access and high-quality digital campus facilities.
* To involve local agents (student, teachers, researchers and institutions) on the initiative of the network.
* To foster collaboration, co-production and resource sharing between teaching and research facilities between priviledged richer countries and lower-income ones. In particular, but not exclusively, and for obvious reasons related to sharing the same language, to foster ''collaboration between European and Latin-american universities'', research initiatives and students through CS-DC.
* To develop at least one ''prototype'' of a MSc level online course (and research network module) around complex cognitive systems that can serve as a model for the other fields of the network.
* To develop a clear conceptual and communicative framework for CS-DC to be able to attract more participants, resources and broader attention and success as pioneering international initiative.


'''Dr. habil. László Barna Iantovics, Professor at “George Emil Palade” Univ. of Medicine, Pharmacy, Science and Technology of Tg. Mures, Romania'''

With the present manifesto, I would like to be a candidate for the CS-DC Executive Committee. I have been the representative of “George Emil Palade” University of Medicine, Pharmacy, Science and Technology of Targu Mures from Romania in CS-DC by many years.

Some of my research and academic activities were related to the complex systems, including: publications; organized conferences (e.g. Symposium on Understanding Intelligent and Complex Systems - UICS 2009; 1st Int. Conf. on Complexity and Intelligence of the Artificial and Natural Complex Systems Medical Applications of the Complex Systems. Biomedical Computing -CANS 2008; 1st Int. Conf. on Bio-Inspired Computational Methods Used for Difficult Problems Solving. Development of Intelligent and Complex Systems - BICS 2008); membership in conference committees (e.g. Int. Conf. Emergent Properties in Natural and Artificial Complex Systems - EPNACS 2007; Workshop on Complex Systems and Self-organization Modeling -CoSSoM 2009); Journal Special Issues (e.g. Special Issue on Complexity in Sciences and Artificial Intelligence; Special Issue on Understanding Complex Systems); membership in Journal’s Editorial Boards (e.g. Complex Adaptive Systems Modeling -CASM, SpringerOpen), and contribution to research performed in projects and projects coordination (Social network of machines- SOON; Hybrid Medical Complex Systems -ComplexMediSys). I am the director of the Research Center on Artificial Intelligence, Data Science and Smart Engineering (Artemis).

I would like to involve myself much deeper in the life and activities of the CS-DC community.

My principal objectives are:

* To involve junior and senior researchers from my university in activities regarding research and education related to complex systems.
* To involve universities and research institutes to actively contribute to the CS-DC development.
* To involve myself in the joint coordination with other CS-DC members of a doctoral and postdoctoral students’ group that will be involved in the CS-DC community works.
* To strengthen the research direction with the theme: applications of intelligent complex systems and machine intelligence measuring. One of the subtopics of interest will be the application of complex systems, artificial intelligence and data science in medicine, pharmacology, and healthcare.


'''Flavia Mori SARTI, Ph.D., Professor and Researcher, University of Sao Paulo, Brazil'''

I would like to present my candidature for the CS-DC Executive Committee in the period 2022-2024 to contribute to the dissemination of Complex Systems Science.

I have been representative of the University of Sao Paulo (USP) at the CS-DC since 2012, and I have been working with complex systems since the creation of the Interdisciplinary Research Group of Complex Systems Modelling at the School of Arts, Sciences and Humanities (EACH-USP) in 2006. Our research group succesfully implemented the first interdisciplinary graduate program (Master) in Complex Systems Modelling in Brazil, in 2010. I have been participating in the coordinating commission of the program since 2010, and I was coordinator of the graduate program from 2010 to 2014.

I have supervised seven students in the Master program, which resulted in thesis, book chapters, and papers published on the subject of complex systems, including models on tax evasion, health systems regulation, food policy and nutrition programs, and complex networks on scientific collaboration and international food trade. I also contributed to the organization of the e-Session &quot;Economics as a Complex Evolutionist System&quot; on the CS-DC'15 World e-conference in 2015, and have been invited to present seminars on complex systems applied to health economics, health technology assessment, and public policy of nutrition and health.

My goals in the CS-DC Executive Committee include:

* To disseminate the role of CS-DC in education and research on Complex Systems, especially in Brazil and other developing countries;
* To support and to engage other research groups working with Complex Systems for participation in the CS-DC;
* To contribute further with management and organization of CS-DC activities during the period of 2022-2024;
* To continue supporting capacity building in Complex Systems through the Complex Systems Modelling Program at USP;
* To participate in innovation, research and development activities based on the application of Complex Systems in public policy and entrepreneurship.


'''Pr Panos Argyarakis, Professor in the University of Thessaloniki, Greece.'''

I have been with the Complex Systems Society since its inception in 2004 by participating in the NEST projects Dysonet and Giacs which created CSS. My experience in the Executive Committee will be to contribute towards the spreading of the Complexity idea to various levels of education throughout the different countries. I am currently the PI in an Erasmus+ network that introduces new models of teaching and investigating how is education been affected for future generations. I can contribute in decision making for such important activities, and also serve as liaison with the European Commission, and the Complex Systems Society, due to my past experience. I have extended organizational experience by organizing several internationally meetings in this field that were attended by large audiences. My research interests are related to Complex systems and Networks. Scale-free, random, and small world networks. Dynamic properties on networks, Diffusion, spreading phenomena on networks, disease spreading. Phase transitions, percolation model, reaction-diffusion processes, trapping processes. Random walks.


'''Ali Moussaoui, Professor, University of Tlemcen, Department of Mathematics, Algeria,'''

I wish to present my candidature to become member of the executive committee of the CS-DC, I wish to develop collaborations with the partner universities in the field of complex systems. I wish to participate in the creation of international mixed laboratories and international masters on complex systems.

In the past, I was responsible for a master's degree entitled: modeling of complex systems in our department, I am currently responsible for a research team entitled: Modeling of complex systems in our laboratory, I was responsible for a Franco-Algerian project on the modeling of complex systems. My research skills are focused on the modeling of complex natural and biological systems.

'''Carlos Gershenson, Research Professor, Universidad Nacional Autónoma de México.'''

I was involved with CS-DC in its initial years in UNESCO's UniTwin, also representing UNAM. I have been editor-in-chief of Complexity Digest since 2007. I co-organized the Conference on Complex Systems in 2017. I am currently vice-president secretary of the Complex Systems Society (CSS).

I am a strong proponent of open online learning. I managed to start a collaboration between UNAM and Coursera, which has led to more than a hundred MOOCs and millions of enrolled students.

I would be interested in strengthening the relationship between CS-DC and CSS, as well as other organizations.

==Elected members to the Executive Committee for 2021 ==
'''Carlos J. BARRIOS H., PhD., Professor, Bucaramanga, Colombia '''

I write to express my interest to candidacy to be part of the CS-DC Executive Committee. I'm very motivated to develop actions to strengthen digital ecosystem supporting research and education proposals of our CS-DC Council. Among these years participating in the CS-DC group, I can see different ways to leverage the impact and the development of our actions with computational strategies, and now, I want to be part of the leadership council joined mutual visions.

My experience leading the Advanced Computing System for Latin America and Caribbean (SCALAC : http://scalac.redclara.net ) and as member of other leadership boards in international projects (mainly between Europe and Latin America) supports my candidature. (linkedin.com/in/carlosjaimebh) Also, my role as professor, director and researcher contributes to build the common vision of the CS-DC Council and the leadership of the CS- DC Executive Committee.


'''Mina TEICHER, Professor of Bar-Ilan University, Israël'''

I submit my candidacy to the Executive Committee of the CS Digital Campus. If elected I will work towards our following needs, using my past experience in Professional international societies, universities managements and the data industry : 

* We need in the near future to build an optimal and effective agreement with the Complex System Society.
* We need to build a business  plan for fund raising.
* We need to build a modular budget for 2021.
* We need to build a strategy for geographically extension. 
* We need to build a strategy for thematic  extension. 
* We need to build partnerships with the big multi national high tech Companies in network and in content. 


'''Yasmin MERALI, Professor of University of Hull, UK'''

This manifesto is connected with the ideals that I had as a founding member of our UniNet which was conceived as part of the FP7 ASSYST project.
CS-DC has come a long way since its initial conception. The way I see it, there are three categories that have grown to emerge as our core activities- 
* Capacity building through education and training in Complex Systems Science
* The application of Complex Systems Science to address global challenges
* The advancement of Complex Systems Science through research and development.

I believe this is a good time to link back to the inception of our UNITWIN which was in part inspired by considerations of issues at a human scale, and the desire to address the inequalities that divided the so-called developed and developing countries. This resonates strongly with the ambition of the  Sustainable Development Goals (SDGs) we are currently grappling with.
In the growth phase of the UNITWIN and CSDC we have been focused on extending the size of the network, and scaling up our educational offerings across the digital campus. In the next phase I believe we need to:
# understand and leverage the diversity and distinctive capabilities and resources (e.g. indigenous knowledge) of the countries in our network to develop a healthy ecosystem, and
# tailor the support that we provide to align with the diverse nature of their relational and social capital and their economic, political and environmental challenges and priorities with regard to the SDGs.  

I am concerned that if we do not explicitly design a social/ideational exchange mechanism that attends to these two imperatives, we will not have full, active participation of all member institutions, and the countries of the South that do not currently have champions in Europe will be marginalized.

If elected I would champion a strategy of organizing ourselves following the Complex Adaptive Systems paradigm, as a hyper network with dynamically connected local clusters. In practical terms I would like to begin by establishing the local (country-based) clusters and establishing a discourse that would allow us to map the diverse profiles, challenges and aspirations for the different countries. This would then form the basis for the development of a mechanism for shaping the meaningful collaborative development of our three core activities to deliver advances that are globally co-ordinated and locally responsive.

Personal Profile: I am Professor of Systems Thinking at the University of Hull and have served as Director of the Centre Systems Studies there. Prior to that I was Co-Director of the Doctoral Training Centre for Complex Systems Science at the University of Warwick. My research is transdisciplinary, focusing on the use of Complex Systems Science to enhance the resilience of socio-economic systems. I am an Expert advisor to the EU and I have significant experience of lecturing internationally as Visiting Professor in Asia, Europe and the USA.

'''Céline ROZENBLAT: Professor, University of Lausanne - Institut de géographie et de durabilité (IGD), Switzerland'''

I'm pleased to applied to become member of the CS-DC council.
As founding member of CS-DC, my university, the university of Lausanne, is very engaged in Complex sciences.
I would not only represent my university, but also social science as geographer and vice-president of the International Geographical Union and member of the International Science Council commission on Urban Health and Well being.
I would act in the council in specific programs to develop the reality of the Digital Campus of the Complex systems.

All these actions combine very ambitious interdisciplinary approaches, and in this perspective, we developed with CS-DC for 3 years the TIMES Flagship  Territorial Intelligence For Multilevel Equity And Sustainability. It comprises four main programs:

 '''SIRE''': Socially Intelligence Roadmap Ecosystem
 '''POLE:''' Personalized Open Lifelong Education
 '''WOSI:''' Worldwide Open Smart Innovation
 '''WOSP:''' Worldwide Open Stochastic Prediction 

In this perspective a MOOC « Healthy Urban system » is now in development, basing the interdisciplinary approach on the CS-DC Road-Map grid. It seems very useful and relevant in this implementation stage.
I would help to develop other programs in this perspective\[Ellipsis]


'''André TINDANO, Director General of CARFS (African Center for Research and Training in Synecoculture)'''

What motivates me to aspire to the position of member of the executive committee of CS DC is my long term participation in the promotion of sustainable development and my commitment to the sharing of knowledge and expertise.

My research interests Sustainable agriculture, ecology, nutrition, life science. I have a strong experience  in:
* Administration and management of development projects and programs;
* Accompaniment of associations and groups;
* Technical capacity building (animation of training sessions and reflection workshops).
* Action research;
* Sociological, socio-economic and economic studies.
* Development of development projects and programs;
* Training of trainers
* Results Based Management Training (RBM)
* Monitoring and evaluation of development programs and projects;
* Management of programs and development projects;
* Institutional development and organizational strengthening;
* Development and implementation of training / awareness / animation program;
* Very good knowledge of participatory methods
'''Guiou KOBAYASHI, Associate Professor at Federal University of ABC in São Paulo State, Brazil. '''

I worked with fault-tolerant computer systems for nuclear power plants and Metro signaling systems and recently my interests have evolved to resilience properties of Complex Systems. Traditionally, redundancy was the main feature for fault-tolerant and fail-safe systems, but the adaptability and the evolution of Complex Systems are the key elements for the resilience of these systems. How to characterize, design and implement these key elements in our future resilient systems? The Complex System - Digital Campus (CS-DC) is a way to create a world-wide community of researchers, philosophers and students to promote and discuss this kind of questions involving Complex Systems. For me, participating in its foundation was a great honor and I am very glad for the opportunity that I have had to contribute since 2012 in the consolidation of CS-DC.

Through this manifest I am applying to be one of the members of the new CS-DC's Executive Committee. I would like to help just a little more to strengthen and structure this fantastic community through which I had the opportunity to meet important people with very interesting works that expanded my knowledge of Complex Systems. Although my University and my personal contribution for CS-DC are very limited and small, I hope to continue to work with this great team.


'''Pierre COLLET, Professor of Computer Science, University of Strasbourg, France. Co-coordinator of the CS-DC UNESCO UniTwin'''

The CS-DC initiated by Paul Bourgine, Jeffrey Johnson, Cyrille Bertelle and many others has been an extraordinary adventure a) to instantiate as a UNESCO UniTwin and b) to develop and run since it was enacted in July 2014. Many a night have been spent on designing its inner workings, so that it can deliver an effective affordance for the scientists who wish to develop the science and teaching of Complex Systems. Indeed, many projects seeded in the CS-DC have come to fruition, showing the enormous potential of this fertile environment not only for research, but also for teaching: the BBB rooms set up by the CS-DC have not only made it possible for the CS-DC to organize conferences, but have also shown their potential as remote teaching rooms in many Universities around the world.

It has been an honour for me to be part of the development of the CS-DC since its beginning, but so much remains to be done!

In this manifesto, I hereby express my strong desire to continue developing the CS-DC in these trying times, when the effects of the pandemics stretch thin the social links that our research and teaching communities need most. My objectives for this new mandate are not only to deliver a new world conference (originally planned in 2020 but unfortunately delayed due to the high toll imposed on us all, teachers and researchers alike, by COVID-19) but also continue on developing not only efficient complex computational ecosystems (cheap powerful PARSEC machines have been installed in several universities) but more specifically remote teaching environments based on complex systems, to mitigate the terrible impact of the pandemic on face to face education, within the POEM CS-DC flagship, on which Paul Bourgine and myself have been working for many years now..


'''Mariana C. BROENS, Professor - UNESP - BRAZIL.'''

As members of the Executive Committee, our main challenge will be to raise, analyse and to discuss possible positive/negative ethical and political implications of the further development of the Complex Systems Science, and their application on studies of everyday social problems. In particular, We believe that the widespread use of complex system models and Big Data analytics can bring important questions about people's privacy, personal and corporate responsibility, widespread surveillance by public or private institutions, among many others, that should be deeply discussed in our community. Our contribution will be to raise and deepen these discussions from an interdisciplinary perspective.


'''Cyrille Bertelle, Professor in Computer Sciences,  University Le Havre, France, co-coordinator of the CS-DC UNESCO UniTwin'''

I am a candidate for the CS-DC Executive Committee to represent the University of Le Havre
Normandy, which is co-coordinator with the University of Strasbourg of the convention of
recognition of the CS-DC as UniTwin by UNESCO. The University of Le Havre has made
available to the community, resources and skills to provide digital collaborative tools for the
organization of the Councils and the CS-DC'15 virtual conference.

The objective I wish to take is to facilitate the involvement of member universities not only
by their representatives on the council but by allowing researchers from these member
institutions to join concrete and accessible actions.

Co-responsible in the past of a master's degree on complex systems and then of the creation
of the institute on complex systems in Normandie (France) in a multidisciplinary framework,
I have participated in the setting up of the project labeled by the French national program of
investments for the future and entitled &quot;Smart Port City&quot;. The aim is to think about the
future of territories in a sustainable development approach supported by new technologies
and concerned about the environment and the well-being of their citizens. My research skills
are focused on implementing the complexity of complex dynamic systems and networks,
crossing behavioral scales from the interaction of human behaviors to the technical
networks of the territory. 

The book &quot;complex systems, smart territories and mobility&quot; from
the Springer's Understanding Complex Systems series, which will be published in January
2021 (https://www.springer.com/gp/book/9783030593018) illustrates the research
coordination actions that I lead in these fields.


'''Slimane Ben Miled, Senior Researcher at Pasteur Institute of Tunis, Professor at ENIT'''

Our Tunisian consortium want to constitute a collaborative Research Training Programs to increase data science capacity related to health research in Africa by building trainings and enhancing institutional capacity at African academic institutions.
The academy/project is based on 4 pillars to build a training ecosystem for Data and Engineering Science in health.
# A platform of federated master’s programs with à la carte optional courses covering informatics/computer science, biomedical informatics, data science, statistics, and public health). Each program will keep its independence, with a mention to the academy label, and this platform will allow to enrich the training with optional modules, seminars, and courses in the partner institutions. New curricula will be created in relation to ethical issues.
# Network of Doctoral programs and Executive programs
# Platform of federated Business incubator and a career center offers training, support and funding for projects related to the project’s topic.

This challenge is in perfect agreement with the Sustainable Development Goal 3 and the CS-DC flagship PHYSIOMES (Personalized Health phYSIcally, sOcially and Mentally for Each in their networkS).


'''Masa Funabashi: researcher of open complex systems at Sony Computer Science Laboratories, Inc.'''

I would like to contribute to the executive committee of CS-DC on the following two pillars:
* Promote the FOOD (From smart agrOecOnomy to smart fooD) flagship project that aims to resolve the health-diet-environment trilemma through the promotion of sustainable food systems, in collaboration with the e-lab &quot;human augmentation of ecosystems&quot; members institution: Sony CSL, Synecoculture Association, CARFS, and those who wish to participate in CS-DC collaboration.
* Construct a basic e-learning content on Synecoculture and ecological literacy as a part of CS-DC MOOCs and perform initial trials, principally in ECOWAS countries, through the Sony CSL-CARFS collaboration.

Through the development of on-going activities in FOOD project and making synergy with other flagship projects, I would like to contribute CS-DC as a member of the executive committee and realize further extension toward the achievement of global sustainability goals such as SDGs.


'''Dr Mohamed Abdellahi (Ould BABAH) Ebbe, Mauritania, '''
* General Director of the Institut du Sahel/CILSS www.insah.org; 
* Executive Director of the Orthopterist Society (400 researchers among the globe) https://orthsoc.org/

By obtaining the honor of having your hoped-for confidence for this post of member of the executive council of the CS-DC, I will work, in priority and in the short term on two main subjects:
# '''The transboundary plague of the Desert Locust (Schistocerca gregaria (Forska l , 1775))''' This plague of the Desert Locust of more than 3000 years that cites all our holy books (the Tourah, the Bible and the Koran) and which continues to be present to this day and to wreak devastating devastation. In case of invasion, it can affect the agriculture and pastures of about 25 countries including those of the poorest countries of the world, from Mauritania to India, while its best and most effective strategy of struggle is preventive struggle by targeting its first centers of gregarization which are very small in space and much better known today. In 2005, the costs of its struggle in the Sahel and North Africa amounted to half a billion dollars, with 8 million farmers and pastoralists affected in the Sahel. It also massively invaded Asia and Africa. 'East Africa in 2020. On this subject, I have spent 30 years studying and fighting and developing a national strategy against this scourge which has made it possible to establish a whole prevention model and an institutional, technical, operational mechanism. and scientific effective in my country that can be adapted and copied and in all other affected countries: Biogeography of the desert locust Schistocerca gregaria, Forskal, 1775: Identification, characterization and originality of a gregarious focus in central Mauritania (HR.HORS COLLEC.) (French Edition) - Babah Ebbe, Mohamed Abdallahi | 9782705670573 | Amazon.com.au | Books http://www.worldbank.org/en/news/feature/2010/01/07/improved-ways-to-prevent-the-desert-locust-in-mauritania-and-the-sahel, http: // whatsnext.blogs.cnn.com/2012/02/02/in-mauritania-sunny-with-a-chance-of-locusts/
# '''The Sahel Institute (INSAH) www.insah.org of the Permanent Interstate Committee for Drought Control (CILSS)''' that I lead and which has been doing extraordinary work for almost half a century in the field of research and development of animal and plant production techniques and also in the field of support for demographic, population and development policies, in favor of the populations of our 13 Sahelian, coastal and island member countries. This work covered the majority of good practice technologies in the field of plant and animal production, natural resource management, land restauration, cultivation techniques, post-harvest, machining, dehulling operations technology. / ginning, Conservation and storage, good resilience practices

Research on the demographic dividend, gender and the empowerment of women and the Population / Development interrelations ... etc
The results of all this work are contained in a database. data, online http://publications.insah.org/, containing more than 1,500 books, scientific and technical articles that will have to be modernized and connected to the CS Meta data.

Finally, I will work actively with my colleagues on the Executive Board on all aspects of other cross-border scourges but also all aspects of improving agro-sylvo-pastoral production tools as well as the fight against poverty and food insecurity and nutrition in line with the goals (SDGs)


'''Dr. Habil. László Barna Iantovics, Associate Professor at “George Emil Palade” University of Medicine, Pharmacy, Science and Technology of Targu Mures, Romania.''' 

With the present manifesto, I would like to candidate as a member of the CS-DC Executive Committee. I am the representative of “George Emil Palade” University of Medicine, Pharmacy, Science and Technology of Targu Mures from Romania in CS-DC.

Some of my research and academic activities are related to the complex systems, including: publications, organization of conferences (e.g. Symposium on Understanding Intelligent and Complex Systems - UICS 2009; 1st Int. Conf. on Complexity and Intelligence of the Artificial and Natural Complex Systems Medical Applications of the Complex Systems. Biomedical Computing -CANS 2008; 1st Int. Conf. on Bio-Inspired Computational Methods Used for Difficult Problems Solving. Development of Intelligent and Complex Systems - BICS 2008), contribution to conference committees (e.g. Int. Conf. Emergent Proprieties in Natural and Artificial Complex Systems - EPNACS 2007; Workshop on Complex Systems and Self-organization Modeling -CoSSoM 2009), preparing journal special issues (e.g. Special Issue on Complexity in Sciences and Artificial Intelligence; Special Issue on Understanding Complex Systems), participating in journal’s editorial board (e.g. Complex Adaptive Systems Modeling -CASM, SpringerOpen), and contribution to research in projects and projects coordination (Social network of machines- SOON; Hybrid Medical Complex Systems -ComplexMediSys). I was the director of the center Advanced Computational Technologies – AdvCompTech in the frame of my university. At present, I am the director of the Center for Advanced Research in Information Technology from my university.

I would like much deeper involve myself in the life and activities of the CS-DC community.

My objectives:

* To involve junior and senior researchers from my university in activities regarding research and education. To motivate universities and research institutes from my country to contribute to CS-DC. I consider also universities and research institutes with that I have collaboration in the past.
* To PROPOSE the formation of a so-called doctoral and postdoctoral students group. In the case of doctoral and postdoctoral students probably in time more students would like to be involved in activities. In this framework, I suggest the organization yearly 3 times (from 4 to 4 months) workshops in that all the interested students could discuss, present their research and research in progress. With this occasion in the frame of workshops if there is interest could be established separate sessions with presentations also by B.Sc. and M.Sc. students.
* To PROPOSE the strengthening of the following research direction with the general topic: intelligent complex systems and machine intelligence measuring. One of the subtopic by interest will be complex systems approaches in medicine and healthcare. To be accomplishable this subject I propose in a first step the formation of a group of interested persons, after then the establishment of the functionality of the group, for example: discussions when are subjects that should be discussed etc.

==Elected members to the Executive Committee &amp; as (Deputy-)Presidents==


'''Jeffrey JOHNSON, Professor of Complexity Science and Design, The Open University, UK'''

I offer myself as a candidate both to be President and to the Executive Committee of  The UNESCO UniTwin Complex Systems Digital Campus (CS-DC) so that I can help to drive it forward to achieve it goals. 

I am particularly committed to our educational efforts. I have made four MOOCs on the FutureLearn Platform for CS-DC  ( https://www.futurelearn.com/partners/unesco-unitwin-complex-systems-digital-campus ): Global Systems Science (2015-16); Systems Thinking and Complexity (2017-18); First Steps in Data Science with Google Analytics (2018-19) and  COVID-19 - Pandemics, Modelling and Policy (2020). CS-DC has a great opportunity to become the global university providing interdisciplinary education for a better world.

I am also committed to our research mission with UNESCO towards the achieving the U.N. Sustainable Development Goals. My own research on representing the dynamics of complex multilevel systems is relevant to many of the research initiatives of CS-DC.

I have extensive experience working within the complex systems community. I have run various coordination actions supporting research programmes funded by the European Commission, I am a founder member and past president of the Complex Systems Society, and I am Deputy-President of the CS-DC. I believe this experience will enable me to make a significant contribution the CS-DC over the next three years.


'''Paul BOURGINE, present President of the UNESCO UniTwin CS-DC, Complex Systems Institute of Paris'''

I offer myself as a candidate both to be President and to the Executive Committee of  The UNESCO UniTwin Complex Systems Digital Campus (CS-DC).

If elected, my main commitment is to create the conditions for a self-organized development of our UniTwin UNESCO CS-DC as autonomous communities of communities. This self-similar development will be the case both for the two main branches the UniTwin branch of our institutional members and the global eCampus branches of our individual scientific members:
* for the UniTwin branch, the communities of communities are a territorial cascade with Smart Continents, smart countries, smart cities for their sustainable development according our flagship TIMES (Territorial Intelligence for Multilevel Equity and Sustainability). The roadmap is always the same, i.e. the cascade of the 17 Sustainable Development Goals and their 169 Targets: but their relative importance and coherence within this cascade vary from one territory to the others. The institutional members of the UniTwin branch have signed their agreement with the Cooperation Programme signed with UNESCO. In 2021, the CS-DC will ask for a cascade of agreements inside each institutional member, in order to have a &quot;one for all&quot; amplification within the other branch, the e-campus branch. 
* for the eCampus branch, the cascade of communities is along the refinement cascades when studying the theoretical and experimental challenges of complex systems. With Smart Continents'21, scientists are proposing their individual challenges that enact basic communities and communities of communities within the e-departments. In the &quot;all for one&quot; return, the roadmap of each university is the cascade of roadmaps within the eCampus where the University has at least one member. Furthermore each community can organise a monthly e-seminar or e-session in workshop as well as in CS-DC'21 for recorded advanced introductions. Such advanced introductions can be the basis for curriculum largely shared by the set of Universities having members in the  community cascade of the curriculum.

This &quot;accelerator of knowledge and knowhow one for all and all for one&quot; will first benefit to the student curriculum through the flagship POEM (Personalized Open Education for the Masses). This accelerator can be extended through the flagship POLE (Personalized Open Lifelong Education) for a lifelong education. This extended accelerator will be open to all, independently of previously achieved academic levels, respectful of the diversity of social and cultural environments and in a higher and higher inclusive way including refugees, migrants and primary people. genders, religions or ways of life.</text>
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      <text bytes="57691" sha1="rup526cd57q71bjgpkaygu80p8kd2zq" xml:space="preserve">&lt;nowiki&gt;**&lt;/nowiki&gt;Please ''login'' in Wikiversity and then use the ''&lt;nowiki/&gt;'edit''' button: your edition mode will be 'WYSIWYG'. Each Candidature with its manifesto can take inspiration from those for previous years.  

''Decision of the 2025 General Assembly 27th April:''

''The 2026 elections are starting the 27th April, the date of the General Assembly:''

''(i) There will be an election for Vice-President starting the 27 April 2026.''

''(ii) To save time and effort - Instead of an election for 1/3 of the Executive Committee'' ''(EC) for 2025 we propose to keep the existing EC and invite members of the Council '''not on the EC''' to offer themselves as members of an enlarged Executive Committee. If less than six people propose themselves, they will be co-opted onto the Executive Committee. If more than six propose themselves we will hold an election for 1/3 EC.''

The deadline for your candidature is Saturday 16th May 2026 at 24:00 CET. 

== Candidature for the New Vice-President for [2026, 2027] ==
 
'''Manifesto – Candidature for New Vice-President of UNESCO UniTwin CS-DC'''


== New Candidatures to the Executive Committee for [2026] ==

=== Manifesto – Candidature for the Executive Committee of UNESCO UniTwin CS-DC ===


== Candidature for the New President for [2025, 2026] ==
 
'''Manifesto – Candidature for New President of UNESCO UniTwin CS-DC'''

'''Prof. Paul Bourgine''' : 

If elected, my main commitment is to create the conditions for a self-organized development of our UNESCO UniTwin CS-DC as autonomous communities of communities for our flagship TIMES and its Knowledge &amp; Knowhow Accelerator one-for-all &amp; all-for-one (KKA). We know now how to realize —for the two above commitments— the 3&lt;sup&gt;rd&lt;/sup&gt; UNESCO commitment, i.e., the ‘computational ecosystem’. It will use the mature part of Web 3.0, especially the InterPlanetary File System (IPFS). Thanks to our previous efforts especially of the two last years, the remaining work amount is ten times less than we were anticipating at the beginning of the 2&lt;sup&gt;nd&lt;/sup&gt; renewal of our UniTwin by UNESCO 2020-2026.  

If elected, my duty will be not only to fulfill entirely the commitments of our Cooperation Program with UNESCO but also starting an exponential increasing development wave for our UniTwin network (through their continent and country Councils) and of our e-Campus (through CS-DC’25 and e-Labs’26 Conferences especially for our Flagships for sustainable development). The Knowledge &amp; Knowhow Accelerator will directly benefit from 1) such conference series, 2) our past and new flagships for sustainable development and 3) a new decentralized strategy for collecting donations in our decentralized network of X-Legal Entities.

== New Candidatures to the Executive Committee for [2025] ==

=== Manifesto – Candidature for the Executive Committee of UNESCO UniTwin CS-DC ===
'''Prof. Silvius STANCIU, PhD in Economics, PhD in Engineering, Habil.'''

Full Professor, “Dunărea de Jos” University of Galați (UDJG), Romania

Editor-in-Chief, ''Journal of Agriculture and Rural Development Studies (JARDS)''

Former Vice-Rector, Former Director of DFCTT and CTT UGAL
----'''Dear Councillors,'''

It is a great honor for me to submit my candidacy for the Executive Committee of the UNESCO UniTwin Complex Systems Digital Campus (CS-DC). With more than '''30 years of experience in academia''', I am currently Full Professor and doctoral advisor at “Dunărea de Jos” University of Galați (UDJG), Romania — a public research university with a strong regional impact and a long-standing tradition in interdisciplinary education and innovation.

I hold two doctoral degrees — one in Economics and one in Engineering — and I am a habilitated professor. I have published '''163 ISI-indexed scientific articles''' and have a '''Clarivate H-index of 14'''. My research focuses on '''food security, circular economy, technological innovation, rural development''', and '''complex systems in agro-food value chains'''.

I am the founder and coordinator of Romania’s first doctoral program in ''Engineering and Management in Agriculture and Rural Development (IMADR)'', with '''9 PhD graduates''' and '''9 doctoral students''' currently under my supervision. I also serve as '''Editor-in-Chief''' of the ''Journal of Agriculture and Rural Development Studies (JARDS)'', dedicated to interdisciplinary research in sustainable food and rural systems.

Over the past decade, I have been involved as director or expert in '''more than 45 national and international research projects''', including Horizon-compatible initiatives and cross-border cooperation programs. I coordinated a '''Romania–Republic of Moldova cross-border project''' (2020–2021) and currently lead '''two new ROMD-funded projects''' entering implementation.

My former institutional leadership roles include:

* '''Vice-Rector for Research and Innovation'''
* '''Director of the Department for Institutional Development (DFCTT)''' and of the '''Technology Transfer Center (CTT UGAL)'''
* '''Member of national and international quality and research bodies''', including CNATDCU, ARACIS, and CMPTJ

----'''If elected, I am committed to:'''

* Expanding the CS-DC network in '''Eastern Europe and the Black Sea region''', enhancing scientific and territorial diversity;
* Supporting '''POEM''' and '''FOOD flagship programs''' through digital education, doctoral/postdoctoral collaboration, and innovation ecosystems;
* Promoting '''open science''', international e-seminars, and interdisciplinary MOOCs;
* Coordinating thematic initiatives in '''agro-complexity, food systems resilience''', and '''sustainable rural innovation'''.

As a representative of a '''UniTwin member institution''', I see this candidacy as a unique opportunity to strengthen UDJG’s role within the CS-DC ecosystem. I fully embrace the CS-DC mission to foster global collaboration, education, and research in complexity science.

I am ready to bring '''vision, experience, and energy''' to the Executive Committee and help shape the future of our UniTwin community.
----'''Sincerely,'''

'''Prof. Silvius STANCIU, PhD in Economics, PhD in Engineering, Habil.'''

Representative of “Dunărea de Jos” University of Galați (UDJG)

'''Professional Identifiers:'''

* Web of Science Author ID: R-8246-2017
* ORCID: 0000-0001-7697-0968
* Scopus ID: 36633317700
* Google Scholar: Silvius Stanciu
* ResearchGate: Silvius Stanciu

== New Candidatures to the Executive Committee for [2024] ==

'''Enver Oruro Puma, Ph.D., Principal Investigator of Neurocomputing, Social Simulation, and Complex Systems Laboratory at the Instituto Científico of Universidad Andina del Cusco, Peru''' 

Dear Councillor of the UNESCO UniTwin CS-DC. I am very honored to place my candidature for the UNESCO UniTwin CS-DC Executive Committee. I am Enver Miguel Oruro Puma, Ph.D., principal investigator of Neurocomputing, Social Simulation, and Complex Systems Laboratory at the Instituto Científico of Universidad Andina del Cusco, Peru (https://sites.google.com/view/orurolab/). Since 2009, I have promoted and organized conferences and academic events on Complex Systems in Latin America. Recently, I have promoted the area of computational neuroscience on infant attachment (https://sites.google.com/view/envermiguel/seminar-in-maternal-infant-relationship-studies).

It would be a great honor for me if given the opportunity to contribute to the Executive Committee of UNESCO UniTwin CS-DC in the integration of Complex System research groups in the Latin American Region. For this, I propose the creation of two periodical activities: 1) A Special Lectures Series on Complex Systems UNESCO UniTwin CS-DC oriented to experts on Complex Systems, and 2) A Invited Advanced Lectures UNESCO UniTwin CS-DC oriented to experts who do not identify explicitly with complex systems

'''Pierre Collet, full professor of Strasbourg University, on secondment to Universidad Andrés Bello, Instituto de Tecnología para la Innovación en Salud y Bienestar, Viña del Mar, Valparaiso, Chile'''.

Since 2012, I have contributed to the elaboration of the CS-DC Unesco UniTwin together with Paul Bourigne, Jeffrey Johnson and many others, and I have been co-coordinator of the CS-DC UniTwin with Cyrille Bertelle since its creation in 2014. Starting part of this great adventure has changed my academic and personal life: thanks to the UniTwin, I have changed my research from stochastic optimisation, artificial evolution and AI in general to complex systems and epistemology.

Participating in this UniTwin allowed me to make new contacts and start incredible projects that I could not have imagined before. It has even changed my life, as I am now living in Chile, having been recruited by ITISB, an institute founded by Carla Taramasco, the CS-DC representative for South America.

Together with Paul and others, we would like to revive UniTwin by preparing another world conference inspired by the great success of [https://cs-dc-15.org CS-DC'15] and also develop flagship projects such as POEM (Personalised Open Education for the Masses) and the [https://en.wikiversity.org/wiki/Portal:Complex_Systems_Digital_Campus/E-Laboratory_on_complex_computational_ecosystems ECCE e-lab], which this year has welcomed a new very active [[Figures of Play/Les figures du Jeu e-team|Figures of Play]] that has started the [https://ludocorpus.org/ Ludocorpus] in France.

As said before, this incredible UniTwin adventure always pays off for those who invest in it and in its great challenge: to develop the new science of complex systems through research and education. Through its projects, it contributes to making the world a better place to live in, despite the constant attacks on science coming from the most unlikely places.

Science is the solution, not the problem, to many of the world's plagues. We must put our energy into developing it and defend it against all its detractors.

That is why I am once again standing for election to the Executive Committee of this great CS-DC UniTwin. Modern science is Complex Systems science. It is important that its beacon continues to illuminate the world, and we must invest our time and energy in it.

== Candidature Deputy President for [2024, 2025] ==
'''Jeffrey JOHNSON, Professor of Complexity Science and Design, The Open University, UK'''

I offer myself as a candidate both to be President and to the Executive Committee of The UNESCO UniTwin Complex Systems Digital Campus (CS-DC) so that I can help to drive it forward to achieve it goals.

I am particularly committed to our educational efforts. I have made four MOOCs on the FutureLearn Platform for CS-DC ( &lt;nowiki&gt;https://www.futurelearn.com/partners/unesco-unitwin-complex-systems-digital-campus&lt;/nowiki&gt; ): Global Systems Science (2015-16); Systems Thinking and Complexity (2017-18); First Steps in Data Science with Google Analytics (2018-19) and COVID-19 - Pandemics, Modelling and Policy (2020). CS-DC has a great opportunity to become the global university providing interdisciplinary education for a better world.

I am also committed to our research mission with UNESCO towards the achieving the U.N. Sustainable Development Goals. My own research on representing the dynamics of complex multilevel systems is relevant to many of the research initiatives of CS-DC.

I have extensive experience working within the complex systems community. I have run various coordination actions supporting research programmes funded by the European Commission, I am a founder member and past president of the Complex Systems Society, and I am Deputy-President of the CS-DC. I believe this experience will enable me to make a significant contribution the CS-DC over the next three years.

== New Elected President for [2023, 2024] ==
Paul BOURGINE, present President of the UNESCO UniTwin CS-DC, Complex Systems Institute of Paris 
I offer myself as a candidate to be President of The UNESCO UniTwin Complex Systems Digital Campus (CS-DC).
My previous commitment two years ago is below. The bad news is that it was not achieved. The good new is that we know now how to create 'autonomous community of autonomous communities' as a social network with IPFS (the InterPlanetary File System) like the new development of Wikipedia. 
If elected, my first commitment is to finish this job as quickly as possible. My second commitment is simultaneously to visit each country of the UniTwin for creating its country.CS-DC and its roadmap with young eTeams shared by their Universities with a senior scientific committee. The eTeam projects will have the opportunity to be submitted to the EU calls or other ones. 

Enver Oruro PhD, Head of Neurocomputing, Social Simulation and Complex Systems Laboratory, Universidad Andina del Cusco, Peru.
'''I would like to nominate Professor Paul Bourgine.'''

== New Elected Members to the Executive Committee for [2022,2023, 2024,2025] ==
'''2Dr Mohamed Abdellahi (Ould BABAH) Ebbe, Mauritania,''' 

* Senior Advisor for the CILSS Executif Secretary for international Partnership and formal General Director of the     Institut du Sahel/CILSS www.insah.org;

·       Commissionaire General of CILSS for Horticulture Universal  Expo of DOHA 2023-2024   &lt;nowiki&gt;https://www.dohaexpo2023.gov.qa/en/&lt;/nowiki&gt; with central thems:

'''CENTRAL THEME: GREEN DESERT, BETTER ENVIRONMENT'''

* Executive Director of the     Orthopterist Society (400 researchers among the globe) &lt;nowiki&gt;https://orthsoc.org/&lt;/nowiki&gt;
* We have organized our last     congress during 16-20 0ctober in Merida Mexique

&lt;nowiki&gt;https://ico2023mexico.com/&lt;/nowiki&gt;

By obtaining the honor of having your hoped-for confidence for continuing this post of member of the executive council of the CS-DC, I will work, in priority and in the short term on two main subjects:

## '''The transboundary plague of the     Desert Locust (Schistocerca gregaria (Forska l , 1775))''' This plague of the Desert     Locust of more than 3000 years that cites all our holy books (the Tourah,     the Bible and the Koran) and which continues to be present to this day and     to wreak devastating devastation. In case of invasion, it can affect the     agriculture and pastures of about 25 countries including those of the     poorest countries of the world, from Mauritania to India, while its best     and most effective strategy of struggle is preventive struggle by     targeting its first centers of gregarization which are very small in space     and much better known today. In 2005, the costs of its struggle in the     Sahel and North Africa amounted to half a billion dollars, with 8 million     farmers and pastoralists affected in the Sahel. It also massively invaded     Asia and Africa. 'East Africa in 2020. On this subject, I have spent 30     years studying and fighting and developing a national strategy against     this scourge which has made it possible to establish a whole prevention     model and an institutional, technical, operational mechanism. and     scientific effective in my country that can be adapted and copied and in     all other affected countries: Biogeography of the desert locust     Schistocerca gregaria, Forskal, 1775: Identification, characterization and     originality of a gregarious focus in central Mauritania (HR.HORS COLLEC.)     (French Edition) - Babah Ebbe, Mohamed Abdallahi | 9782705670573 |     Amazon.com.au | Books &lt;nowiki&gt;http://www.worldbank.org/en/news/feature/2010/01/07/improved-ways-to-prevent-the-desert-locust-in-mauritania-and-the-sahel&lt;/nowiki&gt;,     http: //     whatsnext.blogs.cnn.com/2012/02/02/in-mauritania-sunny-with-a-chance-of-locusts/ I was invited last year by Royal Society 20-21 may 2024 to moderate one session on locust research management (La plasticité des criquets et des abeilles dans un monde en mutation | Société royale) and in a “International Conference on New Technology and Concepts for Sustainable Management of Locusts and Grasshoppers” held from 2 to 7 June 2024 in Jinan, Shandong, China.We are also preparing our Orthopterist congress in Argentina during the next mars 2026  &lt;nowiki&gt;https://ico2026.com.ar&lt;/nowiki&gt;

'''All this is in addition of more than 110 publications or joint publications on the locust, its environment and management''' 

# '''Senior Adviser to the CILSS Executive Secretary for International Partnerships'''] [Assistance to Mauritania (or 3 months) in the preparation of the organisation of the Nouakchott+10 High-Level Forum on pastoralism held in Nouakchott from 6 to 8 November 2024, various advising for the international partnership and the mobilization of resources including preparation of the organization of a round table planned in OPEC Vienna Austria for the mobilization of Arabic and Islamic funds for the financing of the CILSS 2050 strategic plan
# '''The Sahel Institute (INSAH)     www.insah.org of the Permanent Interstate Committee for Drought Control     (CILSS)''' that I     lead and which has been doing extraordinary work for almost half a century     in the field of research and development of animal and plant production     techniques and also in the field of support for demographic, population     and development policies, in favor of the populations of our 13 Sahelian,     coastal and island member countries. This work covered the majority of     good practice technologies in the field of plant and animal production,     natural resource management, land rstauration, cultivation techniques,     post-harvest, machining, dehulling operations technology. / ginning,     Conservation and storage, good resilience practices

Research on the demographic dividend, gender and the empowerment of women and the Population / Development interrelations ... etc The results of all this work are contained in a database. data, online &lt;nowiki&gt;http://publications.insah.org/&lt;/nowiki&gt;, containing more than 1,500 books, scientific and technical articles that will have to be modernized and connected to the CS Meta data.

As General Commissionaire  of CILSS for Horticulture Universal  Expo of DOHA 2023-2024   &lt;nowiki&gt;https://www.dohaexpo2023.gov.qa/en/&lt;/nowiki&gt; with central thems:

'''CENTRAL THEME: GREEN DESERT, BETTER ENVIRONMENT''' I am working  in introducing as detailed below:

'''CILSS ''contribution to the improvement of sustainable horticultural agricultural production in a context of drought''''' 

'''I. PRESENTATION OF THE EXPO'''  

Expo 2023 in Doha is part of the fight against desertification. The Expo will be held from 2 October 2023 to 28 March 2024 under the theme &quot;'''''Green Desert, Better Environment'''''&quot;. The aim is to encourage, inspire and inform people about innovative solutions to reduce desertification. The exhibition will provide an international platform for participants, stakeholders, decision-makers, nongovernmental organizations and experts to address the global challenge of &quot;desertification&quot;, while making a valuable contribution to achieving a sustainable future. During the 6 months of the Expo, nearly 3 million visitors from over 80 countries are expected

The objectives of this Expo are in line with those of the CILSS, which seeks to improve the living conditions of the people of the Sahel in a sustainable manner. This is why the participation of CILSS in this Expo is important for the region and its vulnerable populations. 

'''OBJECTIVES OF EXPO 2023 DOHA, QATAR'''

Expo 2023 Doha, Qatar is defined by the following objectives:

-   Encourage horticultural innovation by focusing on Qatar's climate, water and soil.

-   Promote Expo 2023 in Doha, Qatar, as a catalyst for international investment and business opportunities.

-   To propose innovative actions that would allow humanity to fight against desertification more quickly and decisively before it is too late.

-   To build up useful environmental outputs for future generations.  

'''II. ORGANISATION OF THE CILSS PARTICIPATION'''

'''II.1. GOALS OF CILSS EXHIBITION:''' 

1. Sharing experiences and best practices,

2. Building International Partnership,

3. Promoting technology and innovation

Finally, I will continue to work actively with my colleagues on the Executive Board on all aspects of other cross-border scourges but also all aspects of improving agro-sylvo-pastoral production


'''Dr. Xabier E. Barandiaran, Lecturer at the University of the Basque Country (UPV/EHU), Department of Philosophy, Donostia - San Sebastian, Spain'''

I would like to present [https://xabier.barandiaran.net myself] as a candidate for the Executive Committee. I have been the representative and coordinator between CS-DC and the [https://ehu.eus University of the Basque Country] since 2013. I develop my academic research at the [https://ias-research.net IAS-Research Centre for Life, Mind, and Society], with a focus on the understanding of autonomous and complex adaptive systems (from biology to cognition, from brains to societies). I am the author of over 50 indexed publications on topics related to complex systems, philosophy of mind, complex epistemology, simulation models of the origins of life, minimal agency, evolutionary robotics, complex social network analysis, etc. I recently received the “Award for Distinguished Early-Career Investigator” by the International Society for Artificial Life. Overal I have been awarded with 7 different grants and have actively participated on 15 different research projects. I have also supervised 2 PhD thesis (4 more still in development) and I hold an extensive record of scientific and innovative management experience in different academic and public institutions as founder of research networks ReteCog.Net and FLOK Society – Buen Conocer and head of RDI at Barcelona City Council (2016-2018). I have also organized several national workshops, summer schools and conferences, and 2 international summer schools, 4 international workshops and one international conference. I am currently the Principal Investigator of a founded research project (with more than 30 research-collaborators) on a complex systems' approach to the concept of autonomy beyond its classical conception as an individual bounded property.

As part of my university's goal of fostering international collaboration and opening up e-learning and research initiatives I would like to get more deeply involved on CS-DC with the following goals:

* To desing the infrastructure, learning-experience, research-experience and content for distributed, open access and high-quality digital campus facilities.
* To involve local agents (student, teachers, researchers and institutions) on the initiative of the network.
* To foster collaboration, co-production and resource sharing between teaching and research facilities between priviledged richer countries and lower-income ones. In particular, but not exclusively, and for obvious reasons related to sharing the same language, to foster ''collaboration between European and Latin-american universities'', research initiatives and students through CS-DC.
* To develop at least one ''prototype'' of a MSc level online course (and research network module) around complex cognitive systems that can serve as a model for the other fields of the network.
* To develop a clear conceptual and communicative framework for CS-DC to be able to attract more participants, resources and broader attention and success as pioneering international initiative.


'''Dr. habil. László Barna Iantovics, Professor at “George Emil Palade” Univ. of Medicine, Pharmacy, Science and Technology of Tg. Mures, Romania'''

With the present manifesto, I would like to be a candidate for the CS-DC Executive Committee. I have been the representative of “George Emil Palade” University of Medicine, Pharmacy, Science and Technology of Targu Mures from Romania in CS-DC by many years.

Some of my research and academic activities were related to the complex systems, including: publications; organized conferences (e.g. Symposium on Understanding Intelligent and Complex Systems - UICS 2009; 1st Int. Conf. on Complexity and Intelligence of the Artificial and Natural Complex Systems Medical Applications of the Complex Systems. Biomedical Computing -CANS 2008; 1st Int. Conf. on Bio-Inspired Computational Methods Used for Difficult Problems Solving. Development of Intelligent and Complex Systems - BICS 2008); membership in conference committees (e.g. Int. Conf. Emergent Properties in Natural and Artificial Complex Systems - EPNACS 2007; Workshop on Complex Systems and Self-organization Modeling -CoSSoM 2009); Journal Special Issues (e.g. Special Issue on Complexity in Sciences and Artificial Intelligence; Special Issue on Understanding Complex Systems); membership in Journal’s Editorial Boards (e.g. Complex Adaptive Systems Modeling -CASM, SpringerOpen), and contribution to research performed in projects and projects coordination (Social network of machines- SOON; Hybrid Medical Complex Systems -ComplexMediSys). I am the director of the Research Center on Artificial Intelligence, Data Science and Smart Engineering (Artemis).

I would like to involve myself much deeper in the life and activities of the CS-DC community.

My principal objectives are:

* To involve junior and senior researchers from my university in activities regarding research and education related to complex systems.
* To involve universities and research institutes to actively contribute to the CS-DC development.
* To involve myself in the joint coordination with other CS-DC members of a doctoral and postdoctoral students’ group that will be involved in the CS-DC community works.
* To strengthen the research direction with the theme: applications of intelligent complex systems and machine intelligence measuring. One of the subtopics of interest will be the application of complex systems, artificial intelligence and data science in medicine, pharmacology, and healthcare.


'''Flavia Mori SARTI, Ph.D., Professor and Researcher, University of Sao Paulo, Brazil'''

I would like to present my candidature for the CS-DC Executive Committee in the period 2022-2024 to contribute to the dissemination of Complex Systems Science.

I have been representative of the University of Sao Paulo (USP) at the CS-DC since 2012, and I have been working with complex systems since the creation of the Interdisciplinary Research Group of Complex Systems Modelling at the School of Arts, Sciences and Humanities (EACH-USP) in 2006. Our research group succesfully implemented the first interdisciplinary graduate program (Master) in Complex Systems Modelling in Brazil, in 2010. I have been participating in the coordinating commission of the program since 2010, and I was coordinator of the graduate program from 2010 to 2014.

I have supervised seven students in the Master program, which resulted in thesis, book chapters, and papers published on the subject of complex systems, including models on tax evasion, health systems regulation, food policy and nutrition programs, and complex networks on scientific collaboration and international food trade. I also contributed to the organization of the e-Session &quot;Economics as a Complex Evolutionist System&quot; on the CS-DC'15 World e-conference in 2015, and have been invited to present seminars on complex systems applied to health economics, health technology assessment, and public policy of nutrition and health.

My goals in the CS-DC Executive Committee include:

* To disseminate the role of CS-DC in education and research on Complex Systems, especially in Brazil and other developing countries;
* To support and to engage other research groups working with Complex Systems for participation in the CS-DC;
* To contribute further with management and organization of CS-DC activities during the period of 2022-2024;
* To continue supporting capacity building in Complex Systems through the Complex Systems Modelling Program at USP;
* To participate in innovation, research and development activities based on the application of Complex Systems in public policy and entrepreneurship.


'''Pr Panos Argyarakis, Professor in the University of Thessaloniki, Greece.'''

I have been with the Complex Systems Society since its inception in 2004 by participating in the NEST projects Dysonet and Giacs which created CSS. My experience in the Executive Committee will be to contribute towards the spreading of the Complexity idea to various levels of education throughout the different countries. I am currently the PI in an Erasmus+ network that introduces new models of teaching and investigating how is education been affected for future generations. I can contribute in decision making for such important activities, and also serve as liaison with the European Commission, and the Complex Systems Society, due to my past experience. I have extended organizational experience by organizing several internationally meetings in this field that were attended by large audiences. My research interests are related to Complex systems and Networks. Scale-free, random, and small world networks. Dynamic properties on networks, Diffusion, spreading phenomena on networks, disease spreading. Phase transitions, percolation model, reaction-diffusion processes, trapping processes. Random walks.


'''Ali Moussaoui, Professor, University of Tlemcen, Department of Mathematics, Algeria,'''

I wish to present my candidature to become member of the executive committee of the CS-DC, I wish to develop collaborations with the partner universities in the field of complex systems. I wish to participate in the creation of international mixed laboratories and international masters on complex systems.

In the past, I was responsible for a master's degree entitled: modeling of complex systems in our department, I am currently responsible for a research team entitled: Modeling of complex systems in our laboratory, I was responsible for a Franco-Algerian project on the modeling of complex systems. My research skills are focused on the modeling of complex natural and biological systems.

'''Carlos Gershenson, Research Professor, Universidad Nacional Autónoma de México.'''

I was involved with CS-DC in its initial years in UNESCO's UniTwin, also representing UNAM. I have been editor-in-chief of Complexity Digest since 2007. I co-organized the Conference on Complex Systems in 2017. I am currently vice-president secretary of the Complex Systems Society (CSS).

I am a strong proponent of open online learning. I managed to start a collaboration between UNAM and Coursera, which has led to more than a hundred MOOCs and millions of enrolled students.

I would be interested in strengthening the relationship between CS-DC and CSS, as well as other organizations.

==Elected members to the Executive Committee for 2021 ==
'''Carlos J. BARRIOS H., PhD., Professor, Bucaramanga, Colombia '''

I write to express my interest to candidacy to be part of the CS-DC Executive Committee. I'm very motivated to develop actions to strengthen digital ecosystem supporting research and education proposals of our CS-DC Council. Among these years participating in the CS-DC group, I can see different ways to leverage the impact and the development of our actions with computational strategies, and now, I want to be part of the leadership council joined mutual visions.

My experience leading the Advanced Computing System for Latin America and Caribbean (SCALAC : http://scalac.redclara.net ) and as member of other leadership boards in international projects (mainly between Europe and Latin America) supports my candidature. (linkedin.com/in/carlosjaimebh) Also, my role as professor, director and researcher contributes to build the common vision of the CS-DC Council and the leadership of the CS- DC Executive Committee.


'''Mina TEICHER, Professor of Bar-Ilan University, Israël'''

I submit my candidacy to the Executive Committee of the CS Digital Campus. If elected I will work towards our following needs, using my past experience in Professional international societies, universities managements and the data industry : 

* We need in the near future to build an optimal and effective agreement with the Complex System Society.
* We need to build a business  plan for fund raising.
* We need to build a modular budget for 2021.
* We need to build a strategy for geographically extension. 
* We need to build a strategy for thematic  extension. 
* We need to build partnerships with the big multi national high tech Companies in network and in content. 


'''Yasmin MERALI, Professor of University of Hull, UK'''

This manifesto is connected with the ideals that I had as a founding member of our UniNet which was conceived as part of the FP7 ASSYST project.
CS-DC has come a long way since its initial conception. The way I see it, there are three categories that have grown to emerge as our core activities- 
* Capacity building through education and training in Complex Systems Science
* The application of Complex Systems Science to address global challenges
* The advancement of Complex Systems Science through research and development.

I believe this is a good time to link back to the inception of our UNITWIN which was in part inspired by considerations of issues at a human scale, and the desire to address the inequalities that divided the so-called developed and developing countries. This resonates strongly with the ambition of the  Sustainable Development Goals (SDGs) we are currently grappling with.
In the growth phase of the UNITWIN and CSDC we have been focused on extending the size of the network, and scaling up our educational offerings across the digital campus. In the next phase I believe we need to:
# understand and leverage the diversity and distinctive capabilities and resources (e.g. indigenous knowledge) of the countries in our network to develop a healthy ecosystem, and
# tailor the support that we provide to align with the diverse nature of their relational and social capital and their economic, political and environmental challenges and priorities with regard to the SDGs.  

I am concerned that if we do not explicitly design a social/ideational exchange mechanism that attends to these two imperatives, we will not have full, active participation of all member institutions, and the countries of the South that do not currently have champions in Europe will be marginalized.

If elected I would champion a strategy of organizing ourselves following the Complex Adaptive Systems paradigm, as a hyper network with dynamically connected local clusters. In practical terms I would like to begin by establishing the local (country-based) clusters and establishing a discourse that would allow us to map the diverse profiles, challenges and aspirations for the different countries. This would then form the basis for the development of a mechanism for shaping the meaningful collaborative development of our three core activities to deliver advances that are globally co-ordinated and locally responsive.

Personal Profile: I am Professor of Systems Thinking at the University of Hull and have served as Director of the Centre Systems Studies there. Prior to that I was Co-Director of the Doctoral Training Centre for Complex Systems Science at the University of Warwick. My research is transdisciplinary, focusing on the use of Complex Systems Science to enhance the resilience of socio-economic systems. I am an Expert advisor to the EU and I have significant experience of lecturing internationally as Visiting Professor in Asia, Europe and the USA.

'''Céline ROZENBLAT: Professor, University of Lausanne - Institut de géographie et de durabilité (IGD), Switzerland'''

I'm pleased to applied to become member of the CS-DC council.
As founding member of CS-DC, my university, the university of Lausanne, is very engaged in Complex sciences.
I would not only represent my university, but also social science as geographer and vice-president of the International Geographical Union and member of the International Science Council commission on Urban Health and Well being.
I would act in the council in specific programs to develop the reality of the Digital Campus of the Complex systems.

All these actions combine very ambitious interdisciplinary approaches, and in this perspective, we developed with CS-DC for 3 years the TIMES Flagship  Territorial Intelligence For Multilevel Equity And Sustainability. It comprises four main programs:

 '''SIRE''': Socially Intelligence Roadmap Ecosystem
 '''POLE:''' Personalized Open Lifelong Education
 '''WOSI:''' Worldwide Open Smart Innovation
 '''WOSP:''' Worldwide Open Stochastic Prediction 

In this perspective a MOOC « Healthy Urban system » is now in development, basing the interdisciplinary approach on the CS-DC Road-Map grid. It seems very useful and relevant in this implementation stage.
I would help to develop other programs in this perspective\[Ellipsis]


'''André TINDANO, Director General of CARFS (African Center for Research and Training in Synecoculture)'''

What motivates me to aspire to the position of member of the executive committee of CS DC is my long term participation in the promotion of sustainable development and my commitment to the sharing of knowledge and expertise.

My research interests Sustainable agriculture, ecology, nutrition, life science. I have a strong experience  in:
* Administration and management of development projects and programs;
* Accompaniment of associations and groups;
* Technical capacity building (animation of training sessions and reflection workshops).
* Action research;
* Sociological, socio-economic and economic studies.
* Development of development projects and programs;
* Training of trainers
* Results Based Management Training (RBM)
* Monitoring and evaluation of development programs and projects;
* Management of programs and development projects;
* Institutional development and organizational strengthening;
* Development and implementation of training / awareness / animation program;
* Very good knowledge of participatory methods
'''Guiou KOBAYASHI, Associate Professor at Federal University of ABC in São Paulo State, Brazil. '''

I worked with fault-tolerant computer systems for nuclear power plants and Metro signaling systems and recently my interests have evolved to resilience properties of Complex Systems. Traditionally, redundancy was the main feature for fault-tolerant and fail-safe systems, but the adaptability and the evolution of Complex Systems are the key elements for the resilience of these systems. How to characterize, design and implement these key elements in our future resilient systems? The Complex System - Digital Campus (CS-DC) is a way to create a world-wide community of researchers, philosophers and students to promote and discuss this kind of questions involving Complex Systems. For me, participating in its foundation was a great honor and I am very glad for the opportunity that I have had to contribute since 2012 in the consolidation of CS-DC.

Through this manifest I am applying to be one of the members of the new CS-DC's Executive Committee. I would like to help just a little more to strengthen and structure this fantastic community through which I had the opportunity to meet important people with very interesting works that expanded my knowledge of Complex Systems. Although my University and my personal contribution for CS-DC are very limited and small, I hope to continue to work with this great team.


'''Pierre COLLET, Professor of Computer Science, University of Strasbourg, France. Co-coordinator of the CS-DC UNESCO UniTwin'''

The CS-DC initiated by Paul Bourgine, Jeffrey Johnson, Cyrille Bertelle and many others has been an extraordinary adventure a) to instantiate as a UNESCO UniTwin and b) to develop and run since it was enacted in July 2014. Many a night have been spent on designing its inner workings, so that it can deliver an effective affordance for the scientists who wish to develop the science and teaching of Complex Systems. Indeed, many projects seeded in the CS-DC have come to fruition, showing the enormous potential of this fertile environment not only for research, but also for teaching: the BBB rooms set up by the CS-DC have not only made it possible for the CS-DC to organize conferences, but have also shown their potential as remote teaching rooms in many Universities around the world.

It has been an honour for me to be part of the development of the CS-DC since its beginning, but so much remains to be done!

In this manifesto, I hereby express my strong desire to continue developing the CS-DC in these trying times, when the effects of the pandemics stretch thin the social links that our research and teaching communities need most. My objectives for this new mandate are not only to deliver a new world conference (originally planned in 2020 but unfortunately delayed due to the high toll imposed on us all, teachers and researchers alike, by COVID-19) but also continue on developing not only efficient complex computational ecosystems (cheap powerful PARSEC machines have been installed in several universities) but more specifically remote teaching environments based on complex systems, to mitigate the terrible impact of the pandemic on face to face education, within the POEM CS-DC flagship, on which Paul Bourgine and myself have been working for many years now..


'''Mariana C. BROENS, Professor - UNESP - BRAZIL.'''

As members of the Executive Committee, our main challenge will be to raise, analyse and to discuss possible positive/negative ethical and political implications of the further development of the Complex Systems Science, and their application on studies of everyday social problems. In particular, We believe that the widespread use of complex system models and Big Data analytics can bring important questions about people's privacy, personal and corporate responsibility, widespread surveillance by public or private institutions, among many others, that should be deeply discussed in our community. Our contribution will be to raise and deepen these discussions from an interdisciplinary perspective.


'''Cyrille Bertelle, Professor in Computer Sciences,  University Le Havre, France, co-coordinator of the CS-DC UNESCO UniTwin'''

I am a candidate for the CS-DC Executive Committee to represent the University of Le Havre
Normandy, which is co-coordinator with the University of Strasbourg of the convention of
recognition of the CS-DC as UniTwin by UNESCO. The University of Le Havre has made
available to the community, resources and skills to provide digital collaborative tools for the
organization of the Councils and the CS-DC'15 virtual conference.

The objective I wish to take is to facilitate the involvement of member universities not only
by their representatives on the council but by allowing researchers from these member
institutions to join concrete and accessible actions.

Co-responsible in the past of a master's degree on complex systems and then of the creation
of the institute on complex systems in Normandie (France) in a multidisciplinary framework,
I have participated in the setting up of the project labeled by the French national program of
investments for the future and entitled &quot;Smart Port City&quot;. The aim is to think about the
future of territories in a sustainable development approach supported by new technologies
and concerned about the environment and the well-being of their citizens. My research skills
are focused on implementing the complexity of complex dynamic systems and networks,
crossing behavioral scales from the interaction of human behaviors to the technical
networks of the territory. 

The book &quot;complex systems, smart territories and mobility&quot; from
the Springer's Understanding Complex Systems series, which will be published in January
2021 (https://www.springer.com/gp/book/9783030593018) illustrates the research
coordination actions that I lead in these fields.


'''Slimane Ben Miled, Senior Researcher at Pasteur Institute of Tunis, Professor at ENIT'''

Our Tunisian consortium want to constitute a collaborative Research Training Programs to increase data science capacity related to health research in Africa by building trainings and enhancing institutional capacity at African academic institutions.
The academy/project is based on 4 pillars to build a training ecosystem for Data and Engineering Science in health.
# A platform of federated master’s programs with à la carte optional courses covering informatics/computer science, biomedical informatics, data science, statistics, and public health). Each program will keep its independence, with a mention to the academy label, and this platform will allow to enrich the training with optional modules, seminars, and courses in the partner institutions. New curricula will be created in relation to ethical issues.
# Network of Doctoral programs and Executive programs
# Platform of federated Business incubator and a career center offers training, support and funding for projects related to the project’s topic.

This challenge is in perfect agreement with the Sustainable Development Goal 3 and the CS-DC flagship PHYSIOMES (Personalized Health phYSIcally, sOcially and Mentally for Each in their networkS).


'''Masa Funabashi: researcher of open complex systems at Sony Computer Science Laboratories, Inc.'''

I would like to contribute to the executive committee of CS-DC on the following two pillars:
* Promote the FOOD (From smart agrOecOnomy to smart fooD) flagship project that aims to resolve the health-diet-environment trilemma through the promotion of sustainable food systems, in collaboration with the e-lab &quot;human augmentation of ecosystems&quot; members institution: Sony CSL, Synecoculture Association, CARFS, and those who wish to participate in CS-DC collaboration.
* Construct a basic e-learning content on Synecoculture and ecological literacy as a part of CS-DC MOOCs and perform initial trials, principally in ECOWAS countries, through the Sony CSL-CARFS collaboration.

Through the development of on-going activities in FOOD project and making synergy with other flagship projects, I would like to contribute CS-DC as a member of the executive committee and realize further extension toward the achievement of global sustainability goals such as SDGs.


'''Dr Mohamed Abdellahi (Ould BABAH) Ebbe, Mauritania, '''
* General Director of the Institut du Sahel/CILSS www.insah.org; 
* Executive Director of the Orthopterist Society (400 researchers among the globe) https://orthsoc.org/

By obtaining the honor of having your hoped-for confidence for this post of member of the executive council of the CS-DC, I will work, in priority and in the short term on two main subjects:
# '''The transboundary plague of the Desert Locust (Schistocerca gregaria (Forska l , 1775))''' This plague of the Desert Locust of more than 3000 years that cites all our holy books (the Tourah, the Bible and the Koran) and which continues to be present to this day and to wreak devastating devastation. In case of invasion, it can affect the agriculture and pastures of about 25 countries including those of the poorest countries of the world, from Mauritania to India, while its best and most effective strategy of struggle is preventive struggle by targeting its first centers of gregarization which are very small in space and much better known today. In 2005, the costs of its struggle in the Sahel and North Africa amounted to half a billion dollars, with 8 million farmers and pastoralists affected in the Sahel. It also massively invaded Asia and Africa. 'East Africa in 2020. On this subject, I have spent 30 years studying and fighting and developing a national strategy against this scourge which has made it possible to establish a whole prevention model and an institutional, technical, operational mechanism. and scientific effective in my country that can be adapted and copied and in all other affected countries: Biogeography of the desert locust Schistocerca gregaria, Forskal, 1775: Identification, characterization and originality of a gregarious focus in central Mauritania (HR.HORS COLLEC.) (French Edition) - Babah Ebbe, Mohamed Abdallahi | 9782705670573 | Amazon.com.au | Books http://www.worldbank.org/en/news/feature/2010/01/07/improved-ways-to-prevent-the-desert-locust-in-mauritania-and-the-sahel, http: // whatsnext.blogs.cnn.com/2012/02/02/in-mauritania-sunny-with-a-chance-of-locusts/
# '''The Sahel Institute (INSAH) www.insah.org of the Permanent Interstate Committee for Drought Control (CILSS)''' that I lead and which has been doing extraordinary work for almost half a century in the field of research and development of animal and plant production techniques and also in the field of support for demographic, population and development policies, in favor of the populations of our 13 Sahelian, coastal and island member countries. This work covered the majority of good practice technologies in the field of plant and animal production, natural resource management, land restauration, cultivation techniques, post-harvest, machining, dehulling operations technology. / ginning, Conservation and storage, good resilience practices

Research on the demographic dividend, gender and the empowerment of women and the Population / Development interrelations ... etc
The results of all this work are contained in a database. data, online http://publications.insah.org/, containing more than 1,500 books, scientific and technical articles that will have to be modernized and connected to the CS Meta data.

Finally, I will work actively with my colleagues on the Executive Board on all aspects of other cross-border scourges but also all aspects of improving agro-sylvo-pastoral production tools as well as the fight against poverty and food insecurity and nutrition in line with the goals (SDGs)


'''Dr. Habil. László Barna Iantovics, Associate Professor at “George Emil Palade” University of Medicine, Pharmacy, Science and Technology of Targu Mures, Romania.''' 

With the present manifesto, I would like to candidate as a member of the CS-DC Executive Committee. I am the representative of “George Emil Palade” University of Medicine, Pharmacy, Science and Technology of Targu Mures from Romania in CS-DC.

Some of my research and academic activities are related to the complex systems, including: publications, organization of conferences (e.g. Symposium on Understanding Intelligent and Complex Systems - UICS 2009; 1st Int. Conf. on Complexity and Intelligence of the Artificial and Natural Complex Systems Medical Applications of the Complex Systems. Biomedical Computing -CANS 2008; 1st Int. Conf. on Bio-Inspired Computational Methods Used for Difficult Problems Solving. Development of Intelligent and Complex Systems - BICS 2008), contribution to conference committees (e.g. Int. Conf. Emergent Proprieties in Natural and Artificial Complex Systems - EPNACS 2007; Workshop on Complex Systems and Self-organization Modeling -CoSSoM 2009), preparing journal special issues (e.g. Special Issue on Complexity in Sciences and Artificial Intelligence; Special Issue on Understanding Complex Systems), participating in journal’s editorial board (e.g. Complex Adaptive Systems Modeling -CASM, SpringerOpen), and contribution to research in projects and projects coordination (Social network of machines- SOON; Hybrid Medical Complex Systems -ComplexMediSys). I was the director of the center Advanced Computational Technologies – AdvCompTech in the frame of my university. At present, I am the director of the Center for Advanced Research in Information Technology from my university.

I would like much deeper involve myself in the life and activities of the CS-DC community.

My objectives:

* To involve junior and senior researchers from my university in activities regarding research and education. To motivate universities and research institutes from my country to contribute to CS-DC. I consider also universities and research institutes with that I have collaboration in the past.
* To PROPOSE the formation of a so-called doctoral and postdoctoral students group. In the case of doctoral and postdoctoral students probably in time more students would like to be involved in activities. In this framework, I suggest the organization yearly 3 times (from 4 to 4 months) workshops in that all the interested students could discuss, present their research and research in progress. With this occasion in the frame of workshops if there is interest could be established separate sessions with presentations also by B.Sc. and M.Sc. students.
* To PROPOSE the strengthening of the following research direction with the general topic: intelligent complex systems and machine intelligence measuring. One of the subtopic by interest will be complex systems approaches in medicine and healthcare. To be accomplishable this subject I propose in a first step the formation of a group of interested persons, after then the establishment of the functionality of the group, for example: discussions when are subjects that should be discussed etc.

==Elected members to the Executive Committee &amp; as (Deputy-)Presidents==


'''Jeffrey JOHNSON, Professor of Complexity Science and Design, The Open University, UK'''

I offer myself as a candidate both to be President and to the Executive Committee of  The UNESCO UniTwin Complex Systems Digital Campus (CS-DC) so that I can help to drive it forward to achieve it goals. 

I am particularly committed to our educational efforts. I have made four MOOCs on the FutureLearn Platform for CS-DC  ( https://www.futurelearn.com/partners/unesco-unitwin-complex-systems-digital-campus ): Global Systems Science (2015-16); Systems Thinking and Complexity (2017-18); First Steps in Data Science with Google Analytics (2018-19) and  COVID-19 - Pandemics, Modelling and Policy (2020). CS-DC has a great opportunity to become the global university providing interdisciplinary education for a better world.

I am also committed to our research mission with UNESCO towards the achieving the U.N. Sustainable Development Goals. My own research on representing the dynamics of complex multilevel systems is relevant to many of the research initiatives of CS-DC.

I have extensive experience working within the complex systems community. I have run various coordination actions supporting research programmes funded by the European Commission, I am a founder member and past president of the Complex Systems Society, and I am Deputy-President of the CS-DC. I believe this experience will enable me to make a significant contribution the CS-DC over the next three years.


'''Paul BOURGINE, present President of the UNESCO UniTwin CS-DC, Complex Systems Institute of Paris'''

I offer myself as a candidate both to be President and to the Executive Committee of  The UNESCO UniTwin Complex Systems Digital Campus (CS-DC).

If elected, my main commitment is to create the conditions for a self-organized development of our UniTwin UNESCO CS-DC as autonomous communities of communities. This self-similar development will be the case both for the two main branches the UniTwin branch of our institutional members and the global eCampus branches of our individual scientific members:
* for the UniTwin branch, the communities of communities are a territorial cascade with Smart Continents, smart countries, smart cities for their sustainable development according our flagship TIMES (Territorial Intelligence for Multilevel Equity and Sustainability). The roadmap is always the same, i.e. the cascade of the 17 Sustainable Development Goals and their 169 Targets: but their relative importance and coherence within this cascade vary from one territory to the others. The institutional members of the UniTwin branch have signed their agreement with the Cooperation Programme signed with UNESCO. In 2021, the CS-DC will ask for a cascade of agreements inside each institutional member, in order to have a &quot;one for all&quot; amplification within the other branch, the e-campus branch. 
* for the eCampus branch, the cascade of communities is along the refinement cascades when studying the theoretical and experimental challenges of complex systems. With Smart Continents'21, scientists are proposing their individual challenges that enact basic communities and communities of communities within the e-departments. In the &quot;all for one&quot; return, the roadmap of each university is the cascade of roadmaps within the eCampus where the University has at least one member. Furthermore each community can organise a monthly e-seminar or e-session in workshop as well as in CS-DC'21 for recorded advanced introductions. Such advanced introductions can be the basis for curriculum largely shared by the set of Universities having members in the  community cascade of the curriculum.

This &quot;accelerator of knowledge and knowhow one for all and all for one&quot; will first benefit to the student curriculum through the flagship POEM (Personalized Open Education for the Masses). This accelerator can be extended through the flagship POLE (Personalized Open Lifelong Education) for a lifelong education. This extended accelerator will be open to all, independently of previously achieved academic levels, respectful of the diversity of social and cultural environments and in a higher and higher inclusive way including refugees, migrants and primary people. genders, religions or ways of life.</text>
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An [[Wikipedia:Open access journal|open access journal]] with&lt;br&gt; no publication costs – [[WikiJournal of PPB/About|About]]
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{{WikiJournal/Publishing/Intro}} 

The journal publishes both review articles and original research in [[Wikijournal_of_Medicine/Publishing#Publication_formats|various formats]]. WikiJournals enable academic and medical professionals to contribute expert knowledge to the Wikimedia movement in the [[Wikipedia:Academic publishing|academic publishing]] format that directly rewards them with citable publications. Included works are assigned [[Wikipedia:Digital object identifier|DOI codes]] (permanent links to each work via [[W:Crossref|Crossref]]) and are indexed by [https://scholar.google.co.uk/scholar?hl=en&amp;as_sdt=0%2C5&amp;q=source%3AWikiJournal+source%3AMedicine Google Scholar], [https://doaj.org/toc/2002-4436 DOAJ] and [[{{ROOTPAGENAME}}/About#Indexing_and_impact|others]].

The journal targets a broad population spanning from advanced researchers and clinicians to students and laypersons, wherein the latter can get quick explanations of advanced terms by [[Wikipedia:Hyperlink#Hyperlinks_in_wikis|in-line links to Wikipedia]].

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Further information:
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* [[WikiJournal User Group/Comparison to other journals|Comparison to other journals]]

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=== Publication ===
All accepted articles are published in two forms:
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* A stable [[Wikipedia:PDF|PDF]] version, which is the cite-able &quot;version of record&quot;
Before inclusion in the journal, article submissions are [[{{ROOTPAGENAME}}/Peer reviewers|peer reviewed]] by at least two relevant experts. The quality of the peer reviews is weighted in when the [[WikiJournal of Medicine/Editorial board|editorial board]] makes final decision on including the article in the journal. The journal emphasises transparency throughout the process, from peer reviews to final version and all versions are freely accessible online.

=== Contributors ===
The journal offers its services for free, by donations to the [https://wikimediafoundation.org/wiki/Home Wikimedia Foundation], and fully volunteer [[{{ROOTPAGENAME}}/Peer reviewers|peer reviewers]], [[{{ROOTPAGENAME}}/Editors|editors]] and [[{{ROOTPAGENAME}}/Editors|editorial board]] members. WikiJournal of Psychology, Psychiatry and Behavioral Sciences is part of the [[meta:WikiJournal User Group|WikiJournal Publishing Group]], which is also the formal ''publisher'' of the journal.

Anyone may edit journal pages, but only minor edits to accepted journal articles (e.g. spelling) are permitted. Any edits that change the content of a published article require an additional round of peer-review, and if accepted, are published in an updated version of the page. Suggestions for updates of the main text of published articles may be created as separate drafts that are re-submitted to undergo peer review before being used to update the article.

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=== Guidelines ===
The [[{{ROOTPAGENAME}}/Ethics statement|publication ethics statement]] of the WikiJournal of Psychology, Psychiatry and Behavioral Sciences abides by several international guidelines:
*[[wikipedia:ICMJE Recommendations (Uniform Requirements for Manuscripts)|ICMJE]] Recommendations for the Conduct, Reporting, Editing, and Publication: [http://www.icmje.org/icmje-recommendations.pdf]
*[[Wikipedia:Committee on Publication Ethics|COPE]] code of conduct for journal editors: [http://publicationethics.org/files/Code_of_conduct_for_journal_editors_Mar11.pdf]
*[[Wikipedia:Budapest Open Access Initiative|Budapest Open Access Initiative]] recommendations: [http://www.budapestopenaccessinitiative.org/boai-10-recommendations]

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The journal prioritises diverse impact across a range of metrics, such as [https://www.altmetric.com/explorer/report/ab3b956e-bf4a-4277-8084-6605d397fcb7 its AltMetric score], in addition to traditional Impact Factor (Not yet assigned by Web of Science; unofficial estimation [[/Citation metrics|here]]). Additionally high reach is achieved for those articles integrated into Wikipedia [https://tools.wmflabs.org/massviews/?platform=all-access&amp;agent=user&amp;source=category&amp;target=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FCategory%3AWikipedia_articles_published_in_WikiJournal_of_Medicine&amp;range=latest-356&amp;subjectpage=0&amp;subcategories=0&amp;sort=views in full] or [https://tools.wmflabs.org/massviews/?platform=all-access&amp;agent=user&amp;source=category&amp;target=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FCategory%3AWikipedia_articles_with_sections_published_in_WikiJournal_of_Medicine&amp;range=latest-356&amp;subjectpage=0&amp;subcategories=0&amp;sort=views&amp;direction=1&amp;view=list in part].  

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:''More detailed updates are located at the [[Talk:{{ROOTPAGENAME}}|discussion page]]''

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*2021-01-30: 

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      <text bytes="9508" sha1="gg2y2qad9pz0ajpwx13wi2eu78mhr95" xml:space="preserve">A number of seemingly credible sources are describing an increase in political polarization worldwide. [[w:Maria Ressa|Maria Ressa]] describes how [[w:Rodrigo Duterte|Rodrigo Duterte]], former President of the [[w:Philippines|Philippines]] &quot;started ... with five hundred volunteers&lt;ref&gt;Ressa (2022, pp. 147-8).&lt;/ref&gt; (1) creating “sock puppets,” or fake accounts that attack or praise; (2) “mass reporting,” or organizing to negatively impact a targeted account; and (3) “astroturfing,” or fake posts or lies designed to look like grassroots support or interest.&lt;ref&gt;Ressa (2022, pp. 152-3).&lt;/ref&gt; These actions tricked the algorithms of social media companies like Facebook and Twitter into amplifying fraudulent messages including incitements to violence and criminal prosecutions based on trumped up charges. The results easily overwhelmed honest media. [[w:Leila de Lima|Leila de Lima]], a Senator and former Secretary of Justice of the Philippines, spent years  in pretrial detention before the charges were dropped for lack of evidence.&lt;ref&gt;Ressa (2022, p. 158ff) and Wikipedia, &quot;[[w:Leila de Lima|Leila de Lima]]&quot;, accessed 2024-07-22.&lt;/ref&gt; Ressa's news organization, [[w:Rappler|Rappler]].com, was ordered to close. Ressa herself was convicted on questionable charges. Both continued operating while the legal procedures against them were appealed.&lt;ref&gt;Ressa (2022, pp. 152-3) and Wikipedia, &quot;[[w:Maria Ressa|Maria Ressa]]&quot;, accessed 2024-07-22.&lt;/ref&gt; Ressa says similar procedures are making major contributions to the rise of fascism and far-right nationalist populists in the US, Europe, Turkey, India, Russia, and elsewhere.&lt;ref&gt;Ressa (2022, pp. 152-3).&lt;/ref&gt; [[w:H. R. McMaster|H. R. McMaster]], former President Trump's second National Security advisor, said that &quot;The internet and social media thus provided [Russia] with a low-cost, easy way to divide and weaken America from within.&quot;&lt;ref&gt;McMaster (2020, pp. 47-48).&lt;/ref&gt; The [[w:2021 Facebook leak|2021 Facebook leak]] documented how executives of [[w:Facebook|Facebook]] and [[w:Meta Platforms|Meta]] knowingly prioritized profits over action to limit incitements to violence, even facilitating the [[w:Rohingya genocide|Rohingya genocide]] in [[w:Myanmar|Myanmar]], because doing otherwise would have reduced their profits. 

This &quot;Category:Media reform to improve democracy&quot; include videos of experts and activists working this issue along with 29:00 mm:ss audio files submitted to a ''Media &amp; Democracy'' series syndicated on the [[w:List of Pacifica Radio stations and affiliates|Pacifica radio network]]&lt;ref&gt;&lt;!--Media &amp; Democracy on Audioport--&gt;{{cite Q|Q127839818}}&lt;/ref&gt; plus text and space for moderated discussions. 

Some of this work is cited in the book on ''[[Media Literacy and You]]'', which is being written -- [[w:Crowdsourcing|crowdsourced]] -- to help humans better understand how they can counter the trend toward increasing political polarization and violence by talking politics, calmly, with respect and humility, with others with whom they may vehemently disagree, because the alternative is killing humans over misunderstanding. The goal is ''not'' to convince anyone that they are wrong. Rather it is to build relationships where humans can agree to disagree agreeably and collaborate to improve issues of common concern. 

== Table of episodes == 

{| class=&quot;wikitable sortable&quot;
|+ Episodes of &quot;Media &amp; Democracy&quot; for the [[w:List of Pacifica Radio stations and affiliates|Pacifica Radio Network]]
|-
! 
!! colspan=3 | Date !! 
|- 
! no. || recorded !! broadcasted on [[w:KKFI|KKFI]] !! released to Pacifica !! Episode 
|-
| 51 || 2026-05-06 || 2026-05-12 || 2026-05-16 || [[Online platforms' effects on public health, safety and democracy]]
|-
| 50 || 2026-04-09	|| 2026-04-28 || 2026-05-02 || [[How US media threaten the health of all]]
|- 
| 49 || 2026-04-06 || 2026-04-14 || 2026-04-18 || [[News suppressed for those who control money for the media]] 
|-
| 48 || 2026-03-27 || 2026-03-31 || 2026-04-04 || [[Media and war]]
|-
| 47 || 2026-03-12 || 2026-03-17 || 2026-03-21 || [[Media literacy to dispel myths and improve public policy]]
|-
| 46 || 2026-02-26 || 2026-03-03 || 2026-03-07 || [[Concerns about media, especially in Germany]]
|-
| 45 || 2026-02-12 || 2026-02-17 || 2026-02-21 || [[Underserved serve themselves with low-power FM]]
|-
| 44 || 2026-01-30 || 2026-02-03 || 2026-02-07 || [[Conservative media are different]]
|-
| 43 || 2026-01-15 || 2026-01-20 || 2026-01-24 || [[Medill says you can help yourself by helping improve local media]] 
|-
| 42 || 2026-01-03 || 2026-01-06 || 2026-01-10 || [[Lisa Loving on media literacy and how you can report for your community]]
|- 
| 41 || 2015-12-11 || 2025-12-23 || 2025-12-27 || [[John Maxwell Hamilton on American propaganda]] 
|- 
| 40 || 2025-12-05 || 2025-12-09 || 2025-12-13 || [[You can better protect yourself from Big Tech]]
|- 
| 39 || 2025-11-20 || 2025-11-25 || 2025-11-29 || [[Differences between media outlets including coverage of Gaza]]
|- 
| 38 || 2025-11-06 || 2025-11-11 || 2025-11-15 || [[Media &amp; Democracy lessons for the future]]
|-
| 37 || 2025-10-23 || 2025-20-28 || 2025-11-01 || [[Media reform initiatives in West Africa]]  
|-
| 36 || 2025-10-03 || 2025-10-14 || 2025-10-18 || [[Seth Radwell says that the two Enlightenments tell us how to heal US political polarization]]
|-
| 35 || 2025-09-25 || 2025-09-30 || 2025-10-04 ||
[[Media Reform Coalition challenges anti-democratic media bias in the UK]]
|- 
| 34 || 2025-09-12 || 2025-09-16 || 2025-09-20 || [[Fighting back against the campaign of censorship and control]]
|- 
| 33 || 2025-08-28 || 2025-09-02 || 2025-08-06 || [[The role of the media in conflict]]
|- 
| 32 || 2025-07-31 || 2025-08-19 || 2025-08-21 || [[Evidence-informed public policy]]
|-
| 31 || 2025-08-01 || 2025-08-05 || 2025-08-09 || [[What the Left can learn from Fox]]
|-
| 30 || 2025-07-17 || 2025-07-22 || 2025-07-26 || [[Democratic delusions: Fix the media to fix democracy]]
|- 
| 29 || 2025-07-03 || 2025-07-08 || 2025-07-12 || [[News from Germany 1900-1945 and implications for today]]
|-
| 28 || 2025-06-12 || 2025-06-24 || 2025-06-28 || [[How news impacts democracy per USD Communications Professor Nik Usher]]
|-
| 27 || 2025-06-08 || 2025-06-10 || 2025-06-14 || [[Media concentration per Columbia History Professor Richard John]]
|- 
| 26 || 2025-05-21 || 2025-05-27 || 2025-05-31 || [[Dean Starkman and the watchdog that didn't bark]]
|- 
| 25 || 2025-05-08 || 2025-05-13 || 2025-05-17 || [[Freedom of the Press Foundation says...]]
|- 
| 24 || 2025-04-24 || 2025-04-29 || 2025-05-03 || [[Canadian journalist Marc Edge on media reform to improve democracy]]
|-
| 23 || 2025-04-10 || 2025-04-15 || 2025-04-19 || [[The value of indigenous and community radio]]
|- 
| 22 || 2025-03-28 || 2025-04-01 || 2025-04-05 || [[Trump ordered changes in public data]]
|-
| 21 || 2025-03-06 || 2025-03-11 || 2025-03-22 || [[Vulture capitalists destroying newspapers]]
|- 
| 20 || 2025-02-25 || 2025-02-25 || 2025-03-08 || [[Local newspapers limit malfeasance]]
|-
| 19 || 2025-02-06 || 2025-02-11 || 2025-02-22 || [[Palast says Trump lost, vote suppression won the 2024 elections]]
|-
| 18 || 2025-01-25 || 2025-02-04 || 2025-02-12 || [[Defend free speech hybrid town hall]]
|-
| 17 || 2025-01-13 || 2025-01-14 || 2025-01-25 || [[Media in the Syrian conflict]]
|-
| 16 || 2024-12-20 || 2024-12-31 || 2025-01-04 || [[HR 9495, the nonprofit-killer bill, per Michael Novick]]
|- 
| 15 || 2024-12-13 || 2024-12-24 || 2024-12-21 || [[Information is a public good per communications prof Pickard]]
|-
| 14 || 2024-12-02 || 2024-12-10 || 2024-12-07 || [[Media literacy for the Arab World per Ahmed Al-Rawi]]
|-
| 13 || 2024-11-21 || 2024-11-26 || 2024-11-23 || [[Thom Hartmann on The Hidden History of the American Dream]]
|-
| 12 || 2024-10-25 || 2024-11-05 || 2024-11-09 || [[Legal concerns of Wikimedia Europe]] 
|-
| 11 || 2024-10-26 || 2024-10-19 || 2024-10-27 || [[Project 2025 per Professor Brooks]]
|-
| 10 || 2024-10-01 || 2024-10-01 || 2024-10-12 || [[Jacob Ware on far-right terrorism in the US]]
|-
| 9 || 2024-09-13 || 2024-09-17 || 2024-09-29 || [[Dis- and misinformation and their threats to democracy]]
|- 
| 8 || 2024-09-11 || 2024-11-12 || 2024-09-14 || [[22nd Century Initiative]]
|- 
| 7 || 2024-08-22|| 2024-08-27 || 2024-08-31 || [[Global Project Against Hate &amp; Extremism (GPAHE)]]
|-
| 6 || 2024-08-19 || 2024-08-20 || 2024-08-24 || [[Facebook whistleblower Frances Haugen says]]
|-
| 5 || 2024-08-13 || 2024-08-13 || 2024-08-17 || [[Legal concerns of Free Press including Section 230]]
|-
| 4 || 2024-08-02 || 2024-08-06 || 2024-08-10 || [[How psychological and interpersonal processes are influenced by human-computer interactions]]
|-
| 3 || 2024-07-30 || 2024-07-30 || 2024-08-03 || [[Dean Baker on Internet companies threatening democracy internationally and how to fix that]]
|-
| 2 || 2021-04-29 || 2021-04-29 || 2021-05-16 || [[Media reform per Freepress.net]]
|- 
| 1 || 2021-02-23 || 2021-02-23 || 2021-03-17 ||[[Unrigging the media and the economy]]
|}

== Notes ==
{{reflist}}

== Bibliography == 

* &lt;!-- H. R. McMaster (2020) Battlegrounds: The Fight to Defend the Free World--&gt;{{cite Q|Q104774898}}
* &lt;!--Maria Ressa (2022) How to Stand Up To a Dictator--&gt;{{cite Q|Q117559286}}

[[Category:Interdisciplinary studies]]

[[Category:Political science]]
[[Category:Economics]]
[[Category:Freedom and abundance]]
[[Category:Videoconferences on media and democracy]]</text>
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    <title>Global Audiology/Oceania/Australia</title>
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{{:Global Audiology/Oceania/Header}}
{{CountryHeader|File:Australia (orthographic projection).svg|https://en.wikipedia.org/wiki/Australia}}
{{HTitle|General Information}}
[https://en.wikipedia.org/wiki/Australia Australia], officially the Commonwealth of Australia is a country comprising the mainland of the Australian continent, the island of Tasmania, and numerous smaller islands. Australia has a population of approximately 27.7 million. Aboriginal and Torres Strait Islander peoples make up approximately 3.2% of the Australian population. Although English is not the official language in law, it is the de facto official and national language. The 2021 Census found that 167 Indigenous languages were spoken at home by Indigenous Australians. 

{{HTitle|History of Audiology}}
Audiology emerged as a profession in Australia during the 1940s. Two major events drove its establishment as a medical specialty. First, many servicemen returned from World War II suffering from hearing loss caused by injuries and noise exposure. Second, the 1940-41 rubella epidemic damaged the hearing of numerous children. These circumstances led Australia to establish the [https://www.nal.gov.au/ National Acoustics Laboratory (NAL)] in 1947. The laboratory began testing hearing and fitting hearing aids for veterans and affected children. Twenty years later, in 1967, NAL expanded its services to include pensioners. 

A pivotal moment in establishing audiology as a distinct profession came when NAL's original director determined that audiological functions such as hearing testing, hearing aid fitting, and associated rehabilitation should be performed by professionally qualified personnel rather than technicians. The initial debate centered on whether preschool teachers or psychologists would be better suited for these roles. Psychologists were ultimately selected, reportedly because they were considered better equipped to identify malingering among veterans. The first audiologists in Australia were psychologists who undertook postgraduate training in speech therapy and hearing care.

Throughout the 1950s and 1960s, the number of practicing audiologists grew steadily, driven primarily by NAL's expansion and, in the latter half of this period, by the creation of positions in hospital settings. Around 1960, informal gatherings of audiologists began in Sydney and Melbourne. In Melbourne, these typically consisted of monthly lunches followed by afternoon seminars. These regional networking efforts eventually led to the formation of the Audiological Society of Australia in May 1968, which began with ten foundation members representing four states. The majority of audiologists practicing before 1970 came from psychology backgrounds, though some arrived from fields including teaching, physics, and speech pathology. Significant developments after 1970 transformed the profession considerably.

The establishment of formal audiology training programs in Brisbane, Melbourne, Sydney, and Perth marked a crucial step toward professional standardization. Audiology subjects were also incorporated into speech-language pathology courses at several institutions. The Australian government and other employers granted official recognition to audiology as a distinct profession, lending credibility and structure to the field.

Employment opportunities expanded significantly, particularly in academia and private practice. Audiological research flourished, with notable emphasis on hearing aids and cochlear implants, though research extended into many other areas as well. The scope of audiological services broadened considerably during this period. The profession established regular national conferences beginning in 1974, held biennially, along with other specialized scientific meetings. The Australian Journal of Audiology was founded in 1979, providing a dedicated publication outlet for Australian audiological research. Publications by Australian audiologists in both national and international venues increased substantially, and interactions with international colleagues intensified through conferences, visits, and collaborative research projects.

{{HTitle|Incidence and Prevalence of Hearing Loss}} 
Hearing loss represents a substantial and growing public health burden in Australia. National estimates from the Australian Government Roadmap for Hearing Health indicate that, by 2019, approximately 3.6 million Australians were living with some form of hearing impairment.

The prevalence of moderate and more severe hearing loss (≥ 40 dB HL) in children ranges from 1.04 per 1000 live births at 3 years of age to 1.57 per 1000 live births for children between 9 and 16 years of age. Mild hearing loss (&lt; 40 dB HL) increases from 0.28 per 1000 live births at 3 years to 1.68 per 1000 live births at 9 years and older.&lt;ref&gt;{{Cite journal|title=The Ages of Intervention in Regions With and Without Universal Newborn Hearing Screening and Prevalence of Childhood Hearing Impairment in Australia|url=http://www.portico.org/Portico/article?article=pf16jrb3hm|journal=Australian and New Zealand Journal of Audiology|date=2006-11-01|pages=137–150|volume=28|issue=2|doi=10.1375/audi.28.2.137|first=Teresa Y.C|last=Ching|first2=Ron|last2=Oong|first3=Emma van|last3=Wanrooy}}&lt;/ref&gt;  Among urban Australian school-aged children (5 to 7 years), the prevalence of bilateral hearing loss ≥26 dB was estimated to 2.1% as reprted in 2006.&lt;ref&gt;{{Cite journal|title=Cross-sectional prevalence and risk factors for otitis media and hearing loss in Australian children aged 5 to 7 years: a prospective cohort study|url=https://www.theajo.com/article/view/4259/html|journal=Australian Journal of Otolaryngology|date=2020-03|pages=0–0|volume=3|doi=10.21037/ajo.2020.02.02|first=Christopher G.|last=Brennan-Jones|first2=Hrehan H.|last2=Hakeem|first3=Cheryl Da|last3=Costa|first4=Weijie|last4=Weng|first5=Andrew J. O.|last5=Whitehouse|first6=Sarra E.|last6=Jamieson|first7=Robert H.|last7=Eikelboom}}&lt;/ref&gt; In a national study, the prevalence of bilateral and unilateral hearing loss ≥16 dB HL was 9.3% and 13.3%, respectively. Slight losses (16-25 dB HL) were more prevalent than mild or greater losses (≥26 dB HL). &lt;ref&gt;{{Cite journal|title=Cross-sectional epidemiology of hearing loss in Australian children aged 11–12 years old and 25-year secular trends|url=https://adc.bmj.com/lookup/doi/10.1136/archdischild-2017-313505|journal=Archives of Disease in Childhood|date=2018-06|issn=0003-9888|pages=579–585|volume=103|issue=6|doi=10.1136/archdischild-2017-313505|language=en|first=Jing|last=Wang|first2=Carlijn M P|last2=le Clercq|first3=Valerie|last3=Sung|first4=Peter|last4=Carew|first5=Richard S|last5=Liu|first6=Fiona K|last6=Mensah|first7=Rachel A|last7=Burt|first8=Lisa|last8=Gold|first9=Melissa|last9=Wake}}&lt;/ref&gt; 

In the Blue Mountains Hearing Study, 33% of older adults had some degree of hearing loss at baseline, with a 5-year incidence of 17.9%.&lt;ref&gt;{{Cite journal|title=Five-Year Incidence and Progression of Hearing Impairment in an Older Population|url=https://journals.lww.com/00003446-201103000-00010|journal=Ear &amp; Hearing|date=2011-03|issn=0196-0202|pages=251–257|volume=32|issue=2|doi=10.1097/AUD.0b013e3181fc98bd|language=en|first=Paul|last=Mitchell|first2=Bamini|last2=Gopinath|first3=Jie Jin|last3=Wang|first4=Catherine M.|last4=McMahon|first5=Julie|last5=Schneider|first6=Elena|last6=Rochtchina|first7=Stephen R.|last7=Leeder}}&lt;/ref&gt; In 2022 hearing loss was estimated to affect 74% of people aged over 70 in Australia.&lt;ref&gt;{{Cite journal|title=Hearing loss, cognition, and risk of neurocognitive disorder: evidence from a longitudinal cohort study of older adult Australians|url=https://www.tandfonline.com/doi/full/10.1080/13825585.2020.1857328|journal=Aging, Neuropsychology, and Cognition|date=2022-01-02|issn=1382-5585|pages=121–138|volume=29|issue=1|doi=10.1080/13825585.2020.1857328|language=en|first=Paul A.|last=Strutt|first2=Amanda J.|last2=Barnier|first3=Greg|last3=Savage|first4=Gabrielle|last4=Picard|first5=Nicole A.|last5=Kochan|first6=Perminder|last6=Sachdev|first7=Brian|last7=Draper|first8=Henry|last8=Brodaty}}&lt;/ref&gt;

{{HTitle|Information About Audiology Education in Australia}}
Audiologists in Australia complete a minimum of five years of university education, including a two-year master’s level audiology program accredited by Audiology Australia. Currently, seven Australian universities offer Audiology Australia–accredited postgraduate audiology programs. Following graduation, Audiology Australia members are required to complete a one-year supervised clinical internship.During this period, interns practice under the supervision of an Audiology Australia Accredited Audiologist, facilitating a structured transition into professional practice and ensuring high standards of service delivery (Audiology Australia, n.d.)

Currently, six universities across Australia offer master's programs in audiology. These include Macquarie University, University of Queensland, Flinders University, Melbourne University, La Trobe University, and the University of Western Australia. The University of Western Australia offers a joint master's/PhD in clinical audiology. Australian master's programs require graduates to meet a set of clinical competencies and complete 250 hours of clinical experience before graduation. Graduates must also complete a one-year clinical internship if they want to provide services to clients in the large government-funded sector. During this year, experienced audiologists supervise interns and prepare them for independent clinical practice. After completing the internship, new graduates receive certification from either Audiology Australia or the Australian College of Audiology. Both organizations represent audiologists professionally in Australia. 

{{HTitle|Scope of Practice and Licensing}}
Audiologists need to meet the relevant membership and clinical competency requirements set by Australian Practitioner Professional Bodies to practice in Australia. They need to hold full membership in Audiology Australia with a Certificate of Clinical Practice (CCP) and/or full/ordinary membership in the Australian College of Audiology (ACAud) with Hearing Rehabilitation Specialist (HRS) and Diagnostic Rehabilitation Specialist (DRS) competencies. Audiologists must complete at least the equivalent of an Australian university master's degree in clinical audiology.

Audiologists in Australia work with clients of all ages, from infants to older adults, including clients with complex needs. They assess hearing and auditory function, vestibular function, tinnitus, auditory processing function, and neural function. Audiologists perform diagnostic tests, including advanced tests using electrophysiological methods. They provide aural, vestibular, and tinnitus rehabilitation as well as communication training. Audiologists offer a range of rehabilitation services, including counseling and prescribing and fitting various devices and aids. These include bone conduction aids, FM and other remote sensing systems, hearing aids, hearing assistive technology, and earplugs (custom noise/swim/musician plugs). Audiologists possess knowledge of implantable devices such as cochlear implants, middle ear implantable hearing aids, fully implantable hearing aids, and bone anchored hearing aids.They collaborate with other professionals when applying these devices in rehabilitation. 

In addition to audiologists, hearing services in Australia are also provided by audiometrists. Audiometrists typically complete vocational or industry-based training programs focused on hearing assessment and hearing aid provision. Their scope of practice generally includes conducting hearing tests, fitting and managing hearing aids, and providing basic hearing rehabilitation services, particularly in community and private practice settings.

Compared to audiologists, audiometrists usually have a more limited scope of practice, particularly in areas such as complex diagnostic testing, vestibular assessment, and electrophysiological measures. Audiometrists may be affiliated with professional organizations such as the Australian College of Audiology or other industry bodies.

{{HTitle|Professional and Regulatory Bodies}}
Australia does not have a single statutory licensing authority for audiologists or audiometrists. Instead, hearing care professionals are represented by three Practitioner Professional Bodies: [https://audiology.asn.au/
Audiology Australia], [https://www.acaud.com.au/Australian College of Audiology (ACAud)], and [https://www.haasa.org.au/
 Hearing Aid Audiometrist Society of Australia (HAASA)]. Audiology Australia serves as the leading accrediting body for audiologists and supports the profession through accreditation, education and professional development, advocacy, and networking. ACAud represents both audiologists and audiometrists, while HAASA specifically represents audiometrists.

Audiologists must meet membership and clinical competency requirements set by Audiology Australia and/or ACAud, typically including completion of a master’s-level degree in clinical audiology and a supervised clinical internship. New members of Audiology Australia become accredited audiologists after completing this internship. Accreditation is valid for one year and must be renewed annually through continuing professional development (CPD). Members are required to demonstrate sufficient professional development over the previous 12 months, which may include conferences, seminars, training courses, and research activities.

Audiometrists must meet the relevant membership and competency requirements of ACAud and/or HAASA and typically complete at least a diploma-level Technical and Further Education (TAFE) qualification in audiometry or a bachelor’s degree. These professional bodies define scope of practice, competency standards, and continuing professional development requirements; however, they do not function as statutory regulatory authorities.&lt;ref&gt;{{Cite report|title=Scope of Practice for Audiologists and Audiometrists|url=https://www.acaud.com.au/wp-content/uploads/2019/07/Scope-of-Practice-for-Audiologists-and-Audiometrists.pdf|institution=Audiology Australia; Australian College of Audiology; Hearing Aid Audiometrist Society of Australia|date=2016-09-20}}&lt;/ref&gt;

'''Code of Conduct'''
Audiology Australia members are obliged by a code of conduct. Members must also comply with the Criminal History Policy and Mandatory Declarations Policy.On 12 September 2025, Australia's health ministers agreed to regulate audiology under the National Regulation and Accreditation Scheme. This development requires the Australian Health Practitioner Regulation Agency (AHPRA), which administers NRAS, to provide certification and accreditation to audiologists.

{{HTitle|Ongoing audiology research}}
The NAL leads audiology research in Australia by focusing on unmet needs within the hearing sector.NAL's strategic approach drives meaningful results through targeted research initiatives and collaborative partnerships. 
==== Accessible and Personalized Care ==== 
NAL researchers are working to make hearing services more accessible, acceptable, and personalized for all Australians. They are focusing on understanding community needs and are developing solutions tailored to different populations. Researchers are placing particular emphasis on reducing the impact of otitis media-related hearing loss in Aboriginal and Torres Strait Islander children. Teams are co-developing solutions with communities to address hearing deprivation in these vulnerable populations.

''Artificial Intelligence Applications'': NAL is actively exploring how artificial intelligence can improve hearing healthcare delivery. Researchers are applying AI to enhance screening, diagnosis, management, and ongoing support for people with hearing loss. These innovations are aiming to make hearing services more efficient and effective.

''Targeted Solutions for Listening Difficulties'': NAL is developing targeted solutions for different groups of people who are experiencing listening difficulties during daily communication. Researchers are recognizing that various populations face unique challenges and are requiring customized approaches to hearing rehabilitation.

''Research Methodology and Collaboration'': NAL is employing a focused approach that is driving results across the hearing sector. Experienced teams of researchers, audiologists, and behavioral specialists are regularly identifying unmet needs within the field. They are establishing cross-functional, multidisciplinary project teams using a matrix structure that is leveraging different areas of expertise within the organization. NAL is implementing research and innovation methodologies to develop practical outcomes. Collaboration is forming a core function of their work. Teams are regularly partnering with a global network of research collaborators, volunteers, and industry partners worldwide to deliver various initiatives.

'''Fields of Research'''

''Adult Hearing Loss'': NAL is conducting outstanding people-focused research on adult hearing loss. Researchers are promoting evidence-based outcomes to improve quality of life for people with hearing difficulties. They are providing insights into the needs of adults with hearing loss and are developing solutions and recommendations for affected individual.

''Pediatric Hearing Loss'': NAL is performing evidence-based and patient-centered multidisciplinary research on childhood hearing loss. Researchers are working to understand the long-term effectiveness of early hearing detection and intervention programs. They are informing best practices for preventing and managing childhood hearing loss using various research methods including behavioral assessments, electrophysiological measures, and quantitative and qualitative approaches.

''Technology Development:'' NAL focuses on evaluating, researching, and developing innovative tools that address unmet needs in hearing health. Current work includes focus on advanced signal and audio processing algorithms, machine learning–based diagnostics, and digital health platforms that support clinical decision making and real-world listening assessment. Some of the current ongoing technological research includeds AI-enabled remote hearing screening, online speech testing, speech to text app, smartphone-based ecological momentary assessment, and  novel auditory evoked potential tests for hidden hearing loss. Through close collaboration with hearing aid users, clinicians, and industry partners worldwide, NAL translates research into practical technologies that improve hearing outcomes and quality of life.

{{HTitle|Challenges, Opportunities and Notes}}
Australia's audiology sector struggles with several interconnected problems, including insufficient numbers of hearing specialists in regional areas, difficulties accessing services across vast distances, requirements for culturally appropriate care, hearing loss going undetected in elderly care settings, and limited uptake of remote audiology services, even though strong government support programs exist. Primary concerns involve uneven distribution of audiologists across the country, absence of a unified nationwide approach, social stigma surrounding hearing loss, and technological barriers, all of which restrict healthcare access for people living in rural and remote locations, Indigenous Australians, and senior citizens, highlighting the need for enhanced professional training and improved digital health systems. &lt;ref&gt;{{Cite journal|last=Mui|first=Boaz|last2=Lawless|first2=Michael|last3=Timmer|first3=Barbra H. B.|last4=Gopinath|first4=Bamini|last5=Tang|first5=Diana|last6=Venning|first6=Anthony|last7=May|first7=David|last8=Muzaffar|first8=Jameel|last9=Bidargaddi|first9=Niranjan|date=2025-01-02|title=Australian hearing healthcare stakeholders’ experiences of and attitudes towards teleaudiology uptake: a qualitative study|url=https://www.tandfonline.com/doi/full/10.1080/2050571X.2024.2372171|journal=Speech, Language and Hearing|language=en|volume=28|issue=1|doi=10.1080/2050571X.2024.2372171|issn=2050-571X}}&lt;/ref&gt; 

''Workforce &amp; Access Issues:''
Regional shortages: Audiologists concentrate in cities, leaving rural/remote areas under-serviced
Geographic barriers: Vast distances make consistent care difficult for remote communities
Aged care gaps: Hearing loss frequently missed due to lack of staff training and awareness.&lt;ref&gt;{{Cite journal|last=El-Saifi|first=Najwan|last2=Campbell|first2=Megan E.J.|last3=George|first3=Neha|last4=Keay|first4=Lisa|last5=Kumaran|first5=Sheela|last6=Meyer|first6=Carly|last7=Miller Amberber|first7=Amanda|last8=Newall|first8=John|last9=Dawes|first9=Piers|date=2025-09-05|title=Barriers and enablers to hearing service provision in aged care settings in Australia: perspectives from hearing clinicians|url=https://www.tandfonline.com/doi/full/10.1080/14992027.2025.2554236|journal=International Journal of Audiology|language=en|pages=1–12|doi=10.1080/14992027.2025.2554236|issn=1499-2027}}&lt;/ref&gt;
Poor integration: Limited connection between audiology and other allied health services in aged care.

''Service Delivery &amp; Technology Problems:''

Tele-audiology underutilized: Barriers include poor infrastructure, restrictive policies, inadequate funding, and limited clinician training
Digital literacy gaps: Patients, especially elderly, struggle with comfort and skills for remote care
Digital therapeutics challenges: New technologies need stronger evidence bases before widespread adoption
Technology implementation hurdles: Clinicians lack confidence in integrating new digital tools.

''Patient &amp; Cultural Barriers:''

Stigma: Hearing loss viewed as &quot;invisible disability,&quot; causing delayed treatment
Low public awareness: Limited understanding of hearing loss impacts and treatment options
Multicultural needs: Services must be culturally sensitive and linguistically appropriate

{{HTitle|Audiology Service Providers and Advocacy Groups}}
Australia hosts a robust network of organizations dedicated to supporting individuals with hearing loss, ranging from frontline service providers to advocacy groups and research institutions. 

''Leading Service Providers'':

'''NextSense''' stands as one of Australia's premier not-for-profit organizations addressing both hearing and vision loss. The organization delivers comprehensive clinical services, conducts research, and operates Australia's largest cochlear implant program. Beyond direct services, NextSense provides educational support and therapeutic interventions for children and adults navigating sensory loss.

'''Hearing Australia''' functions as a government-funded authority that delivers subsidized hearing services to eligible Australians, including pensioners, veterans, Aboriginal and Torres Strait Islander people, and young Australians under 26. While not structured as a charity, it serves as the primary hearing healthcare provider across the country, operating clinics nationwide and supplying hearing devices to those who qualify for government support.

''National Advocacy and Peak Bodies'':

[https://www.health.gov.au/topics/ear-health-and-hearing/support-services Deafness Forum Australia] operates as the national peak body representing the interests of all Australians with hearing loss. The organization advocates for policy changes, promotes hearing health awareness, and connects individuals with resources across the spectrum of hearing impairment. Deafness Forum Australia plays a crucial role in ensuring that hearing loss remains visible in national health discussions and that the needs of the deaf and hard-of-hearing community influence government policy.

'''Audiology Australia (AudA)''' serves as the professional association for audiologists, establishing clinical standards and ethical guidelines for hearing healthcare practitioners. Although not a charity, AudA maintains partnerships with various hearing organizations and provides the public with access to qualified audiologists through its member directory.

{{HTitle|References}}
{{reflist}}

==External Links==
*	 https://audiology.asn.au/
*	 https://www.audiology.org/news-and-publications/audiology-today/articles/a-hearing-report-from-australia/
*	 https://audiology.asn.au/standards-guidelines/scope-of-practice/
* 	 https://www.auditdata.com/insights/cases/enhancing-audiology-care-in-australia-territory-hearings-success-with-manage-software
*	 https://www.health.gov.au/topics/ear-health-and-hearing/support-services
*	 https://www.acnc.gov.au/charity/charities/f6db5b5d-3aaf-e811-a963-000d3ad24077/profile

{{Global_Audiology Authors
|name1=Biraj Bhattarai
|name2=Sajana Aryal
|role1=Contributor
|role2=Contributor
|linkedin1=https://www.linkedin.com/in/biraj-bhattarai-3172931a3
|linkedin2=https://www.linkedin.com/in/sajana-aryal-209612187/
}}

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{{CountryHeader|File:Australia (orthographic projection).svg|https://en.wikipedia.org/wiki/Australia}}
{{HTitle|General Information}}
[https://en.wikipedia.org/wiki/Australia Australia], officially the Commonwealth of Australia is a country comprising the mainland of the Australian continent, the island of Tasmania, and numerous smaller islands. Australia has a population of approximately 27.7 million. Aboriginal and Torres Strait Islander peoples make up approximately 3.2% of the Australian population. Although English is not the official language in law, it is the de facto official and national language. The 2021 Census found that 167 Indigenous languages were spoken at home by Indigenous Australians. 

{{HTitle|History of Audiology}}
Audiology emerged as a profession in Australia during the 1940s. Two major events drove its establishment as a medical specialty. First, many servicemen returned from World War II suffering from hearing loss caused by injuries and noise exposure. Second, the 1940-41 rubella epidemic damaged the hearing of numerous children. These circumstances led Australia to establish the [https://www.nal.gov.au/ National Acoustics Laboratory (NAL)] in 1947. The laboratory began testing hearing and fitting hearing aids for veterans and affected children. Twenty years later, in 1967, NAL expanded its services to include pensioners. 

A pivotal moment in establishing audiology as a distinct profession came when NAL's original director determined that audiological functions such as hearing testing, hearing aid fitting, and associated rehabilitation should be performed by professionally qualified personnel rather than technicians. The initial debate centered on whether preschool teachers or psychologists would be better suited for these roles. Psychologists were ultimately selected, reportedly because they were considered better equipped to identify malingering among veterans. The first audiologists in Australia were psychologists who undertook postgraduate training in speech therapy and hearing care.

Throughout the 1950s and 1960s, the number of practicing audiologists grew steadily, driven primarily by NAL's expansion and, in the latter half of this period, by the creation of positions in hospital settings. Around 1960, informal gatherings of audiologists began in Sydney and Melbourne. In Melbourne, these typically consisted of monthly lunches followed by afternoon seminars. These regional networking efforts eventually led to the formation of the Audiological Society of Australia in May 1968, which began with ten foundation members representing four states. The majority of audiologists practicing before 1970 came from psychology backgrounds, though some arrived from fields including teaching, physics, and speech pathology. Significant developments after 1970 transformed the profession considerably.

The establishment of formal audiology training programs in Brisbane, Melbourne, Sydney, and Perth marked a crucial step toward professional standardization. Audiology subjects were also incorporated into speech-language pathology courses at several institutions. The Australian government and other employers granted official recognition to audiology as a distinct profession, lending credibility and structure to the field.

Employment opportunities expanded significantly, particularly in academia and private practice. Audiological research flourished, with notable emphasis on hearing aids and cochlear implants, though research extended into many other areas as well. The scope of audiological services broadened considerably during this period. The profession established regular national conferences beginning in 1974, held biennially, along with other specialized scientific meetings. The Australian Journal of Audiology was founded in 1979, providing a dedicated publication outlet for Australian audiological research. Publications by Australian audiologists in both national and international venues increased substantially, and interactions with international colleagues intensified through conferences, visits, and collaborative research projects.

{{HTitle|Incidence and Prevalence of Hearing Loss}} 
Hearing loss represents a substantial and growing public health burden in Australia. National estimates from the Australian Government Roadmap for Hearing Health indicate that, by 2019, approximately 3.6 million Australians were living with some form of hearing impairment.

The prevalence of moderate and more severe hearing loss (≥ 40 dB HL) in children ranges from 1.04 per 1000 live births at 3 years of age to 1.57 per 1000 live births for children between 9 and 16 years of age. Mild hearing loss (&lt; 40 dB HL) increases from 0.28 per 1000 live births at 3 years to 1.68 per 1000 live births at 9 years and older.&lt;ref&gt;{{Cite journal|title=The Ages of Intervention in Regions With and Without Universal Newborn Hearing Screening and Prevalence of Childhood Hearing Impairment in Australia|url=http://www.portico.org/Portico/article?article=pf16jrb3hm|journal=Australian and New Zealand Journal of Audiology|date=2006-11-01|pages=137–150|volume=28|issue=2|doi=10.1375/audi.28.2.137|first=Teresa Y.C|last=Ching|first2=Ron|last2=Oong|first3=Emma van|last3=Wanrooy}}&lt;/ref&gt;  Among urban Australian school-aged children (5 to 7 years), the prevalence of bilateral hearing loss ≥26 dB was estimated to 2.1% as reprted in 2006.&lt;ref&gt;{{Cite journal|title=Cross-sectional prevalence and risk factors for otitis media and hearing loss in Australian children aged 5 to 7 years: a prospective cohort study|url=https://www.theajo.com/article/view/4259/html|journal=Australian Journal of Otolaryngology|date=2020-03|pages=0–0|volume=3|doi=10.21037/ajo.2020.02.02|first=Christopher G.|last=Brennan-Jones|first2=Hrehan H.|last2=Hakeem|first3=Cheryl Da|last3=Costa|first4=Weijie|last4=Weng|first5=Andrew J. O.|last5=Whitehouse|first6=Sarra E.|last6=Jamieson|first7=Robert H.|last7=Eikelboom}}&lt;/ref&gt; In a national study, the prevalence of bilateral and unilateral hearing loss ≥16 dB HL was 9.3% and 13.3%, respectively. Slight losses (16-25 dB HL) were more prevalent than mild or greater losses (≥26 dB HL). &lt;ref&gt;{{Cite journal|title=Cross-sectional epidemiology of hearing loss in Australian children aged 11–12 years old and 25-year secular trends|url=https://adc.bmj.com/lookup/doi/10.1136/archdischild-2017-313505|journal=Archives of Disease in Childhood|date=2018-06|issn=0003-9888|pages=579–585|volume=103|issue=6|doi=10.1136/archdischild-2017-313505|language=en|first=Jing|last=Wang|first2=Carlijn M P|last2=le Clercq|first3=Valerie|last3=Sung|first4=Peter|last4=Carew|first5=Richard S|last5=Liu|first6=Fiona K|last6=Mensah|first7=Rachel A|last7=Burt|first8=Lisa|last8=Gold|first9=Melissa|last9=Wake}}&lt;/ref&gt; 

In the Blue Mountains Hearing Study, 33% of older adults had some degree of hearing loss at baseline, with a 5-year incidence of 17.9%.&lt;ref&gt;{{Cite journal|title=Five-Year Incidence and Progression of Hearing Impairment in an Older Population|url=https://journals.lww.com/00003446-201103000-00010|journal=Ear &amp; Hearing|date=2011-03|issn=0196-0202|pages=251–257|volume=32|issue=2|doi=10.1097/AUD.0b013e3181fc98bd|language=en|first=Paul|last=Mitchell|first2=Bamini|last2=Gopinath|first3=Jie Jin|last3=Wang|first4=Catherine M.|last4=McMahon|first5=Julie|last5=Schneider|first6=Elena|last6=Rochtchina|first7=Stephen R.|last7=Leeder}}&lt;/ref&gt; In 2022 hearing loss was estimated to affect 74% of people aged over 70 in Australia.&lt;ref&gt;{{Cite journal|title=Hearing loss, cognition, and risk of neurocognitive disorder: evidence from a longitudinal cohort study of older adult Australians|url=https://www.tandfonline.com/doi/full/10.1080/13825585.2020.1857328|journal=Aging, Neuropsychology, and Cognition|date=2022-01-02|issn=1382-5585|pages=121–138|volume=29|issue=1|doi=10.1080/13825585.2020.1857328|language=en|first=Paul A.|last=Strutt|first2=Amanda J.|last2=Barnier|first3=Greg|last3=Savage|first4=Gabrielle|last4=Picard|first5=Nicole A.|last5=Kochan|first6=Perminder|last6=Sachdev|first7=Brian|last7=Draper|first8=Henry|last8=Brodaty}}&lt;/ref&gt;

{{HTitle|Information About Audiology Education in Australia}}
Audiologists in Australia complete a minimum of five years of university education, including a two-year master’s level audiology program accredited by Audiology Australia. Currently, seven Australian universities offer Audiology Australia–accredited postgraduate audiology programs. Following graduation, Audiology Australia members are required to complete a one-year supervised clinical internship.During this period, interns practice under the supervision of an Audiology Australia Accredited Audiologist, facilitating a structured transition into professional practice and ensuring high standards of service delivery (Audiology Australia, n.d.)

Currently, six universities across Australia offer master's programs in audiology. These include Macquarie University, University of Queensland, Flinders University, Melbourne University, La Trobe University, and the University of Western Australia. The University of Western Australia offers a joint master's/PhD in clinical audiology. Australian master's programs require graduates to meet a set of clinical competencies and complete 250 hours of clinical experience before graduation. Graduates must also complete a one-year clinical internship if they want to provide services to clients in the large government-funded sector. During this year, experienced audiologists supervise interns and prepare them for independent clinical practice. After completing the internship, new graduates receive certification from either Audiology Australia or the Australian College of Audiology. Both organizations represent audiologists professionally in Australia. 

{{HTitle|Scope of Practice and Licensing}}
Audiologists need to meet the relevant membership and clinical competency requirements set by Australian Practitioner Professional Bodies to practice in Australia. They need to hold full membership in Audiology Australia with a Certificate of Clinical Practice (CCP) and/or full/ordinary membership in the Australian College of Audiology (ACAud) with Hearing Rehabilitation Specialist (HRS) and Diagnostic Rehabilitation Specialist (DRS) competencies. Audiologists must complete at least the equivalent of an Australian university master's degree in clinical audiology.

Audiologists in Australia work with clients of all ages, from infants to older adults, including clients with complex needs. They assess hearing and auditory function, vestibular function, tinnitus, auditory processing function, and neural function. Audiologists perform diagnostic tests, including advanced tests using electrophysiological methods. They provide aural, vestibular, and tinnitus rehabilitation as well as communication training. Audiologists offer a range of rehabilitation services, including counseling and prescribing and fitting various devices and aids. These include bone conduction aids, FM and other remote sensing systems, hearing aids, hearing assistive technology, and earplugs (custom noise/swim/musician plugs). Audiologists possess knowledge of implantable devices such as cochlear implants, middle ear implantable hearing aids, fully implantable hearing aids, and bone anchored hearing aids.They collaborate with other professionals when applying these devices in rehabilitation. 

In addition to audiologists, hearing services in Australia are also provided by audiometrists. Audiometrists typically complete vocational or industry-based training programs focused on hearing assessment and hearing aid provision. Their scope of practice generally includes conducting hearing tests, fitting and managing hearing aids, and providing basic hearing rehabilitation services, particularly in community and private practice settings.

Compared to audiologists, audiometrists usually have a more limited scope of practice, particularly in areas such as complex diagnostic testing, vestibular assessment, and electrophysiological measures. Audiometrists may be affiliated with professional organizations such as the Australian College of Audiology or other industry bodies.

{{HTitle|Professional and Regulatory Bodies}}
Australia does not have a single statutory licensing authority for audiologists or audiometrists. Instead, hearing care professionals are represented by three Practitioner Professional Bodies:[https://audiology.asn.au/Audiology Australia], [https://www.acaud.com.au/Australian College of Audiology (ACAud)], and [https://www.haasa.org.au/Hearing Aid Audiometrist Society of Australia (HAASA)]. Audiology Australia serves as the leading accrediting body for audiologists and supports the profession through accreditation, education and professional development, advocacy, and networking. ACAud represents both audiologists and audiometrists, while HAASA specifically represents audiometrists.

Audiologists must meet membership and clinical competency requirements set by Audiology Australia and/or ACAud, typically including completion of a master’s-level degree in clinical audiology and a supervised clinical internship. New members of Audiology Australia become accredited audiologists after completing this internship. Accreditation is valid for one year and must be renewed annually through continuing professional development (CPD). Members are required to demonstrate sufficient professional development over the previous 12 months, which may include conferences, seminars, training courses, and research activities.

Audiometrists must meet the relevant membership and competency requirements of ACAud and/or HAASA and typically complete at least a diploma-level Technical and Further Education (TAFE) qualification in audiometry or a bachelor’s degree. These professional bodies define scope of practice, competency standards, and continuing professional development requirements; however, they do not function as statutory regulatory authorities.&lt;ref&gt;{{Cite report|title=Scope of Practice for Audiologists and Audiometrists|url=https://www.acaud.com.au/wp-content/uploads/2019/07/Scope-of-Practice-for-Audiologists-and-Audiometrists.pdf|institution=Audiology Australia; Australian College of Audiology; Hearing Aid Audiometrist Society of Australia|date=2016-09-20}}&lt;/ref&gt;

'''Code of Conduct'''
Audiology Australia members are obliged by a code of conduct. Members must also comply with the Criminal History Policy and Mandatory Declarations Policy.On 12 September 2025, Australia's health ministers agreed to regulate audiology under the National Regulation and Accreditation Scheme. This development requires the Australian Health Practitioner Regulation Agency (AHPRA), which administers NRAS, to provide certification and accreditation to audiologists.

{{HTitle|Ongoing audiology research}}
The NAL leads audiology research in Australia by focusing on unmet needs within the hearing sector.NAL's strategic approach drives meaningful results through targeted research initiatives and collaborative partnerships. 
==== Accessible and Personalized Care ==== 
NAL researchers are working to make hearing services more accessible, acceptable, and personalized for all Australians. They are focusing on understanding community needs and are developing solutions tailored to different populations. Researchers are placing particular emphasis on reducing the impact of otitis media-related hearing loss in Aboriginal and Torres Strait Islander children. Teams are co-developing solutions with communities to address hearing deprivation in these vulnerable populations.

''Artificial Intelligence Applications'': NAL is actively exploring how artificial intelligence can improve hearing healthcare delivery. Researchers are applying AI to enhance screening, diagnosis, management, and ongoing support for people with hearing loss. These innovations are aiming to make hearing services more efficient and effective.

''Targeted Solutions for Listening Difficulties'': NAL is developing targeted solutions for different groups of people who are experiencing listening difficulties during daily communication. Researchers are recognizing that various populations face unique challenges and are requiring customized approaches to hearing rehabilitation.

''Research Methodology and Collaboration'': NAL is employing a focused approach that is driving results across the hearing sector. Experienced teams of researchers, audiologists, and behavioral specialists are regularly identifying unmet needs within the field. They are establishing cross-functional, multidisciplinary project teams using a matrix structure that is leveraging different areas of expertise within the organization. NAL is implementing research and innovation methodologies to develop practical outcomes. Collaboration is forming a core function of their work. Teams are regularly partnering with a global network of research collaborators, volunteers, and industry partners worldwide to deliver various initiatives.

'''Fields of Research'''

''Adult Hearing Loss'': NAL is conducting outstanding people-focused research on adult hearing loss. Researchers are promoting evidence-based outcomes to improve quality of life for people with hearing difficulties. They are providing insights into the needs of adults with hearing loss and are developing solutions and recommendations for affected individual.

''Pediatric Hearing Loss'': NAL is performing evidence-based and patient-centered multidisciplinary research on childhood hearing loss. Researchers are working to understand the long-term effectiveness of early hearing detection and intervention programs. They are informing best practices for preventing and managing childhood hearing loss using various research methods including behavioral assessments, electrophysiological measures, and quantitative and qualitative approaches.

''Technology Development:'' NAL focuses on evaluating, researching, and developing innovative tools that address unmet needs in hearing health. Current work includes focus on advanced signal and audio processing algorithms, machine learning–based diagnostics, and digital health platforms that support clinical decision making and real-world listening assessment. Some of the current ongoing technological research includeds AI-enabled remote hearing screening, online speech testing, speech to text app, smartphone-based ecological momentary assessment, and  novel auditory evoked potential tests for hidden hearing loss. Through close collaboration with hearing aid users, clinicians, and industry partners worldwide, NAL translates research into practical technologies that improve hearing outcomes and quality of life.

{{HTitle|Challenges, Opportunities and Notes}}
Australia's audiology sector struggles with several interconnected problems, including insufficient numbers of hearing specialists in regional areas, difficulties accessing services across vast distances, requirements for culturally appropriate care, hearing loss going undetected in elderly care settings, and limited uptake of remote audiology services, even though strong government support programs exist. Primary concerns involve uneven distribution of audiologists across the country, absence of a unified nationwide approach, social stigma surrounding hearing loss, and technological barriers, all of which restrict healthcare access for people living in rural and remote locations, Indigenous Australians, and senior citizens, highlighting the need for enhanced professional training and improved digital health systems. &lt;ref&gt;{{Cite journal|last=Mui|first=Boaz|last2=Lawless|first2=Michael|last3=Timmer|first3=Barbra H. B.|last4=Gopinath|first4=Bamini|last5=Tang|first5=Diana|last6=Venning|first6=Anthony|last7=May|first7=David|last8=Muzaffar|first8=Jameel|last9=Bidargaddi|first9=Niranjan|date=2025-01-02|title=Australian hearing healthcare stakeholders’ experiences of and attitudes towards teleaudiology uptake: a qualitative study|url=https://www.tandfonline.com/doi/full/10.1080/2050571X.2024.2372171|journal=Speech, Language and Hearing|language=en|volume=28|issue=1|doi=10.1080/2050571X.2024.2372171|issn=2050-571X}}&lt;/ref&gt; 

''Workforce &amp; Access Issues:''
Regional shortages: Audiologists concentrate in cities, leaving rural/remote areas under-serviced
Geographic barriers: Vast distances make consistent care difficult for remote communities
Aged care gaps: Hearing loss frequently missed due to lack of staff training and awareness.&lt;ref&gt;{{Cite journal|last=El-Saifi|first=Najwan|last2=Campbell|first2=Megan E.J.|last3=George|first3=Neha|last4=Keay|first4=Lisa|last5=Kumaran|first5=Sheela|last6=Meyer|first6=Carly|last7=Miller Amberber|first7=Amanda|last8=Newall|first8=John|last9=Dawes|first9=Piers|date=2025-09-05|title=Barriers and enablers to hearing service provision in aged care settings in Australia: perspectives from hearing clinicians|url=https://www.tandfonline.com/doi/full/10.1080/14992027.2025.2554236|journal=International Journal of Audiology|language=en|pages=1–12|doi=10.1080/14992027.2025.2554236|issn=1499-2027}}&lt;/ref&gt;
Poor integration: Limited connection between audiology and other allied health services in aged care.

''Service Delivery &amp; Technology Problems:''

Tele-audiology underutilized: Barriers include poor infrastructure, restrictive policies, inadequate funding, and limited clinician training
Digital literacy gaps: Patients, especially elderly, struggle with comfort and skills for remote care
Digital therapeutics challenges: New technologies need stronger evidence bases before widespread adoption
Technology implementation hurdles: Clinicians lack confidence in integrating new digital tools.

''Patient &amp; Cultural Barriers:''

Stigma: Hearing loss viewed as &quot;invisible disability,&quot; causing delayed treatment
Low public awareness: Limited understanding of hearing loss impacts and treatment options
Multicultural needs: Services must be culturally sensitive and linguistically appropriate

{{HTitle|Audiology Service Providers and Advocacy Groups}}
Australia hosts a robust network of organizations dedicated to supporting individuals with hearing loss, ranging from frontline service providers to advocacy groups and research institutions. 

''Leading Service Providers'':

'''NextSense''' stands as one of Australia's premier not-for-profit organizations addressing both hearing and vision loss. The organization delivers comprehensive clinical services, conducts research, and operates Australia's largest cochlear implant program. Beyond direct services, NextSense provides educational support and therapeutic interventions for children and adults navigating sensory loss.

'''Hearing Australia''' functions as a government-funded authority that delivers subsidized hearing services to eligible Australians, including pensioners, veterans, Aboriginal and Torres Strait Islander people, and young Australians under 26. While not structured as a charity, it serves as the primary hearing healthcare provider across the country, operating clinics nationwide and supplying hearing devices to those who qualify for government support.

''National Advocacy and Peak Bodies'':

[https://www.health.gov.au/topics/ear-health-and-hearing/support-services Deafness Forum Australia] operates as the national peak body representing the interests of all Australians with hearing loss. The organization advocates for policy changes, promotes hearing health awareness, and connects individuals with resources across the spectrum of hearing impairment. Deafness Forum Australia plays a crucial role in ensuring that hearing loss remains visible in national health discussions and that the needs of the deaf and hard-of-hearing community influence government policy.

'''Audiology Australia (AudA)''' serves as the professional association for audiologists, establishing clinical standards and ethical guidelines for hearing healthcare practitioners. Although not a charity, AudA maintains partnerships with various hearing organizations and provides the public with access to qualified audiologists through its member directory.

{{HTitle|References}}
{{reflist}}

==External Links==
*	 https://audiology.asn.au/
*	 https://www.audiology.org/news-and-publications/audiology-today/articles/a-hearing-report-from-australia/
*	 https://audiology.asn.au/standards-guidelines/scope-of-practice/
* 	 https://www.auditdata.com/insights/cases/enhancing-audiology-care-in-australia-territory-hearings-success-with-manage-software
*	 https://www.health.gov.au/topics/ear-health-and-hearing/support-services
*	 https://www.acnc.gov.au/charity/charities/f6db5b5d-3aaf-e811-a963-000d3ad24077/profile

{{Global_Audiology Authors
|name1=Biraj Bhattarai
|name2=Sajana Aryal
|role1=Contributor
|role2=Contributor
|linkedin1=https://www.linkedin.com/in/biraj-bhattarai-3172931a3
|linkedin2=https://www.linkedin.com/in/sajana-aryal-209612187/
}}

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{{HTitle|General Information}}
[https://en.wikipedia.org/wiki/Australia Australia], officially the Commonwealth of Australia is a country comprising the mainland of the Australian continent, the island of Tasmania, and numerous smaller islands. Australia has a population of approximately 27.7 million. Aboriginal and Torres Strait Islander peoples make up approximately 3.2% of the Australian population. Although English is not the official language in law, it is the de facto official and national language. The 2021 Census found that 167 Indigenous languages were spoken at home by Indigenous Australians. 

{{HTitle|History of Audiology}}
Audiology emerged as a profession in Australia during the 1940s. Two major events drove its establishment as a medical specialty. First, many servicemen returned from World War II suffering from hearing loss caused by injuries and noise exposure. Second, the 1940-41 rubella epidemic damaged the hearing of numerous children. These circumstances led Australia to establish the [https://www.nal.gov.au/ National Acoustics Laboratory (NAL)] in 1947. The laboratory began testing hearing and fitting hearing aids for veterans and affected children. Twenty years later, in 1967, NAL expanded its services to include pensioners. 

A pivotal moment in establishing audiology as a distinct profession came when NAL's original director determined that audiological functions such as hearing testing, hearing aid fitting, and associated rehabilitation should be performed by professionally qualified personnel rather than technicians. The initial debate centered on whether preschool teachers or psychologists would be better suited for these roles. Psychologists were ultimately selected, reportedly because they were considered better equipped to identify malingering among veterans. The first audiologists in Australia were psychologists who undertook postgraduate training in speech therapy and hearing care.

Throughout the 1950s and 1960s, the number of practicing audiologists grew steadily, driven primarily by NAL's expansion and, in the latter half of this period, by the creation of positions in hospital settings. Around 1960, informal gatherings of audiologists began in Sydney and Melbourne. In Melbourne, these typically consisted of monthly lunches followed by afternoon seminars. These regional networking efforts eventually led to the formation of the Audiological Society of Australia in May 1968, which began with ten foundation members representing four states. The majority of audiologists practicing before 1970 came from psychology backgrounds, though some arrived from fields including teaching, physics, and speech pathology. Significant developments after 1970 transformed the profession considerably.

The establishment of formal audiology training programs in Brisbane, Melbourne, Sydney, and Perth marked a crucial step toward professional standardization. Audiology subjects were also incorporated into speech-language pathology courses at several institutions. The Australian government and other employers granted official recognition to audiology as a distinct profession, lending credibility and structure to the field.

Employment opportunities expanded significantly, particularly in academia and private practice. Audiological research flourished, with notable emphasis on hearing aids and cochlear implants, though research extended into many other areas as well. The scope of audiological services broadened considerably during this period. The profession established regular national conferences beginning in 1974, held biennially, along with other specialized scientific meetings. The Australian Journal of Audiology was founded in 1979, providing a dedicated publication outlet for Australian audiological research. Publications by Australian audiologists in both national and international venues increased substantially, and interactions with international colleagues intensified through conferences, visits, and collaborative research projects.

{{HTitle|Incidence and Prevalence of Hearing Loss}} 
Hearing loss represents a substantial and growing public health burden in Australia. National estimates from the Australian Government Roadmap for Hearing Health indicate that, by 2019, approximately 3.6 million Australians were living with some form of hearing impairment.

The prevalence of moderate and more severe hearing loss (≥ 40 dB HL) in children ranges from 1.04 per 1000 live births at 3 years of age to 1.57 per 1000 live births for children between 9 and 16 years of age. Mild hearing loss (&lt; 40 dB HL) increases from 0.28 per 1000 live births at 3 years to 1.68 per 1000 live births at 9 years and older.&lt;ref&gt;{{Cite journal|title=The Ages of Intervention in Regions With and Without Universal Newborn Hearing Screening and Prevalence of Childhood Hearing Impairment in Australia|url=http://www.portico.org/Portico/article?article=pf16jrb3hm|journal=Australian and New Zealand Journal of Audiology|date=2006-11-01|pages=137–150|volume=28|issue=2|doi=10.1375/audi.28.2.137|first=Teresa Y.C|last=Ching|first2=Ron|last2=Oong|first3=Emma van|last3=Wanrooy}}&lt;/ref&gt;  Among urban Australian school-aged children (5 to 7 years), the prevalence of bilateral hearing loss ≥26 dB was estimated to 2.1% as reprted in 2006.&lt;ref&gt;{{Cite journal|title=Cross-sectional prevalence and risk factors for otitis media and hearing loss in Australian children aged 5 to 7 years: a prospective cohort study|url=https://www.theajo.com/article/view/4259/html|journal=Australian Journal of Otolaryngology|date=2020-03|pages=0–0|volume=3|doi=10.21037/ajo.2020.02.02|first=Christopher G.|last=Brennan-Jones|first2=Hrehan H.|last2=Hakeem|first3=Cheryl Da|last3=Costa|first4=Weijie|last4=Weng|first5=Andrew J. O.|last5=Whitehouse|first6=Sarra E.|last6=Jamieson|first7=Robert H.|last7=Eikelboom}}&lt;/ref&gt; In a national study, the prevalence of bilateral and unilateral hearing loss ≥16 dB HL was 9.3% and 13.3%, respectively. Slight losses (16-25 dB HL) were more prevalent than mild or greater losses (≥26 dB HL). &lt;ref&gt;{{Cite journal|title=Cross-sectional epidemiology of hearing loss in Australian children aged 11–12 years old and 25-year secular trends|url=https://adc.bmj.com/lookup/doi/10.1136/archdischild-2017-313505|journal=Archives of Disease in Childhood|date=2018-06|issn=0003-9888|pages=579–585|volume=103|issue=6|doi=10.1136/archdischild-2017-313505|language=en|first=Jing|last=Wang|first2=Carlijn M P|last2=le Clercq|first3=Valerie|last3=Sung|first4=Peter|last4=Carew|first5=Richard S|last5=Liu|first6=Fiona K|last6=Mensah|first7=Rachel A|last7=Burt|first8=Lisa|last8=Gold|first9=Melissa|last9=Wake}}&lt;/ref&gt; 

In the Blue Mountains Hearing Study, 33% of older adults had some degree of hearing loss at baseline, with a 5-year incidence of 17.9%.&lt;ref&gt;{{Cite journal|title=Five-Year Incidence and Progression of Hearing Impairment in an Older Population|url=https://journals.lww.com/00003446-201103000-00010|journal=Ear &amp; Hearing|date=2011-03|issn=0196-0202|pages=251–257|volume=32|issue=2|doi=10.1097/AUD.0b013e3181fc98bd|language=en|first=Paul|last=Mitchell|first2=Bamini|last2=Gopinath|first3=Jie Jin|last3=Wang|first4=Catherine M.|last4=McMahon|first5=Julie|last5=Schneider|first6=Elena|last6=Rochtchina|first7=Stephen R.|last7=Leeder}}&lt;/ref&gt; In 2022 hearing loss was estimated to affect 74% of people aged over 70 in Australia.&lt;ref&gt;{{Cite journal|title=Hearing loss, cognition, and risk of neurocognitive disorder: evidence from a longitudinal cohort study of older adult Australians|url=https://www.tandfonline.com/doi/full/10.1080/13825585.2020.1857328|journal=Aging, Neuropsychology, and Cognition|date=2022-01-02|issn=1382-5585|pages=121–138|volume=29|issue=1|doi=10.1080/13825585.2020.1857328|language=en|first=Paul A.|last=Strutt|first2=Amanda J.|last2=Barnier|first3=Greg|last3=Savage|first4=Gabrielle|last4=Picard|first5=Nicole A.|last5=Kochan|first6=Perminder|last6=Sachdev|first7=Brian|last7=Draper|first8=Henry|last8=Brodaty}}&lt;/ref&gt;

{{HTitle| Audiology Education in Australia}}
Audiologists in Australia complete a minimum of five years of university education, including a two-year master’s level audiology program accredited by Audiology Australia. Currently, seven Australian universities offer Audiology Australia–accredited postgraduate audiology programs. Following graduation, Audiology Australia members are required to complete a one-year supervised clinical internship.During this period, interns practice under the supervision of an Audiology Australia Accredited Audiologist, facilitating a structured transition into professional practice and ensuring high standards of service delivery (Audiology Australia, n.d.)

Currently, six universities across Australia offer master's programs in audiology. These include Macquarie University, University of Queensland, Flinders University, Melbourne University, La Trobe University, and the University of Western Australia. The University of Western Australia offers a joint master's/PhD in clinical audiology. Australian master's programs require graduates to meet a set of clinical competencies and complete 250 hours of clinical experience before graduation. Graduates must also complete a one-year clinical internship if they want to provide services to clients in the large government-funded sector. During this year, experienced audiologists supervise interns and prepare them for independent clinical practice. After completing the internship, new graduates receive certification from either Audiology Australia or the Australian College of Audiology. Both organizations represent audiologists professionally in Australia. 

{{HTitle|Scope of Practice and Licensing}}
Audiologists need to meet the relevant membership and clinical competency requirements set by Australian Practitioner Professional Bodies to practice in Australia. They need to hold full membership in Audiology Australia with a Certificate of Clinical Practice (CCP) and/or full/ordinary membership in the Australian College of Audiology (ACAud) with Hearing Rehabilitation Specialist (HRS) and Diagnostic Rehabilitation Specialist (DRS) competencies. Audiologists must complete at least the equivalent of an Australian university master's degree in clinical audiology.

Audiologists in Australia work with clients of all ages, from infants to older adults, including clients with complex needs. They assess hearing and auditory function, vestibular function, tinnitus, auditory processing function, and neural function. Audiologists perform diagnostic tests, including advanced tests using electrophysiological methods. They provide aural, vestibular, and tinnitus rehabilitation as well as communication training. Audiologists offer a range of rehabilitation services, including counseling and prescribing and fitting various devices and aids. These include bone conduction aids, FM and other remote sensing systems, hearing aids, hearing assistive technology, and earplugs (custom noise/swim/musician plugs). Audiologists possess knowledge of implantable devices such as cochlear implants, middle ear implantable hearing aids, fully implantable hearing aids, and bone anchored hearing aids.They collaborate with other professionals when applying these devices in rehabilitation. 

In addition to audiologists, hearing services in Australia are also provided by audiometrists. Audiometrists typically complete vocational or industry-based training programs focused on hearing assessment and hearing aid provision. Their scope of practice generally includes conducting hearing tests, fitting and managing hearing aids, and providing basic hearing rehabilitation services, particularly in community and private practice settings.

Compared to audiologists, audiometrists usually have a more limited scope of practice, particularly in areas such as complex diagnostic testing, vestibular assessment, and electrophysiological measures. Audiometrists may be affiliated with professional organizations such as the Australian College of Audiology or other industry bodies.

{{HTitle|Professional and Regulatory Bodies}}
Australia does not have a single statutory licensing authority for audiologists or audiometrists. Instead, hearing care professionals are represented by two main Practitioner Professional Bodies: [https://audiology.asn.au/Audiology Australia] and [https://www.acaud.com.au/Australian College of Audiology incorporating HAASA (ACAud inc. HAASA)]. Audiology Australia is the peak and largest accrediting professional member body for audiologists in Australia, while ACAud inc. HAASA represents both audiologists and audiometrists.

Audiologists must meet membership and clinical competency requirements set by Audiology Australia and/or ACAud, typically including completion of a master’s-level degree in clinical audiology and a supervised clinical internship. New members of Audiology Australia become accredited audiologists after completing this internship. Accreditation is valid for one year and must be renewed annually through continuing professional development (CPD). Members are required to demonstrate sufficient professional development over the previous 12 months, which may include conferences, seminars, training courses, and research activities.

Audiometrists must meet the relevant membership and competency requirements of ACAud and/or HAASA and typically complete at least a diploma-level Technical and Further Education (TAFE) qualification in audiometry or a bachelor’s degree. These professional bodies define scope of practice, competency standards, and continuing professional development requirements; however, they do not function as statutory regulatory authorities.&lt;ref&gt;{{Cite report|title=Scope of Practice for Audiologists and Audiometrists|url=https://www.acaud.com.au/wp-content/uploads/2019/07/Scope-of-Practice-for-Audiologists-and-Audiometrists.pdf|institution=Audiology Australia; Australian College of Audiology; Hearing Aid Audiometrist Society of Australia|date=2016-09-20}}&lt;/ref&gt;

'''Code of Conduct'''
Audiology Australia members are obliged by a code of conduct. Members must also comply with the Criminal History Policy and Mandatory Declarations Policy.On 12 September 2025, Australia's health ministers agreed to regulate audiology under the National Regulation and Accreditation Scheme. This development requires the Australian Health Practitioner Regulation Agency (AHPRA), which administers NRAS, to provide certification and accreditation to audiologists.

{{HTitle|Ongoing audiology research}}
Audiology research in Australia are conducted across a wide range of institutions, including universities, independent research institutes, and national organizations. Major contributors include the [https://www.nal.gov.au/National Acoustic Laboratories (NAL)], [https://www.earscience.org.au/Ear Science Institute Australia], and leading universities such as the [https://healthsciences.unimelb.edu.au/departments/audiology-and-speech-pathology University of Melbourne], [https://www.mq.edu.au/faculty-of-medicine-health-and-human-sciences/departments-and-schools/department-of-linguistics/our-research/audiology-and-hearing Macquarie University], University of Queensland, La Trobe University, and Flinders University.&lt;ref&gt;{{Cite web|title=Our research|url=https://www.nal.gov.au/our-research/|website=National Acoustic Laboratories|access-date=2026-04-28}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|title=Ear Science Institute Australia|url=https://www.earscience.org.au/|website=Ear
Science Institute Australia|access-date=2026-04-28}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|title=Audiology and hearing research|url=https://www.mq.edu.au/faculty-of-medicine-health-and-human-sciences/departments-and-schools/department-of-linguistics/our-research/audiology-and-hearing|website=Macquarie University|access-date=2026-04-28}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|title=Audiology and Speech Pathology|url=https://healthsciences.unimelb.edu.au/departments/audiology-and-speech-pathology|website=University of Melbourne|access-date=2026-04-28}}&lt;/ref&gt;

Research areas include hearing assessment, hearing devices, cochlear implants, tinnitus, vestibular disorders, speech perception, auditory neuroscience, digital health, artificial intelligence, and public-health approaches to hearing care. NAL continues to play a major role through research on adult and paediatric hearing loss, technology, personalized care, and listening difficulties. Its current work includes improving access to hearing services, developing targeted solutions for everyday listening difficulties, and exploring the use of artificial intelligence to improve hearing screening, diagnosis, management, and ongoing support.&lt;ref&gt;{{Cite web|title=Our research|url=https://www.nal.gov.au/our-research/|website=National Acoustic Laboratories|access-date=2026-04-28}}&lt;/ref&gt;

A major national priority is improving hearing health equity. Australian researchers and service providers are working to reduce the impact of otitis media-related hearing loss among Aboriginal and Torres Strait Islander children, often through community-informed and collaborative approaches. Australia also has a national Hearing Services Program that provides subsidised hearing services and devices to eligible Australians with hearing loss. Eligibility includes children and young adults under 26, Aboriginal and Torres Strait Islander people who meet program criteria, and many older adults who qualify through pensioner or veteran status.&lt;ref&gt;{{Cite web|title=Hearing Services Program|url=https://www.health.gov.au/our-work/hearing-services-program|website=Australian Government Department of Health, Disability and Ageing|access-date=2026-04-28}}&lt;/ref&gt;

{{HTitle|Challenges, Opportunities and Notes}}
Australia's audiology sector struggles with several interconnected problems, including insufficient numbers of hearing specialists in regional areas, difficulties accessing services across vast distances, requirements for culturally appropriate care, hearing loss going undetected in elderly care settings, and limited uptake of remote audiology services, even though strong government support programs exist. Primary concerns involve uneven distribution of audiologists across the country, absence of a unified nationwide approach, social stigma surrounding hearing loss, and technological barriers, all of which restrict healthcare access for people living in rural and remote locations, Indigenous Australians, and senior citizens, highlighting the need for enhanced professional training and improved digital health systems. &lt;ref&gt;{{Cite journal|last=Mui|first=Boaz|last2=Lawless|first2=Michael|last3=Timmer|first3=Barbra H. B.|last4=Gopinath|first4=Bamini|last5=Tang|first5=Diana|last6=Venning|first6=Anthony|last7=May|first7=David|last8=Muzaffar|first8=Jameel|last9=Bidargaddi|first9=Niranjan|date=2025-01-02|title=Australian hearing healthcare stakeholders’ experiences of and attitudes towards teleaudiology uptake: a qualitative study|url=https://www.tandfonline.com/doi/full/10.1080/2050571X.2024.2372171|journal=Speech, Language and Hearing|language=en|volume=28|issue=1|doi=10.1080/2050571X.2024.2372171|issn=2050-571X}}&lt;/ref&gt; 

''Workforce &amp; Access Issues:''
Regional shortages: Audiologists concentrate in cities, leaving rural/remote areas under-serviced
Geographic barriers: Vast distances make consistent care difficult for remote communities
Aged care gaps: Hearing loss frequently missed due to lack of staff training and awareness.&lt;ref&gt;{{Cite journal|last=El-Saifi|first=Najwan|last2=Campbell|first2=Megan E.J.|last3=George|first3=Neha|last4=Keay|first4=Lisa|last5=Kumaran|first5=Sheela|last6=Meyer|first6=Carly|last7=Miller Amberber|first7=Amanda|last8=Newall|first8=John|last9=Dawes|first9=Piers|date=2025-09-05|title=Barriers and enablers to hearing service provision in aged care settings in Australia: perspectives from hearing clinicians|url=https://www.tandfonline.com/doi/full/10.1080/14992027.2025.2554236|journal=International Journal of Audiology|language=en|pages=1–12|doi=10.1080/14992027.2025.2554236|issn=1499-2027}}&lt;/ref&gt;
Poor integration: Limited connection between audiology and other allied health services in aged care.

''Service Delivery &amp; Technology Problems:''

Tele-audiology underutilized: Barriers include poor infrastructure, restrictive policies, inadequate funding, and limited clinician training
Digital literacy gaps: Patients, especially elderly, struggle with comfort and skills for remote care
Digital therapeutics challenges: New technologies need stronger evidence bases before widespread adoption
Technology implementation hurdles: Clinicians lack confidence in integrating new digital tools.

''Patient &amp; Cultural Barriers:''

Stigma: Hearing loss viewed as &quot;invisible disability,&quot; causing delayed treatment
Low public awareness: Limited understanding of hearing loss impacts and treatment options
Multicultural needs: Services must be culturally sensitive and linguistically appropriate

{{HTitle|Audiology Service Providers and Advocacy Groups}}
Australia hosts a robust network of organizations dedicated to supporting individuals with hearing loss, ranging from frontline service providers to advocacy groups and research institutions. 

''Leading Service Providers'':

'''NextSense''' stands as one of Australia's premier not-for-profit organizations addressing both hearing and vision loss. The organization delivers comprehensive clinical services, conducts research, and operates Australia's largest cochlear implant program. Beyond direct services, NextSense provides educational support and therapeutic interventions for children and adults navigating sensory loss.

'''Hearing Australia''' functions as a government-funded authority that delivers subsidized hearing services to eligible Australians, including pensioners, veterans, Aboriginal and Torres Strait Islander people, and young Australians under 26. While not structured as a charity, it serves as the primary hearing healthcare provider across the country, operating clinics nationwide and supplying hearing devices to those who qualify for government support.

''National Advocacy and Peak Bodies'':

[https://www.health.gov.au/topics/ear-health-and-hearing/support-services Deafness Forum Australia] operates as the national peak body representing the interests of all Australians with hearing loss. The organization advocates for policy changes, promotes hearing health awareness, and connects individuals with resources across the spectrum of hearing impairment. Deafness Forum Australia plays a crucial role in ensuring that hearing loss remains visible in national health discussions and that the needs of the deaf and hard-of-hearing community influence government policy.

'''Audiology Australia (AudA)''' serves as the professional association for audiologists, establishing clinical standards and ethical guidelines for hearing healthcare practitioners. Although not a charity, AudA maintains partnerships with various hearing organizations and provides the public with access to qualified audiologists through its member directory.

{{HTitle|References}}
{{reflist}}

==External Links==
*	 https://audiology.asn.au/
*	 https://www.audiology.org/news-and-publications/audiology-today/articles/a-hearing-report-from-australia/
*	 https://audiology.asn.au/standards-guidelines/scope-of-practice/
* 	 https://www.auditdata.com/insights/cases/enhancing-audiology-care-in-australia-territory-hearings-success-with-manage-software
*	 https://www.health.gov.au/topics/ear-health-and-hearing/support-services
*	 https://www.acnc.gov.au/charity/charities/f6db5b5d-3aaf-e811-a963-000d3ad24077/profile

{{Global_Audiology Authors
|name1=Biraj Bhattarai
|name2=Sajana Aryal
|role1=Contributor
|role2=Contributor
|linkedin1=https://www.linkedin.com/in/biraj-bhattarai-3172931a3
|linkedin2=https://www.linkedin.com/in/sajana-aryal-209612187/
}}

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{{CountryHeader|File:Australia (orthographic projection).svg|https://en.wikipedia.org/wiki/Australia}}
{{HTitle|General Information}}
[https://en.wikipedia.org/wiki/Australia Australia], officially the Commonwealth of Australia is a country comprising the mainland of the Australian continent, the island of Tasmania, and numerous smaller islands. Australia has a population of approximately 27.7 million. Aboriginal and Torres Strait Islander peoples make up approximately 3.2% of the Australian population. Although English is not the official language in law, it is the de facto official and national language. The 2021 Census found that 167 Indigenous languages were spoken at home by Indigenous Australians. 

{{HTitle|History of Audiology}}
Audiology emerged as a profession in Australia during the 1940s. Two major events drove its establishment as a medical specialty. First, many servicemen returned from World War II suffering from hearing loss caused by injuries and noise exposure. Second, the 1940-41 rubella epidemic damaged the hearing of numerous children. These circumstances led Australia to establish the [https://www.nal.gov.au/ National Acoustics Laboratory (NAL)] in 1947. The laboratory began testing hearing and fitting hearing aids for veterans and affected children. Twenty years later, in 1967, NAL expanded its services to include pensioners. 

A pivotal moment in establishing audiology as a distinct profession came when NAL's original director determined that audiological functions such as hearing testing, hearing aid fitting, and associated rehabilitation should be performed by professionally qualified personnel rather than technicians. The initial debate centered on whether preschool teachers or psychologists would be better suited for these roles. Psychologists were ultimately selected, reportedly because they were considered better equipped to identify malingering among veterans. The first audiologists in Australia were psychologists who undertook postgraduate training in speech therapy and hearing care.

Throughout the 1950s and 1960s, the number of practicing audiologists grew steadily, driven primarily by NAL's expansion and, in the latter half of this period, by the creation of positions in hospital settings. Around 1960, informal gatherings of audiologists began in Sydney and Melbourne. In Melbourne, these typically consisted of monthly lunches followed by afternoon seminars. These regional networking efforts eventually led to the formation of the Audiological Society of Australia in May 1968, which began with ten foundation members representing four states. The majority of audiologists practicing before 1970 came from psychology backgrounds, though some arrived from fields including teaching, physics, and speech pathology. Significant developments after 1970 transformed the profession considerably.

The establishment of formal audiology training programs in Brisbane, Melbourne, Sydney, and Perth marked a crucial step toward professional standardization. Audiology subjects were also incorporated into speech-language pathology courses at several institutions. The Australian government and other employers granted official recognition to audiology as a distinct profession, lending credibility and structure to the field.

Employment opportunities expanded significantly, particularly in academia and private practice. Audiological research flourished, with notable emphasis on hearing aids and cochlear implants, though research extended into many other areas as well. The scope of audiological services broadened considerably during this period. The profession established regular national conferences beginning in 1974, held biennially, along with other specialized scientific meetings. The Australian Journal of Audiology was founded in 1979, providing a dedicated publication outlet for Australian audiological research. Publications by Australian audiologists in both national and international venues increased substantially, and interactions with international colleagues intensified through conferences, visits, and collaborative research projects.

{{HTitle|Incidence and Prevalence of Hearing Loss}} 
Hearing loss represents a substantial and growing public health burden in Australia. National estimates from the Australian Government Roadmap for Hearing Health indicate that, by 2019, approximately 3.6 million Australians were living with some form of hearing impairment.

The prevalence of moderate and more severe hearing loss (≥ 40 dB HL) in children ranges from 1.04 per 1000 live births at 3 years of age to 1.57 per 1000 live births for children between 9 and 16 years of age. Mild hearing loss (&lt; 40 dB HL) increases from 0.28 per 1000 live births at 3 years to 1.68 per 1000 live births at 9 years and older.&lt;ref&gt;{{Cite journal|title=The Ages of Intervention in Regions With and Without Universal Newborn Hearing Screening and Prevalence of Childhood Hearing Impairment in Australia|url=http://www.portico.org/Portico/article?article=pf16jrb3hm|journal=Australian and New Zealand Journal of Audiology|date=2006-11-01|pages=137–150|volume=28|issue=2|doi=10.1375/audi.28.2.137|first=Teresa Y.C|last=Ching|first2=Ron|last2=Oong|first3=Emma van|last3=Wanrooy}}&lt;/ref&gt;  Among urban Australian school-aged children (5 to 7 years), the prevalence of bilateral hearing loss ≥26 dB was estimated to 2.1% as reprted in 2006.&lt;ref&gt;{{Cite journal|title=Cross-sectional prevalence and risk factors for otitis media and hearing loss in Australian children aged 5 to 7 years: a prospective cohort study|url=https://www.theajo.com/article/view/4259/html|journal=Australian Journal of Otolaryngology|date=2020-03|pages=0–0|volume=3|doi=10.21037/ajo.2020.02.02|first=Christopher G.|last=Brennan-Jones|first2=Hrehan H.|last2=Hakeem|first3=Cheryl Da|last3=Costa|first4=Weijie|last4=Weng|first5=Andrew J. O.|last5=Whitehouse|first6=Sarra E.|last6=Jamieson|first7=Robert H.|last7=Eikelboom}}&lt;/ref&gt; In a national study, the prevalence of bilateral and unilateral hearing loss ≥16 dB HL was 9.3% and 13.3%, respectively. Slight losses (16-25 dB HL) were more prevalent than mild or greater losses (≥26 dB HL). &lt;ref&gt;{{Cite journal|title=Cross-sectional epidemiology of hearing loss in Australian children aged 11–12 years old and 25-year secular trends|url=https://adc.bmj.com/lookup/doi/10.1136/archdischild-2017-313505|journal=Archives of Disease in Childhood|date=2018-06|issn=0003-9888|pages=579–585|volume=103|issue=6|doi=10.1136/archdischild-2017-313505|language=en|first=Jing|last=Wang|first2=Carlijn M P|last2=le Clercq|first3=Valerie|last3=Sung|first4=Peter|last4=Carew|first5=Richard S|last5=Liu|first6=Fiona K|last6=Mensah|first7=Rachel A|last7=Burt|first8=Lisa|last8=Gold|first9=Melissa|last9=Wake}}&lt;/ref&gt; 

In the Blue Mountains Hearing Study, 33% of older adults had some degree of hearing loss at baseline, with a 5-year incidence of 17.9%.&lt;ref&gt;{{Cite journal|title=Five-Year Incidence and Progression of Hearing Impairment in an Older Population|url=https://journals.lww.com/00003446-201103000-00010|journal=Ear &amp; Hearing|date=2011-03|issn=0196-0202|pages=251–257|volume=32|issue=2|doi=10.1097/AUD.0b013e3181fc98bd|language=en|first=Paul|last=Mitchell|first2=Bamini|last2=Gopinath|first3=Jie Jin|last3=Wang|first4=Catherine M.|last4=McMahon|first5=Julie|last5=Schneider|first6=Elena|last6=Rochtchina|first7=Stephen R.|last7=Leeder}}&lt;/ref&gt; In 2022 hearing loss was estimated to affect 74% of people aged over 70 in Australia.&lt;ref&gt;{{Cite journal|title=Hearing loss, cognition, and risk of neurocognitive disorder: evidence from a longitudinal cohort study of older adult Australians|url=https://www.tandfonline.com/doi/full/10.1080/13825585.2020.1857328|journal=Aging, Neuropsychology, and Cognition|date=2022-01-02|issn=1382-5585|pages=121–138|volume=29|issue=1|doi=10.1080/13825585.2020.1857328|language=en|first=Paul A.|last=Strutt|first2=Amanda J.|last2=Barnier|first3=Greg|last3=Savage|first4=Gabrielle|last4=Picard|first5=Nicole A.|last5=Kochan|first6=Perminder|last6=Sachdev|first7=Brian|last7=Draper|first8=Henry|last8=Brodaty}}&lt;/ref&gt;

{{HTitle| Audiology Education in Australia}}
Audiologists in Australia complete a minimum of five years of university education, including a two-year master’s level audiology program accredited by Audiology Australia. Currently, seven Australian universities offer Audiology Australia–accredited postgraduate audiology programs. Following graduation, Audiology Australia members are required to complete a one-year supervised clinical internship.During this period, interns practice under the supervision of an Audiology Australia Accredited Audiologist, facilitating a structured transition into professional practice and ensuring high standards of service delivery (Audiology Australia, n.d.)

Currently, six universities across Australia offer master's programs in audiology. These include Macquarie University, University of Queensland, Flinders University, Melbourne University, La Trobe University, and the University of Western Australia. The University of Western Australia offers a joint master's/PhD in clinical audiology. Australian master's programs require graduates to meet a set of clinical competencies and complete 250 hours of clinical experience before graduation. Graduates must also complete a one-year clinical internship if they want to provide services to clients in the large government-funded sector. During this year, experienced audiologists supervise interns and prepare them for independent clinical practice. After completing the internship, new graduates receive certification from either Audiology Australia or the Australian College of Audiology. Both organizations represent audiologists professionally in Australia. 

{{HTitle|Scope of Practice and Licensing}}
Audiologists need to meet the relevant membership and clinical competency requirements set by Australian Practitioner Professional Bodies to practice in Australia. They need to hold full membership in Audiology Australia with a Certificate of Clinical Practice (CCP) and/or full/ordinary membership in the Australian College of Audiology (ACAud) with Hearing Rehabilitation Specialist (HRS) and Diagnostic Rehabilitation Specialist (DRS) competencies. Audiologists must complete at least the equivalent of an Australian university master's degree in clinical audiology.

Audiologists in Australia work with clients of all ages, from infants to older adults, including clients with complex needs. They assess hearing and auditory function, vestibular function, tinnitus, auditory processing function, and neural function. Audiologists perform diagnostic tests, including advanced tests using electrophysiological methods. They provide aural, vestibular, and tinnitus rehabilitation as well as communication training. Audiologists offer a range of rehabilitation services, including counseling and prescribing and fitting various devices and aids. These include bone conduction aids, FM and other remote sensing systems, hearing aids, hearing assistive technology, and earplugs (custom noise/swim/musician plugs). Audiologists possess knowledge of implantable devices such as cochlear implants, middle ear implantable hearing aids, fully implantable hearing aids, and bone anchored hearing aids.They collaborate with other professionals when applying these devices in rehabilitation. 

In addition to audiologists, hearing services in Australia are also provided by audiometrists. Audiometrists typically complete vocational or industry-based training programs focused on hearing assessment and hearing aid provision. Their scope of practice generally includes conducting hearing tests, fitting and managing hearing aids, and providing basic hearing rehabilitation services, particularly in community and private practice settings.

Compared to audiologists, audiometrists usually have a more limited scope of practice, particularly in areas such as complex diagnostic testing, vestibular assessment, and electrophysiological measures. Audiometrists may be affiliated with professional organizations such as the Australian College of Audiology or other industry bodies.

{{HTitle|Professional and Regulatory Bodies}}
Australia does not have a single statutory licensing authority for audiologists or audiometrists. Instead, hearing care professionals are represented by two main Practitioner Professional Bodies: [https://audiology.asn.au/Audiology Australia] and [https://www.acaud.com.au/Australian College of Audiology incorporating HAASA (ACAud inc. HAASA)]. Audiology Australia is the peak and largest accrediting professional member body for audiologists in Australia, while ACAud inc. HAASA represents both audiologists and audiometrists.

Audiologists must meet membership and clinical competency requirements set by Audiology Australia and/or ACAud, typically including completion of a master’s-level degree in clinical audiology and a supervised clinical internship. New members of Audiology Australia become accredited audiologists after completing this internship. Accreditation is valid for one year and must be renewed annually through continuing professional development (CPD). Members are required to demonstrate sufficient professional development over the previous 12 months, which may include conferences, seminars, training courses, and research activities.

Audiometrists must meet the relevant membership and competency requirements of ACAud and/or HAASA and typically complete at least a diploma-level Technical and Further Education (TAFE) qualification in audiometry or a bachelor’s degree. These professional bodies define scope of practice, competency standards, and continuing professional development requirements; however, they do not function as statutory regulatory authorities.&lt;ref&gt;{{Cite report|title=Scope of Practice for Audiologists and Audiometrists|url=https://www.acaud.com.au/wp-content/uploads/2019/07/Scope-of-Practice-for-Audiologists-and-Audiometrists.pdf|institution=Audiology Australia; Australian College of Audiology; Hearing Aid Audiometrist Society of Australia|date=2016-09-20}}&lt;/ref&gt;

'''Code of Conduct'''
Audiology Australia members are obliged by a code of conduct. Members must also comply with the Criminal History Policy and Mandatory Declarations Policy.On 12 September 2025, Australia's health ministers agreed to regulate audiology under the National Regulation and Accreditation Scheme. This development requires the Australian Health Practitioner Regulation Agency (AHPRA), which administers NRAS, to provide certification and accreditation to audiologists.

{{HTitle|Ongoing audiology research}}
Audiology research in Australia are conducted across a wide range of institutions, including universities, independent research institutes, and national organizations. Key contributors include the [https://www.nal.gov.au/National Acoustic Laboratories (NAL)], [https://www.earscience.org.au/ Ear Science Institute Australia], and universities such as the [https://healthsciences.unimelb.edu.au/departments/audiology-and-speech-pathology University of Melbourne], [https://www.mq.edu.au/faculty-of-medicine-health-and-human-sciences/departments-and-schools/department-of-linguistics/our-research/audiology-and-hearing Macquarie University], University of Queensland, La Trobe University, and Flinders University.&lt;ref&gt;{{Cite web|title=Our research|url=https://www.nal.gov.au/our-research/|website=National Acoustic Laboratories|access-date=2026-04-28}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|title=Ear Science Institute Australia|url=https://www.earscience.org.au/|website=Ear Science Institute Australia|access-date=2026-04-28}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|title=Audiology and hearing research|url=https://www.mq.edu.au/faculty-of-medicine-health-and-human-sciences/departments-and-schools/department-of-linguistics/our-research/audiology-and-hearing|website=Macquarie University|access-date=2026-04-28}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|title=Audiology and Speech Pathology|url=https://healthsciences.unimelb.edu.au/departments/audiology-and-speech-pathology|website=University of Melbourne|access-date=2026-04-28}}&lt;/ref&gt;.

Research areas include hearing assessment, hearing devices, cochlear implants, tinnitus, vestibular disorders, speech perception, auditory neuroscience, digital health, artificial intelligence, and public-health approaches to hearing care. NAL continues to play a major role through research on adult and paediatric hearing loss, technology, personalized care, and listening difficulties. Its current work includes improving access to hearing services, developing targeted solutions for everyday listening difficulties, and exploring the use of artificial intelligence to improve hearing screening, diagnosis, management, and ongoing support.&lt;ref&gt;{{Cite web|title=Our research|url=https://www.nal.gov.au/our-research/|website=National Acoustic Laboratories|access-date=2026-04-28}}&lt;/ref&gt;

A major national priority is improving hearing health equity. Australian researchers and service providers are working to reduce the impact of otitis media-related hearing loss among Aboriginal and Torres Strait Islander children, often through community-informed and collaborative approaches. Australia also has a national Hearing Services Program that provides subsidised hearing services and devices to eligible Australians with hearing loss. Eligibility includes children and young adults under 26, Aboriginal and Torres Strait Islander people who meet program criteria, and many older adults who qualify through pensioner or veteran status.&lt;ref&gt;{{Cite web|title=Hearing Services Program|url=https://www.health.gov.au/our-work/hearing-services-program|website=Australian Government Department of Health, Disability and Ageing|access-date=2026-04-28}}&lt;/ref&gt;

{{HTitle|Challenges, Opportunities and Notes}}
Australia's audiology sector struggles with several interconnected problems, including insufficient numbers of hearing specialists in regional areas, difficulties accessing services across vast distances, requirements for culturally appropriate care, hearing loss going undetected in elderly care settings, and limited uptake of remote audiology services, even though strong government support programs exist. Primary concerns involve uneven distribution of audiologists across the country, absence of a unified nationwide approach, social stigma surrounding hearing loss, and technological barriers, all of which restrict healthcare access for people living in rural and remote locations, Indigenous Australians, and senior citizens, highlighting the need for enhanced professional training and improved digital health systems. &lt;ref&gt;{{Cite journal|last=Mui|first=Boaz|last2=Lawless|first2=Michael|last3=Timmer|first3=Barbra H. B.|last4=Gopinath|first4=Bamini|last5=Tang|first5=Diana|last6=Venning|first6=Anthony|last7=May|first7=David|last8=Muzaffar|first8=Jameel|last9=Bidargaddi|first9=Niranjan|date=2025-01-02|title=Australian hearing healthcare stakeholders’ experiences of and attitudes towards teleaudiology uptake: a qualitative study|url=https://www.tandfonline.com/doi/full/10.1080/2050571X.2024.2372171|journal=Speech, Language and Hearing|language=en|volume=28|issue=1|doi=10.1080/2050571X.2024.2372171|issn=2050-571X}}&lt;/ref&gt; 

''Workforce &amp; Access Issues:''
Regional shortages: Audiologists concentrate in cities, leaving rural/remote areas under-serviced
Geographic barriers: Vast distances make consistent care difficult for remote communities
Aged care gaps: Hearing loss frequently missed due to lack of staff training and awareness.&lt;ref&gt;{{Cite journal|last=El-Saifi|first=Najwan|last2=Campbell|first2=Megan E.J.|last3=George|first3=Neha|last4=Keay|first4=Lisa|last5=Kumaran|first5=Sheela|last6=Meyer|first6=Carly|last7=Miller Amberber|first7=Amanda|last8=Newall|first8=John|last9=Dawes|first9=Piers|date=2025-09-05|title=Barriers and enablers to hearing service provision in aged care settings in Australia: perspectives from hearing clinicians|url=https://www.tandfonline.com/doi/full/10.1080/14992027.2025.2554236|journal=International Journal of Audiology|language=en|pages=1–12|doi=10.1080/14992027.2025.2554236|issn=1499-2027}}&lt;/ref&gt;
Poor integration: Limited connection between audiology and other allied health services in aged care.

''Service Delivery &amp; Technology Problems:''

Tele-audiology underutilized: Barriers include poor infrastructure, restrictive policies, inadequate funding, and limited clinician training
Digital literacy gaps: Patients, especially elderly, struggle with comfort and skills for remote care
Digital therapeutics challenges: New technologies need stronger evidence bases before widespread adoption
Technology implementation hurdles: Clinicians lack confidence in integrating new digital tools.

''Patient &amp; Cultural Barriers:''

Stigma: Hearing loss viewed as &quot;invisible disability,&quot; causing delayed treatment
Low public awareness: Limited understanding of hearing loss impacts and treatment options
Multicultural needs: Services must be culturally sensitive and linguistically appropriate

{{HTitle|Audiology Service Providers and Advocacy Groups}}
Australia hosts a robust network of organizations dedicated to supporting individuals with hearing loss, ranging from frontline service providers to advocacy groups and research institutions. 

''Leading Service Providers'':

'''NextSense''' stands as one of Australia's premier not-for-profit organizations addressing both hearing and vision loss. The organization delivers comprehensive clinical services, conducts research, and operates Australia's largest cochlear implant program. Beyond direct services, NextSense provides educational support and therapeutic interventions for children and adults navigating sensory loss.

'''Hearing Australia''' functions as a government-funded authority that delivers subsidized hearing services to eligible Australians, including pensioners, veterans, Aboriginal and Torres Strait Islander people, and young Australians under 26. While not structured as a charity, it serves as the primary hearing healthcare provider across the country, operating clinics nationwide and supplying hearing devices to those who qualify for government support.

''National Advocacy and Peak Bodies'':

[https://www.health.gov.au/topics/ear-health-and-hearing/support-services Deafness Forum Australia] operates as the national peak body representing the interests of all Australians with hearing loss. The organization advocates for policy changes, promotes hearing health awareness, and connects individuals with resources across the spectrum of hearing impairment. Deafness Forum Australia plays a crucial role in ensuring that hearing loss remains visible in national health discussions and that the needs of the deaf and hard-of-hearing community influence government policy.

'''Audiology Australia (AudA)''' serves as the professional association for audiologists, establishing clinical standards and ethical guidelines for hearing healthcare practitioners. Although not a charity, AudA maintains partnerships with various hearing organizations and provides the public with access to qualified audiologists through its member directory.

{{HTitle|References}}
{{reflist}}

==External Links==
*	 https://audiology.asn.au/
*	 https://www.audiology.org/news-and-publications/audiology-today/articles/a-hearing-report-from-australia/
*	 https://audiology.asn.au/standards-guidelines/scope-of-practice/
* 	 https://www.auditdata.com/insights/cases/enhancing-audiology-care-in-australia-territory-hearings-success-with-manage-software
*	 https://www.health.gov.au/topics/ear-health-and-hearing/support-services
*	 https://www.acnc.gov.au/charity/charities/f6db5b5d-3aaf-e811-a963-000d3ad24077/profile

{{Global_Audiology Authors
|name1=Biraj Bhattarai
|name2=Sajana Aryal
|role1=Contributor
|role2=Contributor
|linkedin1=https://www.linkedin.com/in/biraj-bhattarai-3172931a3
|linkedin2=https://www.linkedin.com/in/sajana-aryal-209612187/
}}

[[Category:Audiology]]
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{{CountryHeader|File:Australia (orthographic projection).svg|https://en.wikipedia.org/wiki/Australia}}
{{HTitle|General Information}}
[https://en.wikipedia.org/wiki/Australia Australia], officially the Commonwealth of Australia is a country comprising the mainland of the Australian continent, the island of Tasmania, and numerous smaller islands. Australia has a population of approximately 27.7 million. Aboriginal and Torres Strait Islander peoples make up approximately 3.2% of the Australian population. Although English is not the official language in law, it is the de facto official and national language. The 2021 Census found that 167 Indigenous languages were spoken at home by Indigenous Australians. 

{{HTitle|History of Audiology}}
Audiology emerged as a profession in Australia during the 1940s. Two major events drove its establishment as a medical specialty. First, many servicemen returned from World War II suffering from hearing loss caused by injuries and noise exposure. Second, the 1940-41 rubella epidemic damaged the hearing of numerous children. These circumstances led Australia to establish the [https://www.nal.gov.au/ National Acoustics Laboratory (NAL)] in 1947. The laboratory began testing hearing and fitting hearing aids for veterans and affected children. Twenty years later, in 1967, NAL expanded its services to include pensioners. 

A pivotal moment in establishing audiology as a distinct profession came when NAL's original director determined that audiological functions such as hearing testing, hearing aid fitting, and associated rehabilitation should be performed by professionally qualified personnel rather than technicians. The initial debate centered on whether preschool teachers or psychologists would be better suited for these roles. Psychologists were ultimately selected, reportedly because they were considered better equipped to identify malingering among veterans. The first audiologists in Australia were psychologists who undertook postgraduate training in speech therapy and hearing care.

Throughout the 1950s and 1960s, the number of practicing audiologists grew steadily, driven primarily by NAL's expansion and, in the latter half of this period, by the creation of positions in hospital settings. Around 1960, informal gatherings of audiologists began in Sydney and Melbourne. In Melbourne, these typically consisted of monthly lunches followed by afternoon seminars. These regional networking efforts eventually led to the formation of the Audiological Society of Australia in May 1968, which began with ten foundation members representing four states. The majority of audiologists practicing before 1970 came from psychology backgrounds, though some arrived from fields including teaching, physics, and speech pathology. Significant developments after 1970 transformed the profession considerably.

The establishment of formal audiology training programs in Brisbane, Melbourne, Sydney, and Perth marked a crucial step toward professional standardization. Audiology subjects were also incorporated into speech-language pathology courses at several institutions. The Australian government and other employers granted official recognition to audiology as a distinct profession, lending credibility and structure to the field.

Employment opportunities expanded significantly, particularly in academia and private practice. Audiological research flourished, with notable emphasis on hearing aids and cochlear implants, though research extended into many other areas as well. The scope of audiological services broadened considerably during this period. The profession established regular national conferences beginning in 1974, held biennially, along with other specialized scientific meetings. The Australian Journal of Audiology was founded in 1979, providing a dedicated publication outlet for Australian audiological research. Publications by Australian audiologists in both national and international venues increased substantially, and interactions with international colleagues intensified through conferences, visits, and collaborative research projects.

{{HTitle|Incidence and Prevalence of Hearing Loss}} 
Hearing loss represents a substantial and growing public health burden in Australia. National estimates from the Australian Government Roadmap for Hearing Health indicate that, by 2019, approximately 3.6 million Australians were living with some form of hearing impairment.

The prevalence of moderate and more severe hearing loss (≥ 40 dB HL) in children ranges from 1.04 per 1000 live births at 3 years of age to 1.57 per 1000 live births for children between 9 and 16 years of age. Mild hearing loss (&lt; 40 dB HL) increases from 0.28 per 1000 live births at 3 years to 1.68 per 1000 live births at 9 years and older.&lt;ref&gt;{{Cite journal|title=The Ages of Intervention in Regions With and Without Universal Newborn Hearing Screening and Prevalence of Childhood Hearing Impairment in Australia|url=http://www.portico.org/Portico/article?article=pf16jrb3hm|journal=Australian and New Zealand Journal of Audiology|date=2006-11-01|pages=137–150|volume=28|issue=2|doi=10.1375/audi.28.2.137|first=Teresa Y.C|last=Ching|first2=Ron|last2=Oong|first3=Emma van|last3=Wanrooy}}&lt;/ref&gt;  Among urban Australian school-aged children (5 to 7 years), the prevalence of bilateral hearing loss ≥26 dB was estimated to 2.1% as reprted in 2006.&lt;ref&gt;{{Cite journal|title=Cross-sectional prevalence and risk factors for otitis media and hearing loss in Australian children aged 5 to 7 years: a prospective cohort study|url=https://www.theajo.com/article/view/4259/html|journal=Australian Journal of Otolaryngology|date=2020-03|pages=0–0|volume=3|doi=10.21037/ajo.2020.02.02|first=Christopher G.|last=Brennan-Jones|first2=Hrehan H.|last2=Hakeem|first3=Cheryl Da|last3=Costa|first4=Weijie|last4=Weng|first5=Andrew J. O.|last5=Whitehouse|first6=Sarra E.|last6=Jamieson|first7=Robert H.|last7=Eikelboom}}&lt;/ref&gt; In a national study, the prevalence of bilateral and unilateral hearing loss ≥16 dB HL was 9.3% and 13.3%, respectively. Slight losses (16-25 dB HL) were more prevalent than mild or greater losses (≥26 dB HL). &lt;ref&gt;{{Cite journal|title=Cross-sectional epidemiology of hearing loss in Australian children aged 11–12 years old and 25-year secular trends|url=https://adc.bmj.com/lookup/doi/10.1136/archdischild-2017-313505|journal=Archives of Disease in Childhood|date=2018-06|issn=0003-9888|pages=579–585|volume=103|issue=6|doi=10.1136/archdischild-2017-313505|language=en|first=Jing|last=Wang|first2=Carlijn M P|last2=le Clercq|first3=Valerie|last3=Sung|first4=Peter|last4=Carew|first5=Richard S|last5=Liu|first6=Fiona K|last6=Mensah|first7=Rachel A|last7=Burt|first8=Lisa|last8=Gold|first9=Melissa|last9=Wake}}&lt;/ref&gt; 

In the Blue Mountains Hearing Study, 33% of older adults had some degree of hearing loss at baseline, with a 5-year incidence of 17.9%.&lt;ref&gt;{{Cite journal|title=Five-Year Incidence and Progression of Hearing Impairment in an Older Population|url=https://journals.lww.com/00003446-201103000-00010|journal=Ear &amp; Hearing|date=2011-03|issn=0196-0202|pages=251–257|volume=32|issue=2|doi=10.1097/AUD.0b013e3181fc98bd|language=en|first=Paul|last=Mitchell|first2=Bamini|last2=Gopinath|first3=Jie Jin|last3=Wang|first4=Catherine M.|last4=McMahon|first5=Julie|last5=Schneider|first6=Elena|last6=Rochtchina|first7=Stephen R.|last7=Leeder}}&lt;/ref&gt; In 2022 hearing loss was estimated to affect 74% of people aged over 70 in Australia.&lt;ref&gt;{{Cite journal|title=Hearing loss, cognition, and risk of neurocognitive disorder: evidence from a longitudinal cohort study of older adult Australians|url=https://www.tandfonline.com/doi/full/10.1080/13825585.2020.1857328|journal=Aging, Neuropsychology, and Cognition|date=2022-01-02|issn=1382-5585|pages=121–138|volume=29|issue=1|doi=10.1080/13825585.2020.1857328|language=en|first=Paul A.|last=Strutt|first2=Amanda J.|last2=Barnier|first3=Greg|last3=Savage|first4=Gabrielle|last4=Picard|first5=Nicole A.|last5=Kochan|first6=Perminder|last6=Sachdev|first7=Brian|last7=Draper|first8=Henry|last8=Brodaty}}&lt;/ref&gt;

{{HTitle| Audiology Education in Australia}}
Audiologists in Australia complete a minimum of five years of university education, including a two-year master’s level audiology program accredited by Audiology Australia. Currently, seven Australian universities offer Audiology Australia–accredited postgraduate audiology programs. Following graduation, Audiology Australia members are required to complete a one-year supervised clinical internship.During this period, interns practice under the supervision of an Audiology Australia Accredited Audiologist, facilitating a structured transition into professional practice and ensuring high standards of service delivery (Audiology Australia, n.d.)

Currently, six universities across Australia offer master's programs in audiology. These include Macquarie University, University of Queensland, Flinders University, Melbourne University, La Trobe University, and the University of Western Australia. The University of Western Australia offers a joint master's/PhD in clinical audiology. Australian master's programs require graduates to meet a set of clinical competencies and complete 250 hours of clinical experience before graduation. Graduates must also complete a one-year clinical internship if they want to provide services to clients in the large government-funded sector. During this year, experienced audiologists supervise interns and prepare them for independent clinical practice. After completing the internship, new graduates receive certification from either Audiology Australia or the Australian College of Audiology. Both organizations represent audiologists professionally in Australia. 

{{HTitle|Scope of Practice and Licensing}}
Audiologists need to meet the relevant membership and clinical competency requirements set by Australian Practitioner Professional Bodies to practice in Australia. They need to hold full membership in Audiology Australia with a Certificate of Clinical Practice (CCP) and/or full/ordinary membership in the Australian College of Audiology (ACAud) with Hearing Rehabilitation Specialist (HRS) and Diagnostic Rehabilitation Specialist (DRS) competencies. Audiologists must complete at least the equivalent of an Australian university master's degree in clinical audiology.

Audiologists in Australia work with clients of all ages, from infants to older adults, including clients with complex needs. They assess hearing and auditory function, vestibular function, tinnitus, auditory processing function, and neural function. Audiologists perform diagnostic tests, including advanced tests using electrophysiological methods. They provide aural, vestibular, and tinnitus rehabilitation as well as communication training. Audiologists offer a range of rehabilitation services, including counseling and prescribing and fitting various devices and aids. These include bone conduction aids, FM and other remote sensing systems, hearing aids, hearing assistive technology, and earplugs (custom noise/swim/musician plugs). Audiologists possess knowledge of implantable devices such as cochlear implants, middle ear implantable hearing aids, fully implantable hearing aids, and bone anchored hearing aids.They collaborate with other professionals when applying these devices in rehabilitation. 

In addition to audiologists, hearing services in Australia are also provided by audiometrists. Audiometrists typically complete vocational or industry-based training programs focused on hearing assessment and hearing aid provision. Their scope of practice generally includes conducting hearing tests, fitting and managing hearing aids, and providing basic hearing rehabilitation services, particularly in community and private practice settings.

Compared to audiologists, audiometrists usually have a more limited scope of practice, particularly in areas such as complex diagnostic testing, vestibular assessment, and electrophysiological measures. Audiometrists may be affiliated with professional organizations such as the Australian College of Audiology or other industry bodies.

{{HTitle|Professional and Regulatory Bodies}}
Australia does not have a single statutory licensing authority for audiologists or audiometrists. Instead, hearing care professionals are represented by two main Practitioner Professional Bodies: [https://audiology.asn.au/Audiology Australia] and [https://www.acaud.com.au/Australian College of Audiology incorporating HAASA (ACAud inc. HAASA)]. Audiology Australia is the peak and largest accrediting professional member body for audiologists in Australia, while ACAud inc. HAASA represents both audiologists and audiometrists.

Audiologists must meet membership and clinical competency requirements set by Audiology Australia and/or ACAud, typically including completion of a master’s-level degree in clinical audiology and a supervised clinical internship. New members of Audiology Australia become accredited audiologists after completing this internship. Accreditation is valid for one year and must be renewed annually through continuing professional development (CPD). Members are required to demonstrate sufficient professional development over the previous 12 months, which may include conferences, seminars, training courses, and research activities.

Audiometrists must meet the relevant membership and competency requirements of ACAud and/or HAASA and typically complete at least a diploma-level Technical and Further Education (TAFE) qualification in audiometry or a bachelor’s degree. These professional bodies define scope of practice, competency standards, and continuing professional development requirements; however, they do not function as statutory regulatory authorities.&lt;ref&gt;{{Cite report|title=Scope of Practice for Audiologists and Audiometrists|url=https://www.acaud.com.au/wp-content/uploads/2019/07/Scope-of-Practice-for-Audiologists-and-Audiometrists.pdf|institution=Audiology Australia; Australian College of Audiology; Hearing Aid Audiometrist Society of Australia|date=2016-09-20}}&lt;/ref&gt;

'''Code of Conduct'''
Audiology Australia members are obliged by a code of conduct. Members must also comply with the Criminal History Policy and Mandatory Declarations Policy.On 12 September 2025, Australia's health ministers agreed to regulate audiology under the National Regulation and Accreditation Scheme. This development requires the Australian Health Practitioner Regulation Agency (AHPRA), which administers NRAS, to provide certification and accreditation to audiologists.

{{HTitle|Ongoing audiology research}}
Audiology research in Australia are conducted across a wide range of institutions, including universities, independent research institutes, and national organizations. Key contributors include the [https://www.nal.gov.au/National Acoustic Laboratories (NAL)], [https://www.earscience.org.au/ Ear Science Institute Australia], and universities such as the [https://healthsciences.unimelb.edu.au/departments/audiology-and-speech-pathology University of Melbourne], [https://www.mq.edu.au/faculty-of-medicine-health-and-human-sciences/departments-and-schools/department-of-linguistics/our-research/audiology-and-hearing Macquarie University], University of Queensland, La Trobe University, and Flinders University.&lt;ref&gt;{{Cite web|title=Our research|url=https://www.nal.gov.au/our-research/|website=National Acoustic Laboratories|access-date=2026-04-28}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|title=Ear Science Institute Australia|url=https://www.earscience.org.au/|website=Ear Science Institute Australia|access-date=2026-04-28}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|title=Audiology and hearing research|url=https://www.mq.edu.au/faculty-of-medicine-health-and-human-sciences/departments-and-schools/department-of-linguistics/our-research/audiology-and-hearing|website=Macquarie University|access-date=2026-04-28}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|title=Audiology and Speech Pathology|url=https://healthsciences.unimelb.edu.au/departments/audiology-and-speech-pathology|website=University of Melbourne|access-date=2026-04-28}}&lt;/ref&gt;.

Research areas include hearing assessment, hearing devices, cochlear implants, tinnitus, vestibular disorders, speech perception, auditory neuroscience, digital health, artificial intelligence, and public-health approaches to hearing care. NAL continues to play a major role through research on adult and paediatric hearing loss, technology, personalized care, and listening difficulties. Its current work includes improving access to hearing services, developing targeted solutions for everyday listening difficulties, and exploring the use of artificial intelligence to improve hearing screening, diagnosis, management, and ongoing support.&lt;ref&gt;{{Cite web|title=Our research|url=https://www.nal.gov.au/our-research/|website=National Acoustic Laboratories|access-date=2026-04-28}}&lt;/ref&gt;

A major national priority is improving hearing health equity. Australian researchers and service providers are working to reduce the impact of otitis media-related hearing loss among Aboriginal and Torres Strait Islander children, often through community-informed and collaborative approaches. Australia also has a national Hearing Services Program that provides subsidised hearing services and devices to eligible Australians with hearing loss. Eligibility includes children and young adults under 26, Aboriginal and Torres Strait Islander people who meet program criteria, and many older adults who qualify through pensioner or veteran status.&lt;ref&gt;{{Cite web|title=Hearing Services Program|url=https://www.health.gov.au/our-work/hearing-services-program|website=Australian Government Department of Health, Disability and Ageing|access-date=2026-04-28}}&lt;/ref&gt;

{{HTitle|Challenges, Opportunities and Notes}}
Australia's audiology sector struggles with several interconnected problems, including insufficient numbers of hearing specialists in regional areas, difficulties accessing services across vast distances, requirements for culturally appropriate care, hearing loss going undetected in elderly care settings, and limited uptake of remote audiology services, even though strong government support programs exist. Primary concerns involve uneven distribution of audiologists across the country, absence of a unified nationwide approach, social stigma surrounding hearing loss, and technological barriers, all of which restrict healthcare access for people living in rural and remote locations, Indigenous Australians, and senior citizens, highlighting the need for enhanced professional training and improved digital health systems. &lt;ref&gt;{{Cite journal|last=Mui|first=Boaz|last2=Lawless|first2=Michael|last3=Timmer|first3=Barbra H. B.|last4=Gopinath|first4=Bamini|last5=Tang|first5=Diana|last6=Venning|first6=Anthony|last7=May|first7=David|last8=Muzaffar|first8=Jameel|last9=Bidargaddi|first9=Niranjan|date=2025-01-02|title=Australian hearing healthcare stakeholders’ experiences of and attitudes towards teleaudiology uptake: a qualitative study|url=https://www.tandfonline.com/doi/full/10.1080/2050571X.2024.2372171|journal=Speech, Language and Hearing|language=en|volume=28|issue=1|doi=10.1080/2050571X.2024.2372171|issn=2050-571X}}&lt;/ref&gt; 

''Workforce &amp; Access Issues:''
Regional shortages: Audiologists concentrate in cities, leaving rural/remote areas under-serviced
Geographic barriers: Vast distances make consistent care difficult for remote communities
Aged care gaps: Hearing loss frequently missed due to lack of staff training and awareness.&lt;ref&gt;{{Cite journal|last=El-Saifi|first=Najwan|last2=Campbell|first2=Megan E.J.|last3=George|first3=Neha|last4=Keay|first4=Lisa|last5=Kumaran|first5=Sheela|last6=Meyer|first6=Carly|last7=Miller Amberber|first7=Amanda|last8=Newall|first8=John|last9=Dawes|first9=Piers|date=2025-09-05|title=Barriers and enablers to hearing service provision in aged care settings in Australia: perspectives from hearing clinicians|url=https://www.tandfonline.com/doi/full/10.1080/14992027.2025.2554236|journal=International Journal of Audiology|language=en|pages=1–12|doi=10.1080/14992027.2025.2554236|issn=1499-2027}}&lt;/ref&gt;
Poor integration: Limited connection between audiology and other allied health services in aged care.

''Service Delivery &amp; Technology Problems:''

Tele-audiology underutilized: Barriers include poor infrastructure, restrictive policies, inadequate funding, and limited clinician training
Digital literacy gaps: Patients, especially elderly, struggle with comfort and skills for remote care
Digital therapeutics challenges: New technologies need stronger evidence bases before widespread adoption
Technology implementation hurdles: Clinicians lack confidence in integrating new digital tools.

''Patient &amp; Cultural Barriers:''

Stigma: Hearing loss viewed as &quot;invisible disability,&quot; causing delayed treatment
Low public awareness: Limited understanding of hearing loss impacts and treatment options
Multicultural needs: Services must be culturally sensitive and linguistically appropriate

{{HTitle|Audiology Service Providers and Advocacy Groups}}
Australia hosts a robust network of organizations dedicated to supporting individuals with hearing loss, ranging from frontline service providers to advocacy groups and research institutions. 

''Leading Service Providers'':

'''NextSense''' stands as one of Australia's premier not-for-profit organizations addressing both hearing and vision loss. The organization delivers comprehensive clinical services, conducts research, and operates Australia's largest cochlear implant program. Beyond direct services, NextSense provides educational support and therapeutic interventions for children and adults navigating sensory loss.

'''Hearing Australia''' functions as a government-funded authority that delivers subsidized hearing services to eligible Australians, including pensioners, veterans, Aboriginal and Torres Strait Islander people, and young Australians under 26. It serves as the primary hearing healthcare provider across the country, operating clinics nationwide and supplying hearing devices to those who qualify for government support.

''National Advocacy and Peak Bodies'':

[https://www.health.gov.au/topics/ear-health-and-hearing/support-services Deafness Forum Australia] operates as the national peak body representing the interests of all Australians with hearing loss. The organization advocates for policy changes, promotes hearing health awareness, and connects individuals with resources across the spectrum of hearing impairment. Deafness Forum Australia plays a crucial role in ensuring that hearing loss remains visible in national health discussions and that the needs of the deaf and hard-of-hearing community influence government policy.

'''Audiology Australia (AudA)''' serves as the professional association for audiologists, establishing clinical standards and ethical guidelines for hearing healthcare practitioners. Although not a charity, AudA maintains partnerships with various hearing organizations and provides the public with access to qualified audiologists through its member directory.

{{HTitle|References}}
{{reflist}}

==External Links==
*	 https://audiology.asn.au/
*	 https://www.audiology.org/news-and-publications/audiology-today/articles/a-hearing-report-from-australia/
*	 https://audiology.asn.au/standards-guidelines/scope-of-practice/
* 	 https://www.auditdata.com/insights/cases/enhancing-audiology-care-in-australia-territory-hearings-success-with-manage-software
*	 https://www.health.gov.au/topics/ear-health-and-hearing/support-services
*	 https://www.acnc.gov.au/charity/charities/f6db5b5d-3aaf-e811-a963-000d3ad24077/profile

{{Global_Audiology Authors
|name1=Biraj Bhattarai
|name2=Sajana Aryal
|role1=Contributor
|role2=Contributor
|linkedin1=https://www.linkedin.com/in/biraj-bhattarai-3172931a3
|linkedin2=https://www.linkedin.com/in/sajana-aryal-209612187/
}}

[[Category:Audiology]]
[[Category:Australia]]</text>
      <sha1>4qepof0z1euwfxv24h8c8hrf9lnxiwp</sha1>
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  <page>
    <title>Mathematics education</title>
    <ns>0</ns>
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      <text bytes="987" sha1="tly2vo0i062nkgmgzy9c7u07rnzl2sv" xml:space="preserve">Mathematics education is a discipline that studies the learning and teaching of mathematics.&lt;ref&gt;{{Cite journal|date=2024-12-21|title=Mathematics education|url=https://en.wikipedia.org/wiki/Mathematics_education|journal=Wikipedia|language=en}}&lt;/ref&gt;

== Mathematics Teacher Education Resources ==

* [[Levels of cognitive demand of tasks]]&lt;ref&gt;{{Cite journal|last=Smith|first=Margaret Schwan|last2=Stein|first2=Mary Kay|date=1998|title=REFLECTIONS on Practice: Selecting and Creating mathematical Tasks: From Research to Practice|url=https://www.jstor.org/stable/41180423|journal=Mathematics Teaching in the Middle School|volume=3|issue=5|pages=344–350|issn=1072-0839}}&lt;/ref&gt;
* [[Lesson plans created by prospective teachers]]
* [[Research: Technology's Support for Learning]]
* [[Research: Manipulatives (Sydney)]]

[[Category:Education]]
[[Category:Mathematics|education]]
&lt;references /&gt;

* Research: Tracking

* [[Research: Problem Based Learning]]
* [[Effects of the Gifted Label]]</text>
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      <text bytes="991" sha1="e6clsa4ilm1h7hlqhp5l9v146ob272f" xml:space="preserve">Mathematics education is a discipline that studies the learning and teaching of mathematics.&lt;ref&gt;{{Cite journal|date=2024-12-21|title=Mathematics education|url=https://en.wikipedia.org/wiki/Mathematics_education|journal=Wikipedia|language=en}}&lt;/ref&gt;

== Mathematics Teacher Education Resources ==

* [[Levels of cognitive demand of tasks]]&lt;ref&gt;{{Cite journal|last=Smith|first=Margaret Schwan|last2=Stein|first2=Mary Kay|date=1998|title=REFLECTIONS on Practice: Selecting and Creating mathematical Tasks: From Research to Practice|url=https://www.jstor.org/stable/41180423|journal=Mathematics Teaching in the Middle School|volume=3|issue=5|pages=344–350|issn=1072-0839}}&lt;/ref&gt;
* [[Lesson plans created by prospective teachers]]
* [[Research: Technology's Support for Learning]]
* [[Research: Manipulatives (Sydney)]]

[[Category:Education]]
[[Category:Mathematics|education]]
&lt;references /&gt;

* [[Research: Tracking]]

* [[Research: Problem Based Learning]]
* [[Effects of the Gifted Label]]</text>
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  <page>
    <title>User:Dc.samizdat/Golden chords of the 120-cell</title>
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      <comment>/* The 8-point regular polytopes */</comment>
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      <text bytes="41312" sha1="9wpsp37oc90oql07v5t7di51ks4tito" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 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 smaller 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 &lt;math&gt;5^2&lt;/math&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 way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point (24-cell), and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&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 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|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 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&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 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this 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, so we obtain 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, so we obtain 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;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;. 

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. 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 regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. 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 has 3 such 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-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, 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;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length:

:&lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552&lt;/math&gt;

none of which chords except &lt;math&gt;r_1=\sqrt{2}&lt;/math&gt; occur in the 16-cell.

The vertex coordinates of the 16-cell form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 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}}

[[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 double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because 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 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic 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 square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in different directions, and the rigid 16-cell assumes a new orientation in 4-space. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&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 cuboctahedron and the 24-cell are also radially equilateral.

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. 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 [[W:Demihypercube|an exact dimensional analogy]] to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]].

The tesseract is also the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 6 distinct instances of the skew octagon. We can build the tesseract the same way we did the 16-cell, by the very same skewing of a planar octagon, but we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, the octagons' common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; that we needed to build the unit-radius 16-cell.

== The 24-cell ==

...

== The 600-cell ==

...

== Finally, the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular 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.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations 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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 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>9wpsp37oc90oql07v5t7di51ks4tito</sha1>
    </revision>
    <revision>
      <id>2806883</id>
      <parentid>2806878</parentid>
      <timestamp>2026-04-28T18:14:45Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 8-point regular polytopes */</comment>
      <origin>2806883</origin>
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      <text bytes="41649" sha1="3xuru0rcoffr3d3jow087rzs3obtofp" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 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 smaller 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 &lt;math&gt;5^2&lt;/math&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 way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point (24-cell), and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&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 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|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 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&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 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this 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, so we obtain 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, so we obtain 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;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;. 

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. 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 regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. 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 has 3 such 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-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, 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;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length:

:&lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552&lt;/math&gt;

none of which chords except &lt;math&gt;r_1=\sqrt{2}&lt;/math&gt; occur in the 16-cell.

The vertex coordinates of the 16-cell form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 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}}

[[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 double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because 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 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic 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 square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in the 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of a geodesic orbit that traces a circular helix in 4-space, and also traces a great circle in each of two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation on the opposite side of the 16-cell, departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&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 cuboctahedron and the 24-cell are also radially equilateral.

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. 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 [[W:Demihypercube|an exact dimensional analogy]] to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]].

The tesseract is also the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 6 distinct instances of the skew octagon. We can build the tesseract the same way we did the 16-cell, by the very same skewing of a planar octagon, but we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, the octagons' common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; that we needed to build the unit-radius 16-cell.

== The 24-cell ==

...

== The 600-cell ==

...

== Finally, the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular 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.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations 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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 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>3xuru0rcoffr3d3jow087rzs3obtofp</sha1>
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    <revision>
      <id>2806884</id>
      <parentid>2806883</parentid>
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      <contributor>
        <username>Dc.samizdat</username>
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      <text bytes="41645" sha1="frq68twn6yeef7ibpsoxo8nqsk5ol1z" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 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 smaller 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 &lt;math&gt;5^2&lt;/math&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 way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point (24-cell), and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&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 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|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 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&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 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this 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, so we obtain 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, so we obtain 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;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;. 

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. 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 regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. 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 has 3 such 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-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, 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;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length:

:&lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552&lt;/math&gt;

none of which chords except &lt;math&gt;r_1=\sqrt{2}&lt;/math&gt; occur in the 16-cell.

The vertex coordinates of the 16-cell form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 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}}

[[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 double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because 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 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic 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 square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of a geodesic orbit that traces a circular helix in 4-space, and also traces a great circle in each of two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation on the opposite side of the 16-cell, departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&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 cuboctahedron and the 24-cell are also radially equilateral.

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. 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 [[W:Demihypercube|an exact dimensional analogy]] to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]].

The tesseract is also the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 6 distinct instances of the skew octagon. We can build the tesseract the same way we did the 16-cell, by the very same skewing of a planar octagon, but we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, the octagons' common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; that we needed to build the unit-radius 16-cell.

== The 24-cell ==

...

== The 600-cell ==

...

== Finally, the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular 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.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations 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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 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>frq68twn6yeef7ibpsoxo8nqsk5ol1z</sha1>
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    <revision>
      <id>2806885</id>
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        <username>Dc.samizdat</username>
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      <text bytes="41674" sha1="rfmib7qlzjeoo4a77oq5ogitstb0mrv" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 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 smaller 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 &lt;math&gt;5^2&lt;/math&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 way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point (24-cell), and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&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 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|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 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&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 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this 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, so we obtain 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, so we obtain 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;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;. 

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. 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 regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. 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 has 3 such 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-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, 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;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length:

:&lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552&lt;/math&gt;

none of which chords except &lt;math&gt;r_1=\sqrt{2}&lt;/math&gt; occur in the 16-cell.

The vertex coordinates of the 16-cell form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 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}}

[[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 double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because 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 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic 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 square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of a geodesic orbit that traces a circular helix in 4-space, and also traces a great circle in each of two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation on the opposite side of the 16-cell, departing from its new vertex position in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&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 cuboctahedron and the 24-cell are also radially equilateral.

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. 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 [[W:Demihypercube|an exact dimensional analogy]] to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]].

The tesseract is also the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 6 distinct instances of the skew octagon. We can build the tesseract the same way we did the 16-cell, by the very same skewing of a planar octagon, but we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, the octagons' common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; that we needed to build the unit-radius 16-cell.

== The 24-cell ==

...

== The 600-cell ==

...

== Finally, the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular 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.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations 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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 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>rfmib7qlzjeoo4a77oq5ogitstb0mrv</sha1>
    </revision>
    <revision>
      <id>2806888</id>
      <parentid>2806885</parentid>
      <timestamp>2026-04-28T18:25:06Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 8-point regular polytopes */</comment>
      <origin>2806888</origin>
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      <text bytes="41676" sha1="4duu0rgjsubx5oy3ng4sb6y1f5tugjs" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 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 smaller 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 &lt;math&gt;5^2&lt;/math&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 way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point (24-cell), and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&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 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|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 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&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 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this 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, so we obtain 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, so we obtain 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;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;. 

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. 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 regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. 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 has 3 such 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-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, 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;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length:

:&lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552&lt;/math&gt;

none of which chords except &lt;math&gt;r_1=\sqrt{2}&lt;/math&gt; occur in the 16-cell.

The vertex coordinates of the 16-cell form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 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}}

[[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 double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because 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 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic 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 square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit that traces a circular helix in 4-space, and also traces a great circle in each of two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation on the opposite side of the 16-cell, departing from its new vertex position in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&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 cuboctahedron and the 24-cell are also radially equilateral.

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. 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 [[W:Demihypercube|an exact dimensional analogy]] to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]].

The tesseract is also the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 6 distinct instances of the skew octagon. We can build the tesseract the same way we did the 16-cell, by the very same skewing of a planar octagon, but we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, the octagons' common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; that we needed to build the unit-radius 16-cell.

== The 24-cell ==

...

== The 600-cell ==

...

== Finally, the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular 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.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations 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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 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>4duu0rgjsubx5oy3ng4sb6y1f5tugjs</sha1>
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    <revision>
      <id>2806890</id>
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      <text bytes="41677" sha1="fwxhvo6aljs9gxc139r80n4mdj61n5c" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 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 smaller 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 &lt;math&gt;5^2&lt;/math&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 way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point (24-cell), and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&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 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|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 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&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 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this 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, so we obtain 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, so we obtain 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;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;. 

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. 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 regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. 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 has 3 such 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-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, 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;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length:

:&lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552&lt;/math&gt;

none of which chords except &lt;math&gt;r_1=\sqrt{2}&lt;/math&gt; occur in the 16-cell.

The vertex coordinates of the 16-cell form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 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}}

[[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 double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because 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 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic 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 square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit which traces a circular helix in 4-space, and also traces a great circle in each of two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation on the opposite side of the 16-cell, departing from its new vertex position in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&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 cuboctahedron and the 24-cell are also radially equilateral.

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. 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 [[W:Demihypercube|an exact dimensional analogy]] to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]].

The tesseract is also the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 6 distinct instances of the skew octagon. We can build the tesseract the same way we did the 16-cell, by the very same skewing of a planar octagon, but we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, the octagons' common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; that we needed to build the unit-radius 16-cell.

== The 24-cell ==

...

== The 600-cell ==

...

== Finally, the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular 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.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations 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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 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>fwxhvo6aljs9gxc139r80n4mdj61n5c</sha1>
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    <revision>
      <id>2806895</id>
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      <contributor>
        <username>Dc.samizdat</username>
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      <text bytes="41681" sha1="oxtu2xyod0ilkyq1vz7dqwtxh049ef0" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 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 smaller 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 &lt;math&gt;5^2&lt;/math&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 way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point (24-cell), and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&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 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|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 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&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 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this 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, so we obtain 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, so we obtain 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;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;. 

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. 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 regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. 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 has 3 such 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-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, 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;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length:

:&lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552&lt;/math&gt;

none of which chords except &lt;math&gt;r_1=\sqrt{2}&lt;/math&gt; occur in the 16-cell.

The vertex coordinates of the 16-cell form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 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}}

[[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 double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because 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 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic 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 square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit. The entire orbit traces a circular helix in 4-space, and also traces a great circle in each of two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell and departs in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&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 cuboctahedron and the 24-cell are also radially equilateral.

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. 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 [[W:Demihypercube|an exact dimensional analogy]] to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]].

The tesseract is also the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 6 distinct instances of the skew octagon. We can build the tesseract the same way we did the 16-cell, by the very same skewing of a planar octagon, but we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, the octagons' common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; that we needed to build the unit-radius 16-cell.

== The 24-cell ==

...

== The 600-cell ==

...

== Finally, the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular 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.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations 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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 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="41684" sha1="pj5o5uk8yg0jrgke0xguvi9sgeqkjcd" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 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 smaller 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 &lt;math&gt;5^2&lt;/math&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 way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point (24-cell), and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&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 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|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 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&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 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this 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, so we obtain 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, so we obtain 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;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;. 

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. 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 regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. 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 has 3 such 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-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, 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;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length:

:&lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552&lt;/math&gt;

none of which chords except &lt;math&gt;r_1=\sqrt{2}&lt;/math&gt; occur in the 16-cell.

The vertex coordinates of the 16-cell form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 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}}

[[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 double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because 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 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic 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 square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit. The entire orbit traces a circular helix in 4-space, and also traces a great circle in each of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&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 cuboctahedron and the 24-cell are also radially equilateral.

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. 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 [[W:Demihypercube|an exact dimensional analogy]] to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]].

The tesseract is also the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 6 distinct instances of the skew octagon. We can build the tesseract the same way we did the 16-cell, by the very same skewing of a planar octagon, but we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, the octagons' common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; that we needed to build the unit-radius 16-cell.

== The 24-cell ==

...

== The 600-cell ==

...

== Finally, the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular 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.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations 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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 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>pj5o5uk8yg0jrgke0xguvi9sgeqkjcd</sha1>
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    <revision>
      <id>2806908</id>
      <parentid>2806898</parentid>
      <timestamp>2026-04-28T19:47:00Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* The 8-point regular polytopes */</comment>
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      <text bytes="41684" sha1="1fbi8pna9zbaqh31mnlk4ny15njw081" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 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 smaller 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 &lt;math&gt;5^2&lt;/math&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 way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point (24-cell), and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&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 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|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 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&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 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this 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, so we obtain 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, so we obtain 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;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;. 

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. 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 regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. 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 has 3 such 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-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, 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;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length:

:&lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552&lt;/math&gt;

none of which chords except &lt;math&gt;r_1=\sqrt{2}&lt;/math&gt; occur in the 16-cell.

The vertex coordinates of the 16-cell form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 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}}

[[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 double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because 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 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic 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 square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in each of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&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 cuboctahedron and the 24-cell are also radially equilateral.

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. 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 [[W:Demihypercube|an exact dimensional analogy]] to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]].

The tesseract is also the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 6 distinct instances of the skew octagon. We can build the tesseract the same way we did the 16-cell, by the very same skewing of a planar octagon, but we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, the octagons' common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; that we needed to build the unit-radius 16-cell.

== The 24-cell ==

...

== The 600-cell ==

...

== Finally, the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular 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.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations 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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 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>1fbi8pna9zbaqh31mnlk4ny15njw081</sha1>
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    <revision>
      <id>2806914</id>
      <parentid>2806908</parentid>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="41683" sha1="ap2n1i47i92vo9mz4oxgqg7jzui5o9s" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 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 smaller 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 &lt;math&gt;5^2&lt;/math&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 way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point (24-cell), and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&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 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|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 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&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 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this 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, so we obtain 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, so we obtain 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;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;. 

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. 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 regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. 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 has 3 such 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-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, 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;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length:

:&lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552&lt;/math&gt;

none of which chords except &lt;math&gt;r_1=\sqrt{2}&lt;/math&gt; occur in the 16-cell.

The vertex coordinates of the 16-cell form 6 [[W:Orthogonal|orthogonal]] central squares lying in 6 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}}

[[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 double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because 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 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic 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 square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&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 cuboctahedron and the 24-cell are also radially equilateral.

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. 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 [[W:Demihypercube|an exact dimensional analogy]] to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]].

The tesseract is also the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 6 distinct instances of the skew octagon. We can build the tesseract the same way we did the 16-cell, by the very same skewing of a planar octagon, but we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, the octagons' common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; that we needed to build the unit-radius 16-cell.

== The 24-cell ==

...

== The 600-cell ==

...

== Finally, the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular 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.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations 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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 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>ap2n1i47i92vo9mz4oxgqg7jzui5o9s</sha1>
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    <revision>
      <id>2806916</id>
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        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="41692" sha1="hujuamljlvdbhtu63r5ox4d20q74ruk" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 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 smaller 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 &lt;math&gt;5^2&lt;/math&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 way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point (24-cell), and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&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 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|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 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&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 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this 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, so we obtain 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, so we obtain 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;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;. 

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. 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 regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. 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 has 3 such 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-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, 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;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length:

:&lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552&lt;/math&gt;

none of which chords except &lt;math&gt;r_1=\sqrt{2}&lt;/math&gt; occur in the 16-cell.

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}}

[[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 double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because 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 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic 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 square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&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 cuboctahedron and the 24-cell are also radially equilateral.

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. 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 [[W:Demihypercube|an exact dimensional analogy]] to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two regular 4-point tetrahedra]].

The tesseract is also the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 6 distinct instances of the skew octagon. We can build the tesseract the same way we did the 16-cell, by the very same skewing of a planar octagon, but we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, the octagons' common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; that we needed to build the unit-radius 16-cell.

== The 24-cell ==

...

== The 600-cell ==

...

== Finally, the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular 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.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations 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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 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>hujuamljlvdbhtu63r5ox4d20q74ruk</sha1>
    </revision>
    <revision>
      <id>2806923</id>
      <parentid>2806916</parentid>
      <timestamp>2026-04-28T22:55:22Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Hypercubes */</comment>
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      <text bytes="41891" sha1="k6zqdviehmnpfjd3rd5jfexjq12vr87" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 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 smaller 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 &lt;math&gt;5^2&lt;/math&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 way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point (24-cell), and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&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 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|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 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&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 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this 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, so we obtain 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, so we obtain 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;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;. 

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. 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 regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. 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 has 3 such 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-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, 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;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length:

:&lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552&lt;/math&gt;

none of which chords except &lt;math&gt;r_1=\sqrt{2}&lt;/math&gt; occur in the 16-cell.

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}}

[[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 double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because 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 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic 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 square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&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 cuboctahedron and the 24-cell are also radially equilateral.

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 6 distinct instances of the skew octagon. We can build the tesseract the same way we did the 16-cell, by the very same skewing of a planar octagon, but we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, the octagons' common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; that we needed to build the unit-radius 16-cell.

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 regular 4-point tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-radius tesseract are the &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges of the unit-radius 16-cells.

== The 24-cell ==

...

== The 600-cell ==

...

== Finally, the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular 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.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations 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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 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>k6zqdviehmnpfjd3rd5jfexjq12vr87</sha1>
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    <revision>
      <id>2806926</id>
      <parentid>2806923</parentid>
      <timestamp>2026-04-28T23:40:52Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="41891" sha1="9anqhmck55jjtyzbovja7sfbbi800el" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 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 smaller 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 &lt;math&gt;5^2&lt;/math&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 way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point (24-cell), and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&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 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|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 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&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 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this 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, so we obtain 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, so we obtain 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;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;. 

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. 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 regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. 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 has 3 such 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-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, 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;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length:

:&lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552&lt;/math&gt;

none of which chords except &lt;math&gt;r_1=\sqrt{2}&lt;/math&gt; occur in the 16-cell.

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}}

[[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 double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because 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 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic 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 square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&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 cuboctahedron and the 24-cell are also radially equilateral.

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 6 distinct instances of the skew octagon. We can build the tesseract the same way we did the 16-cell, by the very same skewing of a planar octagon, but we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, the octagons' common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; that we needed to build the unit-radius 16-cell.

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 regular 4-point tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-radius tesseract are the &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges of two unit-radius 16-cells.

== The 24-cell ==

...

== The 600-cell ==

...

== Finally, the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular 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.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations 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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 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>9anqhmck55jjtyzbovja7sfbbi800el</sha1>
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    <revision>
      <id>2806933</id>
      <parentid>2806926</parentid>
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        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="41871" sha1="7eg66cc7yrscsudvg5mapddr876w0yh" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 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 smaller 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 &lt;math&gt;5^2&lt;/math&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 way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point (24-cell), and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&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 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|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 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&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 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this 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, so we obtain 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, so we obtain 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;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;. 

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. 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 regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. 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 has 3 such 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-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, 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;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length:

:&lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552&lt;/math&gt;

none of which chords except &lt;math&gt;r_1=\sqrt{2}&lt;/math&gt; occur in the 16-cell.

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}}

[[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 double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because 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 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic 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 square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&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 cuboctahedron and the 24-cell are also radially equilateral.

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can build the tesseract the way we built the 16-cell, by skewing a planar octagon, but we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, the octagons' common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; that we needed to build the unit-radius 16-cell.

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 regular 4-point tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-radius tesseract are the &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges of two unit-radius 16-cells.

== The 24-cell ==

...

== The 600-cell ==

...

== Finally, the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular 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.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations 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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 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>7eg66cc7yrscsudvg5mapddr876w0yh</sha1>
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    <revision>
      <id>2806936</id>
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      <text bytes="42277" sha1="njh25u11t70382knc1rdjle794x3gud" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 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 smaller 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 &lt;math&gt;5^2&lt;/math&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 way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point (24-cell), and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&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 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|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 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&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 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this 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, so we obtain 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, so we obtain 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;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;. 

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. 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 regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. 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 has 3 such 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-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, 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;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length:

:&lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552&lt;/math&gt;

none of which chords except &lt;math&gt;r_1=\sqrt{2}&lt;/math&gt; occur in the 16-cell.

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}}

[[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 double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because 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 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic 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 square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&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 cuboctahedron and the 24-cell are also radially equilateral.

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; that we needed to build the unit-radius 16-cell.

For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both in the same direction, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing.

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 regular 4-point tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-radius tesseract are the &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges of two unit-radius 16-cells.

== The 24-cell ==

...

== The 600-cell ==

...

== Finally, the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular 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.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations 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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 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>njh25u11t70382knc1rdjle794x3gud</sha1>
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    <revision>
      <id>2806941</id>
      <parentid>2806936</parentid>
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        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="42268" sha1="ifa9ld60utnkrm08fbrkme3gapuxp62" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 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 smaller 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 &lt;math&gt;5^2&lt;/math&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 way 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) distinct instances of the 24-point (24-cell), and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&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;
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|&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 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[W:16-cell|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 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=1+\sqrt{2}&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 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

If we embed this 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, so we obtain 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, so we obtain 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;small&gt;&lt;math&gt;1/\sqrt{2}&lt;/math&gt;&lt;/small&gt;. 

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. 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 regular skew octagon, its [[W:Petrie polygon|Petrie polygon]]. 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 has 3 such 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-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, 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;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges except opposite pairs. In this convenient unit-radius 4-coordinate system, the original planar octagon we started with had chords of length:

:&lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{4 + \sqrt{8}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552&lt;/math&gt;

none of which chords except &lt;math&gt;r_1=\sqrt{2}&lt;/math&gt; occur in the 16-cell.

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}}

[[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 double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because 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 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic 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 square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its geodesic orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; is &lt;small&gt;&lt;math&gt;\sqrt{n}&lt;/math&gt;&lt;/small&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 cuboctahedron and the 24-cell are also radially equilateral.

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; that we needed to build the unit-radius 16-cell.

For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing.

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 regular 4-point tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-radius tesseract are the &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; edges of two unit-radius 16-cells.

== The 24-cell ==

...

== The 600-cell ==

...

== Finally, the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular 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.

Fontaine and Hurley's discovery of a chordal formula for isoclinic rotations 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 higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chords to subsumption relations among 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="19674" sha1="8igg899bk3eqac3io5oychqcir8n69i" xml:space="preserve">[[File:L'impératrice Eugénie à la Marie-Antoinette.png|alt=Old painting of a young woman richly dressed|thumb|Empress Eugénie as Marie Antoinette, by Winterhalter, 1854]]

==Overview==
Empress Eugénie was politically conservative and active as regent in France when Napoleon III went to join his army, as he often did.

She identified with Marie Antoinette, collecting her possessions, including jewelry, and having Winterhalter paint her as the queen of the ''ancien régime'' in 1854 (right).

Eugénie was a [[Social Victorians/People/Eugenie of France#Fashion Leader|leader in the fashion world]]. She adopted the cage in the mid-1850s and then abandoned it in the late 1860s.&lt;ref name=&quot;:0&quot; /&gt; She appointed Charles Frederick Worth to be official dressmaker in 1869.&lt;ref name=&quot;:0&quot; /&gt;

Queen Victoria recommended Miss Shaw as Louis-Napoléon, the Prince Imperial's tutor.&lt;ref name=&quot;:0&quot; /&gt; She &quot;intervened&quot; so that he could join the British Army fighting the Zulu.&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|date=2025-11-29|title=Louis-Napoléon, Prince Imperial|url=https://en.wikipedia.org/w/index.php?title=Louis-Napol%C3%A9on,_Prince_Imperial&amp;oldid=1324821881|journal=Wikipedia|language=en}}&lt;/ref&gt; Victoria and Eugénie intervened in the court marshall of one of the survivors of the battle, Carey, who was &quot;returned to his regiment.&quot;&lt;ref name=&quot;:3&quot; /&gt;

The villa Eugénie and Napoleon III built in Biarritz after 1854 also brought Queen Victoria and King Alphonso XIII to the area.&lt;ref name=&quot;:0&quot; /&gt;

After the deaths of Napoleon III and her son, she visited Queen Victoria at Osborne House on the Isle of Wight.&lt;ref name=&quot;:0&quot; /&gt;

==Also Known As==
* Family name: Palafox y Kirkpatrick
* Eugénie de Montijo
* María Eugenia Ignacia Agustina

==Acquaintances, Friends and Enemies==
===Acquaintances===

* [[Social Victorians/People/Mary Todd Lincoln|Mary Todd Lincoln]]
* [[Social Victorians/People/Elisabeth of Austria|Empress Elisabeth of Austria]]
* Isabella II of Spain&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|date=2025-11-18|title=Eugénie de Montijo|url=https://en.wikipedia.org/w/index.php?title=Eug%C3%A9nie_de_Montijo&amp;oldid=1322973534|journal=Wikipedia|language=en}}&lt;/ref&gt;
* Alexandra Feodorovna, married to Nicholas II of Russia&lt;ref name=&quot;:0&quot; /&gt;

===Friends===

* [[Social Victorians/People/Queen Victoria|Queen Victoria]]
* [[Social Victorians/People/Sophie of Wurttemberg|Sophie of Württemberg, Queen of the Netherlands]]
*Princess Pauline von Metternich
*James, Duke of Alba&lt;ref name=&quot;:32&quot; /&gt; (146 of 909)

==== The von Metternichs ====
Goldstone says,&lt;blockquote&gt;Recalling Baron Hübner and replacing him with Prince Metternich was perhaps the most astute decision Franz Joseph made during his entire reign. The thirty-one-year-old prince hailed from the highest nobility and the most important political family in Austria. His father had been the greatest statesman of the first half of the nineteenth century. Handsome, cultured, and intelligent, &quot;his manners were of the most polished description, and his attitude… most gallant and charming,&quot; remembered Amélie Carette, one of Eugénie’s ladies-in-waiting. &quot;He was enabled to fulfill the most delicate diplomatic tasks with unfailing honor and credit.&quot;

But it was his twenty-four-year-old wife, Princess Pauline, &quot;that wondrously brilliant and popular woman,&quot; as Amélie raved, who was the new ambassador’s greatest asset. &quot;The Princesse de Metternich left in Parisian society a memory which never will be effaced, and which was for many years unequaled by that of any other woman,&quot; the lady-in-waiting testified solemnly.&lt;ref name=&quot;:32&quot; /&gt; (453 of 909)&lt;/blockquote&gt;

=== Enemies ===

* [[Social Victorians/People/Bourke#Comtesse de Castiglione as Salammbô|Virginia, Countess of Castiglioni]], a mistress of Napoleon III

== Organizations and Social Networks ==

=== Queens and Political Leaders ===

*[[Social Victorians/People/Queen Victoria|Queen Victoria]]
*[[Social Victorians/People/Sophie of Wurttemberg|Sophie of Württemberg, Queen of the Netherlands]]
*[[Social Victorians/People/Mary Todd Lincoln|Mary Todd Lincoln]]
* [[Social Victorians/People/Elisabeth of Austria|Empress Elisabeth of Austria]]

[[File:Winterhalter Franz Xavier The Empress Eugenie Surrounded by her Ladies in Waiting.jpg|alt=Old painting of 9 women in very frilly dresses, seated on the grass|thumb|Empress Eugénie and Her Ladies in Waiting, Winterhalter, 1855]]

=== Dames du Palais ===
What would be called Ladies in Waiting in the U.K.
*''Grand-Maitresse'' Anne Debelle, Princesse d'Essling&lt;ref name=&quot;:0&quot; /&gt;
* ''Dame d'honneur'', Pauline de Bassano&lt;ref name=&quot;:0&quot; /&gt;
Winterhalter's 1855 painting (right) shows Eugénie in white in the back row, with a crown and branch of honeysuckle. Her head is the highest. On her right is Anne Debelle, Princess of Essling (1802–1887); on her left is Pauline van der Linden d'Hooghvorst, Duchess of Bassano (1814–1867).&lt;blockquote&gt;Below are the ladies of the palace [Dames du Palais], who followed the empress daily: on the left, Jane Thorne, baroness of Pierres (1821–1873), ... viscountess of Lezay-Marnésia (1826–1891) ...; in the center, Adrienne de Villeneuve-Bargemont, countess of Montebello (1826–1870) ..., and on the right, Anne Eve Mortier de Trévise, marquise of Latour-Maubourg (1829–1900) ..., Claire Emilie MacDonnel, Marquise de Las Marismas de Guadalquivir (1817–1905) ..., and behind them, standing, Nathalie de Ségur, Baroness of Malaret (1827–1910).&lt;ref&gt;{{Cite journal|date=2025-10-26|title=Empress Eugénie Surrounded by her Ladies in Waiting|url=https://en.wikipedia.org/w/index.php?title=Empress_Eug%C3%A9nie_Surrounded_by_her_Ladies_in_Waiting&amp;oldid=1318941052|journal=Wikipedia|language=en}}&lt;/ref&gt;&lt;/blockquote&gt;

=== Fashion Leader ===
Around 15 November 1852 Napoleon celebrated the Senate abolishing the republic and recommending a national referendum to make him emperor by throwing a hunting party, at which Eugénie was able to shine:&lt;blockquote&gt;Her fashion sense was only outdone by her riding. Her accessories were a marvel. According to an admiring spectator, along with her tight-fitting habit (consisting, provocatively, of a skirt “over grey trousers”) she wore “patent leather boots with high heels and spurs.” Her shining red-gold hair was set off to perfection by “a dainty felt hat, from which waved a magnificent long ostrich feather fastened by a diamond clasp,” and she carried a pearl-handled crop. “She sat her horse like a knight, and despised the saddle ordinarily used by ladies&quot; [i.e., she straddled the horse rather than riding side saddle].&lt;ref name=&quot;:32&quot; /&gt; (182 of 909)&lt;/blockquote&gt;Eugénie was leader of the social world in Paris, hosting many regular as well as one-time events. George Sand says,&lt;blockquote&gt;&quot;These solid, conventional women strive to copy the beautiful Eugénie; they powder their hair, real or false, with gold and copper; they put on make-up, they become redheads.&quot; And it was not only her own subjects who responded strongly to the empress’s appeal. Within months of the wedding, Madame Tussaud’s in London had added a Eugénie exhibit &quot;in her beautiful Bridal Dress of exquisite lace of great value,&quot; and the English papers ran regular advertisements for &quot;RIMMEL’S GOLDEN HAIR POWDER (as worn by the Empress Eugénie and the élite of the French nobility),&quot; which, [229 / 230] for a mere 10 shillings and sixpence per box, &quot;gives a most enchanting and unique brilliancy to female beauty, and is adopted at all the balls of the 'Haut ton.'&quot;&lt;ref name=&quot;:32&quot; /&gt; (229–230 of 909)&lt;/blockquote&gt;Queen Victoria and the cage, 1855.

[[Social Victorians/People/Elisabeth of Austria|Elisabeth of Austria]] modeled her stylish look on Eugénie.&lt;ref name=&quot;:32&quot; /&gt; (322 of 909)

Empress Eugénie's influence on fashion: &quot;when [[Social Victorians/People/Mary Todd Lincoln|Mrs. Lincoln]] first arrived in Washington, she made a point of patterning her gowns after the empress’s wardrobe.&quot;&lt;ref name=&quot;:32&quot;&gt;Goldstone, Nancy. ''The Rebel Empresses: Elisabeth of Austria and Eugénie of France, Power and Glamour in the Struggle for Europe''. Little, Brown, 2025.&lt;/ref&gt;{{rp|566, n. iii}}

The most fashion-forward woman in Paris was Princess Pauline von Metternich, whose savvy maid convinced her in about 1860 to meet with Marie Vernet Worth and consider a ball dress designed for her by the new and unknown Charles Frederick Worth.&lt;ref name=&quot;:32&quot; /&gt; (454–455 of 909) Pauline paid 300 francs for the ballgown, the last time she got a Worth dress for that low a price. When the 2nd-most fashion-forward Empress Eugénie saw the dress at the ball, she asked who had made it and replaced Madame Palmyre by Worth as Eugénie's designer and dressmaker.&lt;ref&gt;{{Cite journal|date=2025-12-09|title=Pauline von Metternich|url=https://en.wikipedia.org/w/index.php?title=Pauline_von_Metternich&amp;oldid=1326545383|journal=Wikipedia|language=en}}&lt;/ref&gt;

==Timeline==
'''1849 February 14''', Paca (María Francisca de Sales de Palafox Portocarrero y Kirkpatrick) married the 15th Duke of Alba.&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|date=2025-11-08|title=María Francisca de Sales Portocarrero, 16th Duchess of Peñaranda|url=https://en.wikipedia.org/w/index.php?title=Mar%C3%ADa_Francisca_de_Sales_Portocarrero,_16th_Duchess_of_Pe%C3%B1aranda&amp;oldid=1321009988|journal=Wikipedia|language=en}}&lt;/ref&gt;

'''1853 January 30''', Eugénie and Charles-Louis Napoléon Bonaparte married in Notre Dame.&lt;ref name=&quot;:0&quot; /&gt; Goldstone says, &quot;Eugénie wore a white velvet gown, its bodice glittering in diamonds, with a matching diamond crown from which fell a sweeping veil of delicate, handwoven English lace,&quot;&lt;ref name=&quot;:32&quot; /&gt; (192 of 909) perhaps Honiton lace, influenced by Victoria, although they had not met yet?

'''1855 April 15–20 or so''', Empress Eugénie and Napoleon III of France began a 5-day visit to the U.K.&lt;ref name=&quot;:32&quot; /&gt;{{rp|276}} Lucy Worsley says, &quot;‘Carter’s Crinoline Saloon’ opened soon afterwards, offering London ladies not only the crinoline but also the new ‘elastic stays … as worn by the Empress of the French’.&lt;sup&gt;20&lt;/sup&gt;{{rp|&quot;20 “Adburgham (1964) p. 93&quot;}} &lt;ref name=&quot;:5&quot;&gt;Worsley, Lucy. ''Queen Victoria: Twenty-Four Days That Changed Her Life''. St. Martin's Press, Hodder &amp; Stoughton, 2018.&lt;/ref&gt;{{rp|361–362, n. 20, p. 696}} Eugénie's hairdresser did not make the train to Windsor after the channel crossing, and while he was quite distraught and said he felt like cutting his throat, she was gracious and good natured.&lt;ref name=&quot;:32&quot; /&gt; (279 of 909)

A. N. Wilson says,&lt;blockquote&gt;When Napoleon and Eugénie went into London the next day, the crowds were immense. The beauty and stylishness of the Empress were immediately attractive. This was the first time London had seen a woman in a crinoline dress, grey with black lace and pink bows, and a wreath of pink chrysanthemums in her red hair.&lt;ref name=&quot;:4&quot;&gt;Wilson, A. N. ''Victoria: A Life''. Penguin, 2014. Apple Books: https://books.apple.com/us/book/victoria/id828766078.&lt;/ref&gt; (361 of 1204)&lt;/blockquote&gt;'''1855 August 18–28 or so''', Queen Victoria, Prince Albert, Bertie and Vicky visited Paris and the Exposition Universelle.&lt;ref name=&quot;:32&quot; /&gt; (287 of 909) Eugénie was weak and not able to do much, but she had organized the visit for Victoria and her family and hosted a ball at Versailles personally.

'''1856 March''', Eugénie bore Napoléon Eugène Louis Jean Joseph Bonaparte, after a 2-day labor.

'''1857 August 6–''', Eugénie and Napoleon visit the UK again. Queen Victoria describes 2 of Eugénie's is dresses with admiring precision:&lt;blockquote&gt;Eugénie looked ‘lovely in a light organdie dress, embroidered all over with violets, a wreath to match, in her hair, and pearls’. ... [388–389] On the Sunday of his visit (9 August 1857), Napoleon III went into Newport for Mass at the Roman Catholic church, accompanied by his much more pious wife this time looking 'lovely in simple white embroidered cambric dress, with lilac ribbons'.&lt;ref name=&quot;:4&quot; /&gt; (388–389 of 1204)&lt;/blockquote&gt;'''1858 January 14''', Felice Orsini and a small group bombed with pocket grenades Eugénie and Louis Napoleon on their way to the opera. Eugénie and Louis Napoleon got out of their carriage and helped the wounded. Elisabeth and Franz Joseph sent a telegram and the next day Elisabeth sent a letter to Eugénie, praising her courage. Albert's brother was in Paris near the opera and described it for Victoria.&lt;ref name=&quot;:32&quot; /&gt; (351 of 909)

'''1859''', Napoleon III was away with the army, and Eugénie was regent.&lt;ref name=&quot;:0&quot; /&gt;

'''1860s''', Eugénie &quot;often attended meetings of the Council of Ministers, even leading the meetings for a brief space of time in 1866 when her husband was away from Paris.&quot;&lt;ref name=&quot;:0&quot; /&gt;

'''1860 April 24''', Eugénie hosted a masked costume ball, &quot;the highlight of the social season&quot;:&lt;blockquote&gt;&quot;The most extravagant expectations had been excited,&quot; the newspapers observed, and by ten o’clock in the evening, the jam of carriages waiting to reach the venue testified to the veracity of this report. Inside the main salon, delicate Louis XV–styled shepherdesses clad in extravagant silks mingled with diamond-bedecked countesses dressed to represent fairy-tale characters or primal elements such as water, earth, and night. The banquet hall boasted a huge fountain, upon which shone the new invention of electric light, &quot;so adjusted as to give the watery spray as it rose and fell the appearance of frosted silver,&quot; one correspondent reported in awe. &quot;The scene was most beautiful, and called vividly to mind those abodes only read of in the Arabian Nights.&quot; At six the next morning, the last of the guests found their happy way to the coaches still waiting on the street outside. &quot;Too much praise cannot be attributed to her Majesty,… who planned so magnificent an entertainment,&quot; was the general consensus. &quot;It was worthy of an Empress.&quot;&lt;ref name=&quot;:32&quot; /&gt; (412 of 909)&lt;/blockquote&gt;'''1860 November 14''', distraught over her sister's death and Napoleon's infidelities, Eugénie asked Victoria for help and went to the UK, visiting Victoria twice, at the beginning and the end of her trip:&lt;blockquote&gt;Abruptly and without her son, she fled suddenly on November 14, 1860, to the home of friends in Scotland. Scrounging around for an excuse for her absence, the court gave out that the empress, on the advice of her physicians, was journeying to the drizzly cold of the northern country for health reasons. (&quot;Eugénie’s expedition is most astonishing,&quot; the king of the Belgians commented drily when he first heard the news. &quot;I have never heard Scotland recommended for winter excursions.&quot;) Her one preparation had been to write personally to Victoria, begging to be allowed to visit the realm incognito.&lt;ref name=&quot;:32&quot; /&gt; (465 of 909)&lt;/blockquote&gt;Victoria found Eugénie likable in their meetings. 

'''1865''', Napoleon III was away with the army, and Eugénie was regent.&lt;ref name=&quot;:0&quot; /&gt;

'''1865 April 15''', Abraham Lincoln's assassination. Eugénie's was among the first letters of condolence from a head of state that Mary Todd Lincoln got; Victoria's was dated the day after Eugénie's. Eugénie's letter reads, in part:&lt;blockquote&gt;Madame: The Emperor sends to Washington his official expressions of the indignation and sadness which have gripped the nation of France, for the fatal blow that struck President Lincoln. ... However, aside from this national misfortune, your personal tragedy arouses profound feelings in my heart. ... I would like … [sic ellipsis] to offer my own condolences, as well as the assurance of my prayers that Heaven will grant you the strength to bear this cruel ordeal. Please believe, Madame, in my deep sympathy and my most sincere sentiments.&lt;ref name=&quot;:32&quot; /&gt; (555 of 909)&lt;/blockquote&gt;'''1869''', Eugénie appointed Charles Frederick Worth to be official court dressmaker.&lt;ref name=&quot;:0&quot; /&gt;

'''1869 November 15–17''', Eugénie was regent and officially opened the Suez Canal.&lt;ref&gt;{{Cite journal|date=2026-01-09|title=Suez Canal|url=https://en.wikipedia.org/w/index.php?title=Suez_Canal&amp;oldid=1331946563|journal=Wikipedia|language=en}}&lt;/ref&gt; [[Social Victorians/People/Elisabeth of Austria|Franz Joseph I]] was present. Eugénie commissioned 250 new dresses from Worth for the event.

'''1870 July 28 – 1870 September 4''', Eugénie was head of state of France.&lt;ref name=&quot;:0&quot; /&gt;

'''1870 September 4''', Napoleon III and the 2nd Empire were overthrown.&lt;ref name=&quot;:2&quot;&gt;{{Cite journal|date=2026-01-08|title=Napoleon III|url=https://en.wikipedia.org/w/index.php?title=Napoleon_III&amp;oldid=1331844421|journal=Wikipedia|language=en}}&lt;/ref&gt;

'''1870 September 5 – 1871 March 19''', Napoleon was imprisoned in Schloss Wilhelmshöhe with his military leaders, held captive by the Germans.&lt;ref name=&quot;:2&quot; /&gt;  His soldiers were also captured, but they were housed less comfortably.

'''1871 March 20''', Eugénie and Louis, the Prince Imperial, met Napoleon III at Dover.&lt;ref name=&quot;:32&quot; /&gt; (743 of 909)

'''1879 February 27''', Louis, who wanted to join the war in South Africa, left the UK and arrived at Capetown on 26 March 1879.&lt;ref name=&quot;:32&quot; /&gt; (753 of 909)

'''1879 June 1''', Louis-Napoléon, Prince Imperial was killed in a battle with Zulu. Queen Victoria took part in the funeral procession

'''1887''', Eugénie was godmother of Princess Beatrice's daughter, Victoria Eugenie of Battenberg, born 24 October 1887,&lt;ref&gt;Van der Kiste, John. ''Queen Victoria's Children''. The History Press, 2011 (1986).&lt;/ref&gt; (177 of 300) later Queen Consort of King Alphonso XIII of Spain.

'''1894''', Eugénie was at a dinner with Queen Victoria where she met Cecil Rhodes.&lt;ref name=&quot;:4&quot; /&gt; (939 of 1204)

==Demographics==

=== Nationality ===
* Eugénie, born in Granada, Spain&lt;ref name=&quot;:0&quot; /&gt;
* Charles-Louis, born in France
===Residences===

==== Eugénie ====

* Saint-Cloud, with a model Petit Trianon&lt;ref name=&quot;:0&quot; /&gt;
* After 1854: Villa Eugénie, Biarritz (today the Hôtel du Palais)&lt;ref name=&quot;:0&quot; /&gt;
* After 19 March 1871: Camden Place in Chislehurst, Kent&lt;ref name=&quot;:0&quot; /&gt;
* 1885: Farnborough Hill in Farnborough, Hampshire&lt;ref name=&quot;:0&quot; /&gt;
* Retirement: Villa Cyrnos, France

==Family==

* Eugénie de Montijo (5 May 1826 – 11 July 1920)&lt;ref name=&quot;:0&quot; /&gt;
* Charles-Louis Napoléon Bonaparte (20 April 1808 – 9 January 1873)&lt;ref name=&quot;:2&quot; /&gt;
*#Louis-Napoléon, Prince Imperial (16 March 1856 – 1 June 1879)&lt;ref name=&quot;:3&quot; /&gt;

===Relations===

* Charles-Louis Napoléon Bonaparte was the paternal nephew of Napoleon I.&lt;ref name=&quot;:2&quot; /&gt;
* Eugénie's sister, Paca (María Francisca de Sales de Palafox Portocarrero y Kirkpatrick) married the [[Social Victorians/People/Alva|15th Duke of Alba]] in 1849 (also, the Dukes of Alba and Berwick?).&lt;ref name=&quot;:1&quot; /&gt;

=== Notes ===

* A. N. Wilson says, &quot;When Napoleon III himself speculated that she might be the daughter of Lord Clarendon, Eugénie's mother replied, '''Mais, sire, les dates ne correspondent pas''.'&quot;&lt;ref name=&quot;:4&quot; /&gt; (325 of 1204)

==Questions and Notes==
==Bibliography==

# Adburgham, Alison. ''Liberty's: A Biography of a Shop''. Routledge, 2023.
# Adburgham, Alison. ''Shops and Shopping, 1800–1914: Where, and in What Manner the Well-Dressed Englishwoman Bought Her Clothes''.  2nd ed. Allen and Unwin, 1981. (First published 1964.)
# Dolan, Therese. &quot;The Empress's New Clothes: Fashion and Politics in Second Empire France.&quot; ''Woman's Art Journal''. 15:1 (1994): 23.
# von Metternich-Winneburg, Pauline Clementine Marie Walburga (Sándor von Szlavnicza), Fürstin. ''My Years in Paris''. E. Nash &amp; Grayson: London, 1922.

==References==
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      <text bytes="25845" sha1="7qk6l0ugq7gnyvy4y50dsf8kt0268o4" xml:space="preserve">[[File:Kaiserin Elisabeth im Morgenlicht.jpg|alt=Old painting of a beautiful white woman with her very long hair tied at below her neck like a scarf|thumb|Winterhalter, &quot;Empress Elisabeth in the Morning Light,&quot; 1864]]

==Overview==
Empress Elisabeth was politically progressive, especially compared to her husband, Franz Josef I, and his mother (and her aunt) Archduchess Sophia, who were committed to absolute and hereditary monarchy and were never able to see the potential value of a compromise or the development of a constitutional monarchy. Elisabeth had been raised in an unstructured and permissive environment with a lot of personal freedom. She was more alienated from the very conservative court than she was from the people in the Austrian and Hungarian empire who were demanding a constitutional monarchy.

At the beginning of her married life, Elisabeth lived in at the Habsburg court in Vienna in what was for her a toxic environment that undermined her health and agency. She was surprised to find herself pregnant and developed a fear of going down steep stairs.&lt;ref name=&quot;:4&quot; /&gt; She had a persistent cough and, after her daughter Sophie died at 2 in 1857, was unable to eat, to the point that sometimes she was too weak and her feet and legs were so swollen she could not walk without help. In November 1860, when Elisabeth was very ill, Queen Victoria sent her yacht to take her from Antwerp to Madeira, where she did in fact get better.

Franz Josef I was a Hapsburg but not the Holy Roman Emperor: the Holy Roman Empire was &quot;formally brought ... to an end&quot; by Napoleon in 1806.&lt;ref name=&quot;:9&quot;&gt;Wilson, A. N. ''Victoria: A Life''. Penguin, 2014. Apple Books:  https://books.apple.com/us/book/victoria/id828766078.&lt;/ref&gt;{{rp|68 of 1204}} He, King Edward VII, and Emperor William II of Germany were the most powerful monarchs in Europe, but Kaiser William was &quot;showy, vainglorious, unstable&quot; and Franz Josef was &quot;imposssibly remote.&quot;&lt;ref name=&quot;:8&quot;&gt;Van der Kiste, John. ''Queen Victoria's Children''. The History Press, 2011 (1986).&lt;/ref&gt;{{rp|247 of 300}} According to A. N. Wilson, Prince &quot;Albert wanted to be friends with the Emperor of Austria, perhaps one day to see him as the figurehead of a united Germany.&quot;&lt;ref name=&quot;:9&quot; /&gt;{{rp|246 of 1204}}

After the death of her first child, Elisabeth's next two children were also taken away from her and raised by the Archduchess Sophia, so their relationship with their mother was never close. Elisabeth herself raised her last child, Valerie, born in 1868 in Buda, Hungary.&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|date=2026-01-13|title=Archduchess Marie Valerie of Austria|url=https://en.wikipedia.org/w/index.php?title=Archduchess_Marie_Valerie_of_Austria&amp;oldid=1332726352|journal=Wikipedia|language=en}}&lt;/ref&gt; Although he was politically reactionary, not faithful and unable to withstand his mother's influence, Franz Joseph actually seems to have loved Elisabeth, who found him incompatible. Their relationship improved over the course of their lives, and their letters suggest it had developed into a genuine friendship when Elisabeth was assassinated in 1898.&lt;ref name=&quot;:4&quot; /&gt;

She read widely and collected the letters of Heinrich Heine.&lt;ref name=&quot;:4&quot; /&gt; She wrote personal poetry, which she published, that was inspired by Heine. She wrote a &quot;humorous&quot; poem about Albert Edward, Prince of Wales called &quot;There is somebody coming upstairs.&quot;&lt;ref name=&quot;:10&quot;&gt;Káli-Rozmis, Barbara. &quot;The Visits of Empress Elisabeth of Austria in Britain and its Impact on the English and Irish People.&quot; Freeside Europe Academic Journal 2020, 1 [https://www.freesideeurope.com/articles/the-visits-of-empress-elisabeth-of-austria-in-britain-and-its-impact-on-the-english-and-irish-people-2 DOI 10.51313/alumni-2020-2].

https://www.freesideeurope.com/images/pdf/pdf_2.pdf&lt;/ref&gt;{{rp|4}} She also traveled widely, spending much of her time away from Vienna once her first 3 children were born. She was not constrained by what Queen Victoria would have considered her &quot;duty&quot; as monarch, spent a great deal of time away from court and did not take part in state occasions, which she found boring. Her English was very good.

Elisabeth's affinity for Hungary made her a successful advocate and negotiator between it and Austria.

She may have suffered from depression and possibly disordered eating — anorexia, compulsive exercising and bulimia can be inferred. Later in her life, she took up smoking.[[File:Winterhalter Elisabeth.jpg|alt=Old painting of a wealthy queen wearing a fluffy pink dress and lots of jewelry|thumb|Empress Elisabeth by Winterhalter, 1865, in a dress by Maison Worth]]

As an adult Elisabeth was a client of Charles Frederick Worth of Paris.&lt;ref&gt;{{Cite journal|title=Charles Frederick Worth|url=https://en.wikipedia.org/w/index.php?title=Charles_Frederick_Worth&amp;oldid=1322100685|journal=Wikipedia|date=2025-11-14|language=en}}&lt;/ref&gt; The 1865 Franz Xaver Winterhalter portrait of her (top right) shows her in a Worth pink tulle ball gown with her signature diamond stars in her hair.  She practiced tight lacing, getting her waist down to 16 inches in 1859–1860. One lady in waiting said in 1863 she was no longer tight lacing (perhaps she had stopped using French leather corsets), but her waist was 18&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt;–19&lt;sup&gt;1&lt;/sup&gt;/&lt;sub&gt;2&lt;/sub&gt; inches for the rest of her life.&lt;ref name=&quot;:4&quot; /&gt; Her youngest daughter was afraid of Queen Victoria when she met her because of the size of the older woman's body and, presumably, her mother's obsesssions.

The leather corsets, which are known to have been used by Parisian sex workers, may have been part of a &quot;bordello-culture&quot; interest of Sisi's,&lt;ref name=&quot;:11&quot;&gt;{{Cite web|url=https://www.andreanolen.com/home/empress-sisi-and-prostitution|title=Empress 'Sisi' and Prostitution|date=2021-08-17|website=Andrea Nolen|language=en-US|access-date=2026-03-27}}&lt;/ref&gt; but the tight lacing is also consistent with disordered eating, in this case actively attempting to be extremely thin.

Elisabeth dressed opulently, although as a very skilled horsewoman, she often wore riding habits, and was very fashion forward, owning a cage early and giving them up for the more fitted look as soon as Worth and fashion did.&lt;ref name=&quot;:4&quot; /&gt; She was &quot;possibly the best-known [horsewoman] of her time&lt;ref name=&quot;:4&quot; /&gt; and, like Alexandra, Princess of Wales, wore closely fitting riding habits as normal day wear. Her clothing fit her so perfectly that it was said that she was sewn into it,&lt;ref name=&quot;:10&quot; /&gt;{{rp|4}} and certainly she had the staff for such a practice. Her riding habit was dark blue with gold buttons, accessorized with a blue veil and an orange fan.&lt;ref name=&quot;:10&quot; /&gt;{{rp|4}}

Elisabeth modeled her stylish look on Eugénie:&lt;blockquote&gt;Her spectacular crinoline evening gowns, modeled on Eugénie’s, were so bouffant that on one occasion the archbishop of Milan became entangled with her dress merely by passing her on a staircase, and to her barely repressed giggles, could only be liberated by tearing off some of the outer folds.&lt;ref name=&quot;:6&quot;&gt;Goldstone, Nancy. ''The Rebel Empresses: Elisabeth of Austria and Eugénie of France, Power and Glamour in the Struggle for Europe''. Little, Brown, 2025.&lt;/ref&gt;{{rp|322 of 909}}&lt;/blockquote&gt;

Sisi collected photographs of beautiful women from all classes, including sex workers, in &quot;albums of beauty.&quot; Christina Catherine Martinez says she had 2,000 photographs in her collection.&lt;ref name=&quot;:11&quot; /&gt;

Elisabeth's relationship with Victoria was not very warm or close, although Elisabeth spent a lot of time in the UK and Ireland. Perhaps her many trips to Ireland and few visits to Victoria had an effect on the older woman.

==Also Known As==
* Family name
** Hers: House of Wittelsbach, Bavaria
** His: Habsburg
* Elisabeth Amalie Eugenie
* Sisi
* Elisabeth of Bavaria
* For incognito travel: Countess Hohenembs&lt;ref name=&quot;:4&quot; /&gt;

==Acquaintances, Friends and Enemies==
===Elisabeth===

==== Acquaintances ====
* [[Social Victorians/People/Mary Todd Lincoln|Mary Todd Lincoln]]
* [[Social Victorians/People/Queen Victoria|Queen Victoria]]
* Vicky?

==== Friends ====
* Count Gyula Andrássy
* Albert Edward, Prince of Wales&lt;ref name=&quot;:10&quot; /&gt; (2)
* Alexandra, Princess of Wales
* Lord Spencer, former Lord Lieutenant of Ireland&lt;ref name=&quot;:10&quot; /&gt; (3)
* Captain William George (Bay) Middleton, Scots horseman and equerry to Spencer&lt;ref name=&quot;:10&quot; /&gt; (3)

==== Enemies ====
* Princess Pauline von Metternich, close friend of [[Social Victorians/People/Eugenie of France|Eugénie of France]]

==== Employees and Members of Her Retinue ====

* Franziska (Fanny) Feifalik, hairdresser, &quot;enge Vertraute [close confidant]&quot; and &quot;ihr Double [body double]&quot; (1863–1896)&lt;ref&gt;{{Cite journal|date=2025-08-19|title=Franziska Feifalik|url=https://de.wikipedia.org/w/index.php?title=Franziska_Feifalik&amp;oldid=259011659|journal=Wikipedia|language=de}}&lt;/ref&gt;

===Franz Joseph===

==== Friends and Allies ====

* Albert Edward, Prince of Wales&lt;ref name=&quot;:8&quot; /&gt; (244 of 300)
* William II, Kaiser of Germany

==== Enemies ====

==Organizations and Social Networks==

=== Queens and Political Leaders ===
* [[Social Victorians/People/Queen Victoria|Queen Victoria]]
* [[Social Victorians/People/Sophie of Wurttemberg|Sophie of Württemberg, Queen of the Netherlands]]
* [[Social Victorians/People/Mary Todd Lincoln|Mary Todd Lincoln]]
* [[Social Victorians/People/Eugenie of France|Empress Eugenie of France]]

=== Couturiers ===

* Maison Worth, Paris
* Antal Alter, Budapest (still in existence, as Alter és Kiss)&lt;ref name=&quot;:4&quot; /&gt;

==Timeline==
[[File:Kaiserin Elisabeth von Österreich und Königin von Ungarn.jpg|alt=Old painting of a white woman dressed opulently with jewels and a crown on her head|thumb|Empress Elisabeth of Austria, 1854]]
'''1853 August 18''', Franz Joseph I of Austria's birthday, the ball where he chose Elisabeth over her sister Helene, whom Archduchess Sophie had arranged for him to meet. While Helene was dressed and coiffed in a sophisticated way to attract Franz Joseph,&lt;blockquote&gt;Sisi, by contrast, was attired in a simple, pink-and-white cotton tarlatan frock. To give a sense of the disparity between the two sisters’ costumes, cotton tarlatan was what Meg from ''Little Women'' was intending to wear to the neighborhood dance until kind friends took pity on her and lent her a nicer gown.&lt;ref name=&quot;:6&quot; /&gt; (84 of 909)&lt;/blockquote&gt;'''1854 April 21''', Elisabeth of Bavaria and Franz Joseph I married.&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|date=2026-01-09|title=Empress Elisabeth of Austria|url=https://en.wikipedia.org/w/index.php?title=Empress_Elisabeth_of_Austria&amp;oldid=1332040784|journal=Wikipedia|language=en}}&lt;/ref&gt; Goldstone says, &quot;“Elisabeth’s dress was of shimmering white silk, richly embroidered in gold and silver. The bodice was adorned with white roses, and on her head was perched her mother-in-law’s diamond-and-opal wedding crown, draped in a veil of gold lace.&quot;&lt;ref name=&quot;:6&quot; /&gt; (216 of 909)

'''1855 March 5''', Elisabeth gave birth to Sophie Friederike Dorothea Maria Josepha, named by Archduchess Sophia, her mother in law; Archduchess Sophia generally denied Elisabeth access to Sophie and did not permit her to breastfeed the child.&lt;ref name=&quot;:4&quot; /&gt;

'''1856 July 12''', Elisabeth gave birth to Gisela Louise Marie, also taken to be raised by Archduchess Sophia.&lt;ref name=&quot;:4&quot; /&gt;

'''1857 January 5''', Elisabeth and Franz Joseph began a tour of Italy from Venice to Milan, meeting hostility from the people, who were very close to outright rebellion.&lt;ref name=&quot;:6&quot; /&gt; (320–322 of 909)

'''1857 May 29''', while the family was traveling in Hungary, the two daughters of Elisabeth and Franz Joseph got sick, and Sophie — the youngest one — died, at two years old.&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|date=2026-01-13|title=Archduchess Sophie of Austria|url=https://en.wikipedia.org/w/index.php?title=Archduchess_Sophie_of_Austria&amp;oldid=1332725850|journal=Wikipedia|language=en}}&lt;/ref&gt;

'''1857 July 27''', Maximilian (Maximilian I of Mexico) and Charlotte, daughter of Leopold I of Belgium, married.&lt;ref name=&quot;:7&quot;&gt;{{Cite journal|title=Maximilian I of Mexico|url=https://en.wikipedia.org/w/index.php?title=Maximilian_I_of_Mexico&amp;oldid=1333693281|journal=Wikipedia|date=2026-01-19|language=en}}&lt;/ref&gt; Charlotte was Queen Victoria's cousin, so the marriage caused an alliance between the House of Habsburg and the House of Saxe-Corburn and Gotha. Prince Albert and many European royals attended the wedding.

'''1858 August 21''', Elisabeth gave birth to Rudolf Franz Karl Josef, the fact that he was male and thus an heir occasioning a 101-gun salute.&lt;ref name=&quot;:4&quot; /&gt;

'''1860 November 19''', Elisabeth sailed south from Antwerp to Madeira on Queen Victoria's yacht. Somehow Victoria knew how sick she was.&lt;ref name=&quot;:6&quot; /&gt; (476 of 909, n. ii) (447 of 909) Her health began to improve immediately, and she ate well on the ship during the rough passage. She had ladies in waiting with her, and as always she was gossiped about maliciously.&lt;ref name=&quot;:6&quot; /&gt; (478–482 of 909)
[[File:Kaiserin Elisabeth 1862.jpg|alt=Old photograph of a beautiful white woman dressed stylishly|thumb|Elisabeth of Austria, 1862]]
'''1861 April 28''', Elisabeth left Madeira to go back to Vienna. She began to be ill again almost immediately and was worse than she had been before she had gone to Madeira. This time, Franz Joseph had his brother Max (Maximilian I of Mexico) take her to Corfu on his yacht.&lt;ref name=&quot;:6&quot; /&gt; (482–483 of 909)

'''1862''', Sisi had a belt as an exhibit in the Great London Exposition or International Exhibition. David Kunzle, in his 1982 ''Fashion and Fetishism'', says,&lt;blockquote&gt;The pro-corset party called upon an exceedingly prestigious example — that of the beautiful Empress Elizabeth of Austria, reputed to possess the smallest waist ever seen. Together with her portrait, her girdle had been on view at the Great Exhibition. A query as to the exact length of this girdle was answered by Miss Turnour, who, when she visited the Exhibition, had seen the exhibitor hold it up in his hand, “so beautiful, purple velvet stiff with rich gold thread, looking like a dog-collar when clasped.” Miss Turnour was even allowed to make a “frantic and futile effort” to close it round her own slender waist, and was enabled to certify the measurement to be exactly 16 inches; and the Empress was tall, 5 foot 6 inches.

The unspoken inference was, of course, that the Empress indulged in tight-lacing, which thereby acquired a social cachet otherwise totally lacking …. ...

Why was an object with such scandalous associations put on public display? With her horror of publicity, especially as regards details of her personal life, it seems inexplicable that the Empress would have encouraged gossip around so intimate a matter as a waist-measurement. If the numerous biographies remain silent on this curious episode, is it because domestically the matter was hushed up? After all, in order to protect the imperial dignity the police actively suppressed stories of her equine acrobatics, and destroyed photographs pertaining to it. If the 16 inch belt was displayed with her permission and knowledge (and it seems hard to conceive otherwise) or, worse, on her personal initiative, was it intended as a provocation? Was it the bizarre symbol of or satire upon the exhibitionism to which the most adulated woman in Europe was subject?&lt;sup&gt;Qtd. in&lt;/sup&gt; &lt;ref name=&quot;:11&quot; /&gt;&lt;/blockquote&gt;'''1863 September 3''', Franz Josef and Queen Victoria met at the Ehrenburg Palace (Schloss Ehrenberg) in Coburg, Franconia, what is now Germany.&lt;ref name=&quot;:9&quot; /&gt; (499 of 1204) Elisabeth does not seem to have been there, and for Victoria, this was too soon after Albert's death. She had no ministers but Princesses Alice and Helena with her, and she was shy and out of her depth.

'''1867''', because of Elisabeth's negotiations, including a decision to have another child and thus to reopen relations with Franz Joseph I, the Austro-Hungarian Compromise of 1867 made Hungary and Austria dual monarchies.&lt;ref name=&quot;:4&quot; /&gt; Franz Josef and Elisabeth were crowned king and queen of Hungary, and she set up &quot;her 'own' court in the Royal Palace of Gödöllő, Hungary .... She asked Franz Joseph to finance her a trainer from the Renz Circus and two of its famous equestriennes, Emilie Loiset and Elise Petzold taught her Circus stunts [sic punctuation].&quot;&lt;ref name=&quot;:10&quot; /&gt; (1)

'''1867 June 19''', Franz Joseph's brother, Maximilian (then Maximilian I of Mexico) was executed by a firing squad of revolutionaries resisting monarchy.&lt;ref name=&quot;:7&quot; /&gt;

'''1867 December 24''', Elisabeth's 30th birthday, after which she did not permit any photographs to be taken, although paparazzi sometimes succeeded in getting an image.&lt;ref name=&quot;:10&quot; /&gt; (4)

'''1868 April 22,''' Elisabeth gave birth to Marie Valerie Mathilde Amalie, the &quot;Hungarian child.&quot;&lt;ref name=&quot;:4&quot; /&gt;

'''1869 June''', Franz Joseph and Elisabeth were crowned King and Queen of Hungary.&lt;ref name=&quot;:4&quot; /&gt;

'''1869 November 15–17''', [[Social Victorians/People/Elisabeth of Austria|Franz Joseph I]] was at the official opening of the Suez Canal.&lt;ref&gt;{{Cite journal|date=2026-01-09|title=Suez Canal|url=https://en.wikipedia.org/w/index.php?title=Suez_Canal&amp;oldid=1331946563|journal=Wikipedia|language=en}}&lt;/ref&gt; Was Elisabeth there?

'''1872 May 28''', Archduchess Sophia died.&lt;ref&gt;{{Cite journal|title=Princess Sophie of Bavaria|url=https://en.wikipedia.org/w/index.php?title=Princess_Sophie_of_Bavaria&amp;oldid=1333066696|journal=Wikipedia|date=2026-01-15|language=en}}&lt;/ref&gt;

'''1873''', the Great Exhibition in Vienna attracted people like Albert Edward, Prince of Wales, who &quot;persuaded [Elisabeth] to visit England during the hunting season.&quot;&lt;ref name=&quot;:10&quot; /&gt; (2)

'''1873 April 20''', Elisabeth's daughter Gisela married Prince Leopold of Bavaria. Elisabeth did not attend the wedding.&lt;ref name=&quot;:2&quot;&gt;{{Cite journal|date=2026-01-13|title=Archduchess Gisela of Austria|url=https://en.wikipedia.org/w/index.php?title=Archduchess_Gisela_of_Austria&amp;oldid=1332726064|journal=Wikipedia|language=en}}&lt;/ref&gt;

'''1874 January 8''', Gisela's first child was born; Elisabeth attended the christening.&lt;ref name=&quot;:2&quot; /&gt;

'''1874 August 1''', Elisabeth, her daughter Marie Valerie, Countess Marie Festetics and Elisabeth's entourage, horses and a dog arrived at Ventnor, Steephill Castle, on the Isle of Wight, near Osborne House. (The castle had been rented for 2 months, and some renovations were made before Elisabeth arrived.)&lt;ref name=&quot;:10&quot; /&gt; (2)

'''1874 August 16''', Queen Victoria visited Elisabeth from Osborne House. Countess Marie Festetics wrote,&lt;blockquote&gt;I was not surprised. [The sight of the Queen] standing next to the Empress was startling, though. Archduchess Valerie was rather scared from the sight. She has never seen anybody with such a strong body. She [Queen Victoria] was quite warm-hearted, though. [interpolations Káli-Rozmis's]&lt;ref name=&quot;:10&quot; /&gt; (2)&lt;/blockquote&gt;Queen Victoria invited Elisabeth to Osborne House, but Elisabeth declined on the grounds of &quot;delicate health,&quot; although she told her mother she declined because she was bored by &quot;that kind of thing.&quot;&lt;ref name=&quot;:10&quot; /&gt; (2)

'''1875''', Elisabeth went to Fecamp, Normandy, France with &quot;her daughter Marie Valerie and a court of 70 people.&quot;&lt;ref name=&quot;:4&quot; /&gt;

'''1876, late winter–early spring''', Elisabeth visited the UK for the hunting season, spending about 6 weeks.

'''1876 March''', Elisabeth visited Victoria at Windsor Castle but spent her time with Baron Ferdinand Rothschild.&lt;ref name=&quot;:10&quot; /&gt; (2) Victoria did feel these as snubs.

'''1876 Autumn''', Elisabeth and Franz Josef were at the Royal Palace of Gödöllő, and Bay Middleton was there.

'''1877, late winter–early spring''', Elisabeth visited the UK for the hunting season with Bay Middleton as her &quot;pilot,&quot; spending about 6 weeks.

'''1877 Autumn''', Elisabeth and Franz Josef were at the Royal Palace of Gödöllő, and Bay Middleton was there.

'''1878 January''', Elisabeth and her son Rudolph were in the UK, she riding to hounds and he &quot;toured factories and towns, charmed Victoria and her daughter Princess Beatrice, and made the acquaintance [sic; this had already happened] of Edward, Prince of Wales, the queen’s rakish heir.&quot;&lt;ref name=&quot;:6&quot; /&gt; (760 of 909) On this trip, he believed some untrue gossip that his mother was having an affair with Captain Bay Middleton, who was her &quot;pilot&quot; on the hunting trips; it took her years to forgive him his doubting of her.&lt;ref name=&quot;:6&quot; /&gt; (761 of 909)

'''1878 Autumn''', Elisabeth and Franz Josef were at the Royal Palace of Gödöllő, and Bay Middleton was there.

'''1879, late winter–early spring''', Elisabeth visited the UK for the hunting season with Bay Middleton as her &quot;pilot,&quot; spending about 6 weeks.

'''1879 Autumn''', Elisabeth and Franz Josef were at the Royal Palace of Gödöllő, and Bay Middleton was there.

'''1880, late winter–early spring''', Elisabeth visited the UK for the hunting season with Bay Middleton as her &quot;pilot,&quot; spending about 6 weeks.

'''1881''', Elisabeth purchased a country house in the UK &quot;and had a spiral staircase built from her sitting room into the kitchen, so that she could reach it in private,&quot; suggesting some bulimic practices.&lt;ref name=&quot;:4&quot; /&gt; (Not clear how this relates to Combermere Abbey, which she rented.)

'''1881, late winter–early spring''', Elisabeth rented Combermere Abbey from Colonel Wellington Henry Stapleton-Cotton, 2nd Viscount Combermere.&lt;ref name=&quot;:4&quot; /&gt; She was there for the hunting season with Bay Middleton as her &quot;pilot,&quot; spending about 6 weeks.

'''1881 February 25''', a letter by Ellen Harriet Tollet about Elisabeth hunting:&lt;blockquote&gt;We had a great meet at Woore to see the Empress jump. ... her manner Mrs C says was ‘most queenly.’ She is still a pretty looking woman with a fine figure and an awfully tight habit, so tight, she descended the stairs at Woore Hall sideways, she could not walk straight in her habit. She was ... talking English perfectly. The funniest thing was her enormous orange fan which she used out hunting, when at a check. Where she kept it, I don’t know. She gave £200 to the United Hunt races near Whitchurch yesterday and was present galloping about with her fan up. (''Shropshire Archives'', quoted by Webb) [sic punctuation]&lt;ref name=&quot;:10&quot; /&gt; (4)&lt;/blockquote&gt;She may have carried the fan to prevent people from seeing her aging face?

'''1881 May 10''', Rudolf married Princess Stéphanie of Belgium, a daughter of King Leopold II of Belgium (whose father had married George IV of England's daughter, Charlotte, Princess of Wales).&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|date=2026-01-13|title=Rudolf, Crown Prince of Austria|url=https://en.wikipedia.org/w/index.php?title=Rudolf,_Crown_Prince_of_Austria&amp;oldid=1332726212|journal=Wikipedia|language=en}}&lt;/ref&gt;

'''1882, late winter–early spring''', Elisabeth again rented Combermere Abbey from Colonel Wellington Henry Stapleton-Cotton, 2nd Viscount Combermere.&lt;ref name=&quot;:4&quot; /&gt; She was there for the hunting season, spending about 6 weeks.

'''1882 March''', Elisabeth was in Paris to visit her sister, Duchess Sophie Charlotte in Bavaria.&lt;ref name=&quot;:4&quot; /&gt;

'''1883''', Elisabeth seems to have given up hunting after the season in 1882. Bay Middleton was getting married and could no longer pilot her, and she had sciatica. Elisabeth said of her giving up hunting, &quot;Suddenly and without all true reason I lost my heart, and I, who did not consider any danger the previous day, then found it in every bush and I could not get rid of its image.&quot;&lt;ref name=&quot;:10&quot; /&gt; (5)

'''1888 March''', Elisabeth was in London, and Bay Middleton visited her there.&lt;ref name=&quot;:10&quot; /&gt; (3)

'''1888 September''', Albert Edward, Prince of Wales was a guest of Franz Josef and embroiled in a fight with Wilhelm II, whose father Fritz had just died.&lt;ref name=&quot;:9&quot; /&gt; (835 of 1204)

'''1889 January 30''', Rudolph and his mistress, the 17-year-old Baroness Marie von Vetsera, were found dead of a murder-suicide in his hunting lodge Mayerling.&lt;ref name=&quot;:3&quot; /&gt; Elisabeth wore mourning for the rest of her life, black or pearl grey.

'''1890 July 31''', Valerie and Franz Salvator married, after her renunciation of rights to the Austrian throne.&lt;ref name=&quot;:0&quot; /&gt;

'''1892 April 9''', Bay Middleton died in a hunting accident. 

'''1896-1897''', Elisabeth was staying in the Grand Hôtel du Cap-Martin, in Roquebrune-Cap-Martin, in Provence.&lt;ref name=&quot;:4&quot; /&gt; Elisabeth and Victoria met ther in 1896.

'''1898 September''', Elisabeth was assassinated by an anarchist, stabbed in the heart by Italian Luigi Lucheni.&lt;ref name=&quot;:4&quot; /&gt;

==Demographics==

=== Nationality ===
* Elisabeth: Bavarian
* Franz Joseph II: Austrian
===Residences===

* 1890: Elisabeth had the palace Achilleion built on Corfu&lt;ref name=&quot;:4&quot; /&gt;

==Family==

* Elisabeth of Bavaria (24 December 1837 – 10 September 1898)&lt;ref&gt;{{Cite journal|date=2026-01-09|title=Empress Elisabeth of Austria|url=https://en.wikipedia.org/w/index.php?title=Empress_Elisabeth_of_Austria&amp;oldid=1332040784|journal=Wikipedia|language=en}}&lt;/ref&gt;
* Franz Josef I (18 August 1830 – 21 November 1916)&lt;ref name=&quot;:5&quot;&gt;{{Cite journal|date=2026-01-22|title=Franz Joseph I|url=https://en.wikipedia.org/w/index.php?title=Franz_Joseph_I&amp;oldid=1334319684|journal=Wikipedia|language=en}}&lt;/ref&gt;
*# Archduchess Sophie (5 March 1855 – 29 May 1857)&lt;ref name=&quot;:1&quot; /&gt;
*# Princess Gisela of Bavaria (12 July 1856 – 27 July 1932)&lt;ref name=&quot;:2&quot; /&gt;
*# Rudolf, Crown Prince of Austria (21 August 1858 – 30 January 1889)&lt;ref name=&quot;:3&quot; /&gt;
*# Archduchess Valerie (Marie Valerie) (22 April 1868 – 6 September 1924)&lt;ref name=&quot;:0&quot; /&gt;

===Relations===

==Questions and Notes==
==Bibliography==
{{reflist}}</text>
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    <title>Communications Law in Malta</title>
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      <text bytes="35763" sha1="jv0zbbu15il2np0bwlxl9pjxezajo7f" xml:space="preserve">== 1. History of Malta and Maltese Communication Law ==
{{law}}

==== History of Malta ====
The Mediterranean island of [https://www.gov.mt/en/Pages/Home.aspx Malta] is located directly in the crossroads of several formidable historical forces.  The impacts of this are clearly seen in the country's dramatic history, with effects still present in its &quot;mixed&quot; legal system.  Situated south of Sicily, Malta was [[wikipedia:Capture_of_Malta_(218_BC)#:~:text=The%20capture%20of%20Malta%20was,Punic%20War%20in%20218%20BC.|conquered by Rome]] in 218 B.C. in the Second Punic War.  This forced the small island to be placed under Roman law for the majority of time between 218 B.C. and 1530, when Holy Roman Emperor Charles V granted the island of Malta to the Order of St. John (also known as the &quot;Knights Hospitallers&quot;). The tiny island was then ruled shortly by France from 1789 - 1813 and then England from 1813 forward,&lt;ref&gt;{{Cite news|url=https://www.bbc.com/news/world-europe-17598542|title=Malta profile - Timeline|date=2012-04-03|work=BBC News|access-date=2026-02-21|language=en-GB}}&lt;/ref&gt; only gaining its sovereign independence in 1964.&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=Aquilina|first=Kevin|date=2013|title=The Nature and Sources of The Maltese Mixed Legal System: A Strange Case of Dr. Jekyll and Mr. Hyde?|url=https://heinonline.org/HOL/Page?handle=hein.journals/complv4&amp;id=&amp;collection=journals&amp;div=8|journal=Comparative law review|pages=1-38|via=HeinOnline}}&lt;/ref&gt;
[[File:Malta, 2010 - panoramio - Bengt Nyman (23).jpg|thumb]]Malta's &quot;mixed system&quot; is based most clearly on the fact that it maintains both civil and common law.&lt;ref name=&quot;:0&quot; /&gt; In practice, this means it follows a legal system predicated on both statutes and past court decisions.  Malta has been influenced greatly by international and European systems - the small country has a &quot;Westminster&quot; styled Constitution, but does not hold Constitutional Court decisions as binding on any other than the parties in dispute.&lt;ref name=&quot;:0&quot; /&gt;

==== Maltese Communication Law History ====
Malta was owned by the British as the &quot;Crown Colony of Malta&quot; from 1813 - 1964, a time period that included many critical changes in the modernization of communication.  As a colony of the British Empire, Malta was subject to British influence on both the implementation of the modern means of communication, as well as the laws surrounding the regulation of these means.&lt;ref&gt;{{Cite web|url=https://broadcasting-authority.azurewebsites.net/ourhistory?utm_source=chatgpt.com|title=Our History {{!}} Broadcasting Authority|website=broadcasting-authority.azurewebsites.net|access-date=2026-03-18}}&lt;/ref&gt; This included the creation of the Broadcast Authority, established in 1961 to regulate all broadcasting in the nation.&lt;ref&gt;{{Cite web|url=https://ba.org.mt/en/about-us|title=About Us|website=ba.org.mt|access-date=2026-03-19}}&lt;/ref&gt; 

Radio broadcasting first came to Malta in the form of cable radio through a private British company called &quot;Rediffusion&quot; in 1935.&lt;ref name=&quot;:6&quot;&gt;{{Cite web|url=https://timesofmalta.com/article/radio-broadcasting-turns-90.1119330|title=Radio broadcasting turns 90|last=Borg|first=Fr Joe|date=2025-11-11|website=Times of Malta|language=en-gb|access-date=2026-03-18}}&lt;/ref&gt;  This was especially important at the time, as Malta's cultural and physical closeness to Italy put them at risk of infiltration by the ideas of the expanding Fascist Mussolini regime through wireless radio broadcasts.&lt;ref name=&quot;:6&quot; /&gt; To combat this, British Colonial Authorities strategically placed loudspeakers with approved programs across the island, and banned the use of non-wire radio.&lt;ref name=&quot;:6&quot; /&gt;  The easy public access to radio programs, paired with the auditory nature of the news bypassing the necessity to be literate, caused radio to surpass newspapers as the main source of information to the public.&lt;ref name=&quot;:6&quot; /&gt; 

The perceived danger to Maltese citizens from Italian and Sicilian influence continued to pressure the island and those who ruled it to continually adopt updated communication systems.  In the 1950s, Maltese reception of Italian TV signals led the country to first create the television network that would become TVM, and later, the adoption of color broadcast.&lt;ref name=&quot;:6&quot; /&gt;  The geographical proximity to Italy created a situation such that, when the one-station system censored differing political ideas, the Nationalist party was able to resort to Sicilian radio and television stations to broadcast their opposing viewpoints.&lt;ref name=&quot;:6&quot; /&gt;

The small nation's reliance on Rediffusion lasted up until 1971, when the Mintoff government decided to nationalize radio and television; this was ultimately not successful until 1975.&lt;ref name=&quot;:6&quot; /&gt;  This led to the establishment of Xandir Malta, or &quot;Broadcasting Malta,&quot; a Socialist-run broadcasting network that goverened the communications sector until it was pushed out by the Nationalist party in the late 1980s.  The Nationalist party then established a public broadcasting network and issued &quot;The 1990 White Paper Commitment to Pluralism,&quot; promoting pluralism in broadcasting.&lt;ref name=&quot;:6&quot; /&gt;  The Church and the Government were the first to have their own radio stations, followed by additions in 1992.  Radio broadcasting is reportedly consumed by two-thirds of Maltese citizens to this day.&lt;ref name=&quot;:6&quot; /&gt;

When Malta joined the European Union (EU) in 2004, they were required to extend their communications legislation to meet EU standards, including promoting European and independent works.&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/legal-content/EN/TXT/?uri=legissum:e20112|website=eur-lex.europa.eu|access-date=2026-03-19}}&lt;/ref&gt;  Malta has since joined multiple international agreements, such that the Maltese lawmakers must ensure that both the law of Malta and the terms of multiple treaties and directives are followed when writing legislation.  Maltese communication law is emblematic of the legal situation of many European countries: though the country maintains its sovereign independence, the law is riddled with influence from past rulers, as well as attempts to catch up with technological progress and the ever-expanding process of globalization.  

== 2. Sources of Maltese Communications Law ==

==== National ====
[[File:Malta ali 2009224 lrg.jpg|thumb]]
The primary statute governing communications law in Malta is the [https://legislation.mt/eli/cap/399/eng?utm_source=chatgpt.com Electronic Communications (Regulation) Act].&lt;ref&gt;{{Cite web|url=https://legislation.mt/eli/cap/399/eng?utm_source=chatgpt.com|title=Electronic Communications (Regulation) Act|date=Dec. 31st, 1997|website=Legislation Malta}}&lt;/ref&gt;  This statute lays the groundwork for how communications law is to be upheld in Malta: it governs access to networks, authorizations and license requirements, radio communications, and the regulatory framework that telecommunications companies will have to adhere to.  

[https://legislation.mt/eli/cap/418/eng Shortly after, the Malta Communications Authority Act]&lt;ref&gt;{{Cite web|url=https://legislation.mt/eli/cap/418/eng|title=MALTA COMMUNICATIONS AUTHORITY ACT|date=Aug. 1, 2000}}&lt;/ref&gt; was passed and functioned to establish the [https://www.mca.org.mt/ Malta Communication Authority (MCA).]  The Act gave the MCA regulating authority to enforce Maltese communication law, established the regulatory goals of the entity, and expressed the requirements for members wishing to serve on the MCA board.  

The MCA's authority was challenged in European Court of Justice case [https://www.maltatoday.com.mt/news/court_and_police/87626/vodafone_loses_challenge_on_financing_legal_interceptions Vodafone Malta Ltd v. Malta Communication Authority.]&lt;ref name=&quot;:2&quot;&gt;{{Cite web|url=https://infocuria.curia.europa.eu/tabs/document?source=document&amp;docid=138859&amp;doclang=EN|title=Vodafone Malta Ltd and Mobisle Communications Ltd v Attorney General and Others (Case C-71/12) — Judgment of the Court (Third Chamber), 27 June 2013.|website=CURIA}}&lt;/ref&gt; Vodafone claimed that the 3% excise tax the Maltese government had levied on mobile providers was a breach of EU law.&lt;ref&gt;{{Cite web|url=https://www.mca.org.mt/articles/judgement-court-vodafone-malta-limited-et-vs-avukati-generali-et-case-c%E2%80%917112|title=Judgement of the Court - Vodafone Malta Limited et vs Avukati Generali et - Case C‑71/12 {{!}} MCA|last=Authority|first=Malta Communication|website=www.mca.org.mt|language=en|access-date=2026-02-27}}&lt;/ref&gt;  The court affirmed the MCA's authority to regulate communications law, ruled that the MCA was acting fully within their regulatory authority, and the national tax to offset interception costs was valid.&lt;ref name=&quot;:2&quot; /&gt;

Another statute governing communications law in Malta is the aforementioned [https://ba.org.mt/legislation Broadcasting Act.]&lt;ref name=&quot;:3&quot;&gt;{{Cite web|url=https://ba.org.mt/legislation|title=CHAPTER 350
BROADCASTING ACT|date=1 June 1991|website=Malta Broadcasting Authority}}&lt;/ref&gt; Where the ECR act was focused on the infrastructure of communications systems in Malta, the Broadcasting Act was directed at the content that was being pushed through television, radio, and audiovisual programming.  This Act also established the regulator the [https://ba.org.mt Malta Broadcasting Authority,] which seeks to enforce the statutes contained in the act.&lt;ref name=&quot;:3&quot; /&gt;     

==== International ====
Being an EU member state, Maltese communication law is heavily impacted on the supranational decisions of the European Union. The MCA lists 8 articles of EU legislation on their website divided into 3 subsections: e-commerce, postal service, and regulatory framework for communications.&lt;ref&gt;{{Cite web|url=https://www.mca.org.mt/general/legislation|title=Legislation {{!}} MCA|last=Authority|first=Malta Communication|website=www.mca.org.mt|language=en|access-date=2026-02-21}}&lt;/ref&gt; The MCA cites an EU directive from 2000&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/legal-content/EN/ALL/?uri=celex%3A32000L0031|title=Directive 2000/31/EC of the European Parliament and of the Council of 8 June 2000 on certain legal aspects of information society services, in particular electronic commerce, in the Internal Market ('Directive on electronic commerce')|date=8 June 2000|website=EUR-Lex}}&lt;/ref&gt; as well as a 2014 EU regulation&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/eli/reg/2014/910/oj/eng|title=Regulation (EU) No 910/2014 of the European Parliament and of the Council of 23 July 2014 on electronic identification and trust services for electronic transactions in the internal market and repealing Directive 1999/93/EC|date=23 July 2014|website=EUR-Lex}}&lt;/ref&gt; as controlling for Maltese E-commerce.  The MCA cites a 2008 EU Directive&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/eli/dir/2008/6/oj/eng|title=Directive 2008/6/EC of the European Parliament and of the Council of 20 February 2008 amending Directive 97/67/EC with regard to the full accomplishment of the internal market of Community postal services|date=20 February 2008|website=EUR-Lex}}&lt;/ref&gt; as the applicable EU law for Maltese postal services, and  a 2018 EU Directive,&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/eli/dir/2018/1972/oj/eng|title=Directive (EU) 2018/1972 of the European Parliament and of the Council of 11 December 2018 establishing the European Electronic Communications Code (Recast) (Text with EEA relevance)|date=11 December 2018|website=EUR-Lex}}&lt;/ref&gt; as well as four 2002 EU Directives&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/eli/dir/2002/22/oj/eng|title=Directive 2002/22/EC of the European Parliament and of the Council of 7 March 2002 on universal service and users' rights relating to electronic communications networks and services (Universal Service Directive)|date=7 March 2002|website=EUR-Lex}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/eli/dir/2002/20/oj/eng|title=Directive 2002/20/EC of the European Parliament and of the Council of 7 March 2002 on the authorisation of electronic communications networks and services (Authorisation Directive)|date=7 March 2002|website=EUR-Lex}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/eli/dir/2002/19/oj/eng|title=Directive 2002/19/EC of the European Parliament and of the Council of 7 March 2002 on access to, and interconnection of, electronic communications networks and associated facilities (Access Directive)|date=7 March 2002|website=EUR-Lex}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/eli/dir/2002/58/oj/eng|title=Directive 2002/58/EC of the European Parliament and of the Council of 12 July 2002 concerning the processing of personal data and the protection of privacy in the electronic communications sector (Directive on privacy and electronic communications)|website=EUR-Lex}}&lt;/ref&gt; as the applicable law for electronic communications. 

Additionally, Maltese membership in the European Union requires that the country adheres to three principles of EU membership:   

# As a member state, citizens of Malta have the ability to uphold certain EU rights in court.  This is known as the &quot;Principle of Direct Effect,&quot; and was established by the 1963 case Van Gend en Loos v Nederlandse Administratie der Belastingen.&lt;ref&gt;{{Cite web|url=https://www.lawyersnjurists.com/article/direct-effect-of-europian-union-law/|title=DIRECT EFFECT OF EUROPEAN UNION LAW|website=The Lawyers &amp; Jurists|access-date=2026-03-19}}&lt;/ref&gt;  This means that any Maltese citizen or company can invoke European Union communication rights in the Maltese domestic court system. 
# Further, Malta is obligated to interpret its domestic laws ''as much as possible'' in light of EU directives.  This is known as the &quot;Principle of Conforming Interpretation&quot; or &quot;Principle of Indirect Effect,&quot; and was established by the 1984 case of ''V''on Colson v Land Nordrhein-Westfalen.&lt;ref&gt;{{Cite web|url=https://employmentrightsireland.com/von-colson-and-kamann-establishing-the-principle-of-indirect-effect/#:~:text=The%20*Von%20Colson%20and%20Kamann*%20case%20established,which%20will%20give%20effect%20to%20EU%20directives.|title=Von Colson and Kamann-Establishing the Principle of Indirect Effect – Employment Rights Ireland|website=employmentrightsireland.com|access-date=2026-03-19}}&lt;/ref&gt;  Therefore, all of the cases relating to communications law in Malta must be decided utilizing a legislative framework that takes into account the directives of the EU. 
# Lastly, the member states of the EU must ensure that their regulatory measures are: 1) suitable 2) necessary 3) does not impose excessive burdens.  This is known as the &quot;Principle of Proportionality,&quot; and was established by the case ''I''nternationale Handelsgesellschaft mbH v Einfuhr- und Vorratsstelle für Getreide und Futtermittel.&lt;ref&gt;{{Cite web|url=https://opil.ouplaw.com/display/10.1093/law:epil/9780199231690/law-9780199231690-e1459|title=Proportionality|last=Crawford|first=Emily|date=May 2011|website=Oxford Public International Law}}&lt;/ref&gt;  This therefore requires that the Maltese Communication Authority and Broadcast Authority regulations align with the Proportionality standards. 

==== Regional ====
Malta is a member party to the [https://www.echr.coe.int/documents/d/echr/Convention_ENG European Convention on Human Rights (ECHR),]&lt;ref name=&quot;:7&quot;&gt;{{Cite web|url=https://www.echr.coe.int/documents/d/echr/Convention_ENG|title=European Convention on Human Rights|date=Nov. 4th, 1950}}&lt;/ref&gt; and therefore must abide by the ECHR's protection of freedom of expression.  This protection is enshrined in Article 10 of the convention,&lt;ref&gt;{{Cite web|url=https://fra.europa.eu/en/law-reference/european-convention-human-rights-article-10|title=European Convention on Human Rights - Article 10|website=European Union Agency for Fundamental Rights}}&lt;/ref&gt; which states the following:  

&quot;Everyone has the right to freedom of expression. This right shall include freedom to hold opinions and to receive and impart information and ideas without interference by public authority and regardless of frontiers. This Article shall not prevent States from requiring the licensing of broadcasting, television or cinema enterprises.&quot;  

Further, the ECHR established the [https://www.echr.coe.int European Court of Human Rights], which has jurisdiction over Maltese law in Malta-specific cases, and persuasive authority for cases dealing with other member states.&lt;ref name=&quot;:7&quot; /&gt;  

== 3. Censorship and Violent Content ==

==== Hate Speech ====
Malta takes a no-nonsense approach to hate speech, penalizing those convicted with 6-18 months.&lt;ref name=&quot;:1&quot;&gt;{{Cite web|url=https://legislation.mt/eli/cap/9/eng/pdf?utm_source=chatgpt.com|title=Criminal Code Chapter 9|website=LEĠIŻLAZZJONI MALTA}}&lt;/ref&gt; Hate speech is governed primarily by the [https://legislation.mt/eli/cap/9/eng/pdf?utm_source=chatgpt.com Maltese Criminal Code]&lt;ref name=&quot;:4&quot;&gt;{{Cite web|url=https://legislation.mt/eli/cap/9/eng/pdf?utm_|title=CHAPTER 9 CRIMINAL CODE To amend and consolidate the Penal Laws and the Laws of Criminal Procedure.|website=LEĠIŻLAZZJONI MALTA}}&lt;/ref&gt;, which states the following: 

# &quot;Whosoever uses any threatening, abusive or insulting words or behaviour, or displays any written or printed material which is threatening, abusive or insulting, or otherwise conducts himself in such a manner, with intent thereby to stir up violence or hatred against another person or group of persons on the grounds of gender, gender identity, sexual orientation, race, colour, language, ethnic origin, age, disability, religion or belief or political or other opinion or whereby such violence or hatred is likely, having regard to all the circumstances, to be stirred up shall, on conviction, be liable&quot;

The Code goes on in Sections 82B and 82C to crim&lt;ref&gt;{{Cite web|url=https://globalfreedomofexpression.columbia.edu/cases/garaudy-v-france/|title=Garaudy v. France|website=Global Freedom of Expression|language=en-US|access-date=2026-03-19}}&lt;/ref&gt;inalize support or denial of crimes against humanity, genocide, war crimes and crimes against peace. 

* ''(''a) likely to incite to violence or hatred against such a group or a member of such a group;

* (b) likely to disturb public order or which is threatening, abusive or insulting

==== Case law: ====
[https://fra.europa.eu/en/databases/criminal-detention/case-law/6700 Police v. Norman Lowell]&lt;ref&gt;{{Cite web|url=https://fra.europa.eu/en/databases/criminal-detention/case-law/6700|title=Malta / Criminal Court of Appeal / 98/2011 / Judgement / Police vs Norman Lowell|website=European Union Agency for Fundamental Rights}}&lt;/ref&gt; 

In this 2013 case, a political activist was convicted for the charge of &quot;incitement of racial hatred&quot; against immigrants and Jews for delivering speeches and publishing allegedly racist views including incitements to violence online. Lowell was charged under Article 82A of the Maltese Criminal Code and received a 2 year sentence (later suspended) and a €500 fine.  

[https://globalfreedomofexpression.columbia.edu/cases/garaudy-v-france/ Garaudy v. France]&lt;ref&gt;{{Cite web|url=https://globalfreedomofexpression.columbia.edu/cases/garaudy-v-france/|title=Garaudy v. France|website=Global Freedom of Expression|language=en-US|access-date=2026-03-19}}&lt;/ref&gt; 

Being bound by the European Convention of Human Rights, this case is a persuasive authority for Maltese decision making. This 2003 case ruled against French Philosopher Roger Garaudy who was convicted in France for his 1998 book that allegedly denied the Holocaust.  The Court ruled that Article 17 of the Convention preempted Article 10 - making his writing fall outside the realm of protected speech. 

==== Illegal Pornography and Child Protection ====
The Maltese government has strict laws governing pornography.  Under Section 208 of the Code Chapter 9, displays of pornographic material in public places will result in fines &quot;of not less than one thousand euro (€1,000) and not more than three thousand euro (€3,000).&quot;&lt;ref name=&quot;:4&quot; /&gt; Section 208 of the statute goes further to strictly outlaw all forms of child pornography and &quot;revenge porn&quot; (illicit photos intended for one person and then used as blackmail).&lt;ref name=&quot;:4&quot; /&gt;

==== Case Law ====
[https://hudoc.echr.coe.int/eng#{%22itemid%22:&amp;#x5B;%22001-57499%22&amp;#x5D;} Handyside v. United Kingdom]&lt;ref&gt;{{Cite web|url=https://hudoc.echr.coe.int/eng#%7B%22itemid%22:%5B%22001-57499%22%5D%7D|title=HUDOC - European Court of Human Rights|website=hudoc.echr.coe.int|access-date=2026-03-19}}&lt;/ref&gt;

This 1976 case established that while the European Union protects &quot;freedom of expression&quot; in Article 10 of the European Convention of Human Rights, the European Union recognizes the authority of countries to limit this due to protecting the morality of their citizens. The case focused on the seizure and prohibition of a pornographic schoolbook set to be disseminated throughout the United Kingdom. The court stated, 

&quot;the Court first finds that the 1959/1964 Acts have an aim that is legitimate under Article 10 para. 2 (art. 10-2), namely, the protection of morals in a democratic society. Only this latter purpose is relevant in this case since the object of the said Acts - to wage war on 'obscene' publications, defined by their tendency to 'deprave and corrupt' - is linked far more closely to the protection of morals than to any of the further purposes permitted by Article 10 para. 2 (art. 10-2).&quot; 

This decision showed to member states that while there are certain supranational requirements they must maintain, individual countries still had rights to uphold their individual public morality.  

==== Defamation and Libel Reform ====
Though previously illegal, the [https://parlament.mt/13th-leg/acts/act-xi-of-2018/ Media and Defamation] act abolished the crime of libel and relegated it to civil courts. Section 3 of the act states that &quot;defamatory words in written media&quot; constitutes libel, &lt;ref&gt;{{Cite web|url=https://parlament.mt/media/93813/act-xi-media-and-defamation-act.pdf|title=ACT No. XI of 2018 The Media and Defamation Act|date=24 April 2018|website=Parliament of Malta}}&lt;/ref&gt;  and prior to this Act, libel could be criminally prosecuted.  

==== Case law ====
[https://globalfreedomofexpression.columbia.edu/cases/falzon-v-malta/?utm_source=chatgpt.com Falzon v. Malta]&lt;ref&gt;{{Cite web|url=https://globalfreedomofexpression.columbia.edu/cases/falzon-v-malta/|title=Falzon v. Malta|website=Global Freedom of Expression|language=en-US|access-date=2026-03-14}}&lt;/ref&gt;

This 2018 ECtHR court decision directly implicated Maltese communication law, as the Maltese court system had ruled against Falzon in a civil libel case for dramatizing a negative story of the Maltese Police Force.  The ECtHR ruled that this was a violation of Article 10 of the Convention, due to the purpose of the article being political commentary based on civic engagement, rather than true defamatory allegations against individuals; the Maltese court was thus overruled.  

[https://globalfreedomofexpression.columbia.edu/cases/cuschieri-v-galizia/?utm_source=chatgpt.com Caruana Galizia v. Cuschieri] &lt;ref&gt;{{Cite web|url=https://globalfreedomofexpression.columbia.edu/cases/cuschieri-v-galizia/|title=Caruana Galizia v. Cuschieri|website=Global Freedom of Expression|language=en-US|access-date=2026-03-14}}&lt;/ref&gt;

In this case, a woman was sued for defamation for falsely alleging in an independent blog that a politician was an antisemite and had employed seditious tactics.  The court ruled that this was not protected speech, as it went beyond mere opinion or political commentary and into character attacks on the politician that were ill-founded. 

== 4. Truth, Honor &amp; Tolerance ==
Malta's membership in both the European Union and European Convention on Human Rights requires the country to provide equal protection to citizens under the law.  Malta itself has also established additional national laws instituting this. 
[[File:Basílica de San Pablo, Rabat, isla de Malta, Malta, 2021-08-25, DD 167.jpg|thumb]]
With 85% of the Maltese Population identifying as Catholic, and 2,000 years of Catholic history,&lt;ref&gt;{{Cite web|url=https://www.ewtnnews.com/world/europe/facing-declining-mass-attendance-malta-s-catholics-hope-papal-visit-will-revitalize-the-faith|title=Facing a decline in Mass attendance, Catholics in one country hope papal visit will revitalize the faith|website=Catholic News Agency|language=en|access-date=2026-03-14}}&lt;/ref&gt; it is unsurprising that Article 2 of the Maltese Constitution established Catholicism as the state religion.&lt;ref name=&quot;:5&quot;&gt;{{Cite web|url=https://legislation.mt/eli/const/eng/pdf|title=Constitution of Malta|date=21st September, 1964|website=LEGISLATION MALTA}}&lt;/ref&gt; However, under Article 40 of the Maltese Constitution,&lt;ref name=&quot;:5&quot; /&gt; as well as Article 10 of the EU Charter of Human Rights,&lt;ref&gt;{{Cite web|url=https://fra.europa.eu/en/eu-charter/article/10-freedom-thought-conscience-and-religion|title=Article 10 - Freedom of thought, conscience and religion {{!}} European Union Agency for Fundamental Rights|date=2015-04-25|website=fra.europa.eu|language=en|access-date=2026-03-19}}&lt;/ref&gt; and Article 9 of the ECtHR,&lt;ref name=&quot;:7&quot; /&gt; Maltese citizens enjoy freedom of religious practice. 

==== Constitutional Rights to Freedom of Expression ====
[https://www.constituteproject.org/constitution/Malta_2016 Article 41] of the Maltese Constitution&lt;ref name=&quot;:8&quot;&gt;{{Cite web|url=https://www.constituteproject.org/constitution/Malta_2016|title=Malta 1964 (rev. 2016) Constitution - Constitute|website=www.constituteproject.org|language=en|access-date=2026-03-16}}&lt;/ref&gt; lays out the groundwork for the parameters of freedom of expression in the country. The first section of the article grants broad &quot;freedom of expression&quot; with the exceptions of citizens consenting to be censored, and parental discipline.  

The second section of Article 41 expresses the occasions in which the government may legally impose limitations on this freedom of expression.  This includes situations in which &quot;it is reasonably required&quot; for:   

&quot;defense, public safety, public order, public morality or decency, or public health; or for the purpose of protecting the reputations, rights and freedoms of other persons, or the private lives of persons concerned in legal proceedings, preventing the disclosure of information received in confidence, maintaining the authority and independence of the courts, protecting the privileges of Parliament, or regulating telephony, telegraphy, posts, wireless broadcasting, television or other means of communication, public exhibitions or public entertainments.&quot;&lt;ref name=&quot;:8&quot; /&gt;   

The second article also explains that the government can limit the freedom of expression for public officers.&lt;ref name=&quot;:8&quot; /&gt;  

The third applicable section of this article explains the limits the government can impose on publisher's freedom of expression.  This first includes the prohibition or restriction of both editing or printing of publications by people under 21 years old; and the mandate that printers and editors inform the proper authorities of their ages and locations.&lt;ref name=&quot;:8&quot; /&gt;   

The final section of Article 41 explains the limits of seizure in response to freedom of expression.   The section first states that when the police seize a newspaper that had been used to commit a criminal offense, they are required to go to a competent court within 24 hours to see if the prima facie case of the offense was satisfied.  The section states that if the prima facie case is not satisfied, the paper must be returned to its owner.  The section ends with stating that, &quot;No person shall be deprived of his citizenship under any provisions made under article 30(1) (b) of this Constitution or of his juridical capacity by reason only of his political opinions.&quot;&lt;ref name=&quot;:8&quot; /&gt;  

==== Tolerance and Anti-Discrimination ====
The Maltese government put forth the [https://parlament.mt/en/9th-leg/acts-9th/act-no-i-of-2003/ Equality for Men and Women Act] &lt;ref&gt;{{Cite web|url=https://parlament.mt/media/1050/i-of-2003-equality-for-men-and-women-act.pdf|title=ACT No. I of 2003, AN ACT to promote equality for men and women|date=4 February 2003|website=Parliament of Malta}}&lt;/ref&gt; in order to protect the right of both sexes to be treated as equals under the law -- especially protecting women from prejudicial discrimination.  While this does not explicitly mention communications law, it does require men and women to be treated non-discriminatorily in every area, therefore including communication platforms.  Additionally, in the [https://parlament.mt/media/129409/att-xxv-gender-identity.pdf Gender Identity, Gender Expression and Sex Characteristics Act],&lt;ref&gt;{{Cite web|url=https://parlament.mt/media/129409/att-xxv-gender-identity.pdf|title=Act No. XXV of 2024 - Gender Identity, Gender Expression and Sex Characteristics (Amendment) Act|date=12th July 2024|website=Parliament of Malta}}&lt;/ref&gt; the Maltese government instituted rights of transgender individuals to legally change and be known by their preferred gender/name, and to be treated in accordance with these preferences.  This act also protects transgender individuals from discrimination for these changes. 

==== Case law: ====
[https://timesofmalta.com/article/from-bench-is-hate-speech-freedom-expression.1051613?utm_source=chatgpt.com Police v. Michael Leonard Paul Hammond]&lt;ref&gt;{{Cite web|url=https://timesofmalta.com/article/from-bench-is-hate-speech-freedom-expression.1051613|title=From the Bench: Is it hate speech or freedom of expression?|last=Contributors|date=2023-08-26|website=Times of Malta|language=en-gb|access-date=2026-03-14}}&lt;/ref&gt;

In this 2023 case, a man was convicted for posting a hateful comment on a pro-LGBT Facebook page.  The court ruled that his comment under a Gay Pride parade post of, &quot;should bomb all u (sic) freaks,&quot;&lt;ref&gt;{{Cite web|url=https://timesofmalta.com/article/from-bench-is-hate-speech-freedom-expression.1051613|title=From the Bench: Is it hate speech or freedom of expression?|last=Contributors|date=2023-08-26|website=Times of Malta|language=en-gb|access-date=2026-03-14}}&lt;/ref&gt; constituted a violation of Section 82A of the Maltese Criminal Code.  Hammond later tried to appeal this conviction, but the court upheld the lower court's decision. 

[https://www.independent.com.mt/articles/2018-05-29/local-news/Transgender-inmates-win-damages-in-constitutional-case-against-prison-authorities-6736190683?utm_source=chatgpt.com Transgender Inmates v. Prison Authorities]&lt;ref&gt;{{Cite web|url=httpss://www.independent.com.mt/articles/2018-05-29/local-news/Transgender-inmates-win-damages-in-constitutional-case-against-prison-authorities-6736190683?utm_source=chatgpt.com|title=Updated: Transgender inmates win damages in constitutional case against prison authorities - The Malta Independent|website=www.independent.com.mt|access-date=2026-03-14}}&lt;/ref&gt;

While this case does not specifically speak to communications law, it does further show the development of the law in treating transgender individuals as a protected class. In this 2018 case, 7 inmates who identified as transgender women sued for being placed in the male section of the prison in accordance with the sex listed on their official documentation. The inmates stated that being placed in this section, while living as women, created a situation in which they were placed in constant abuse and harassment.  The Maltese court ruled that the prison system was inhumane for forcing these individuals to stay in the male-section of the prison, and ruled for the inmates, granting them damages in €5,000 for their mistreatment. 

== 5.Cultural and Religious Expressions ==
- Put here how Malta is culturally Catholic, look back at the module as to what this should go into 

== 6. Privacy &amp; Data Protection ==
For purposes of this section, privacy is defined as “the quality or state of being apart from company or observation; freedom from unauthorized intrusion.&lt;ref&gt;{{Cite web|url=https://www.merriam-webster.com/dictionary/privacy|title=Definition of PRIVACY|date=2026-04-26|website=www.merriam-webster.com|language=en|access-date=2026-04-28}}&lt;/ref&gt;&quot; Many believe that there are certain aspects of life that should rightly be protected from unnecessary government intrusion. However, this creates a tension between the rights of citizens to privacy, and the rights of these same citizens to the free access of information. This tension can clearly be seen in the legislation regarding this subject in Malta. 

'''International Laws Governing Maltese Rights to Privacy:''' 

Because Malta is member of the European Union, it is under the General Data Protection Regulation (GDPR).  This is a 2018 EU regulation relating to how personal data of citizens is collected, stored, shared, and used.  Importantly, the statute not only provides the framework for how this data is treated, but provides the necessary definitions of what falls into the category of “personal data.”  The regulation states in article 4 that personal data refers to: 

''Any information'' relating to an identified or identifiable natural person (‘data subject’); an identifiable natural person is one who can be identified, directly or indirectly, in particular by reference to an identifier such as a name, an identification number, location data, an online identifier or to one or more factors specific to the physical, physiological, genetic, mental, economic, cultural or social identity of that natural person. (emphasis added)

The legislation speaks in Article 5 on the principles of how this information ought to be processed. The first line speaks to transparency, stating that the “data subject’s” (more clearly defined as a “natural” person), information should be “processed lawfully, fairly and in a transparent manner” in relation to that person.  Article 5 goes on to emphasize that all of the processing ought to be done for legitimate purposes, and only processed to the extent necessary. This “processing” is defined essentially as any use of the personal data in Article 4(2). 

The lawfulness of this use, or “processing,” of this personal data is explicated in Article 6 of the GDPR. The main points of this Ariticle can be summarized in the following: the processing of personal data must be necessary and consented to by the person. 

Significantly, Article 1 of the GDPR states that, “the free movement of personal data within the Union shall be neither restricted nor prohibited for reasons connected with the protection of natural persons with regard to the processing of personal data.” 
[[File:Surveillance video cameras, Gdynia.jpeg|thumb]]
Further, as a member of the ECHR, Malta is required to protect the rights found in Article 8, titled “Right to respect for private and family life.” These rights include the right to “respect for his private and family life, his home and his correspondence,” and grants that, &quot;there shall be no interference by a public authority with the exercise of this right except such as is in accordance with the law and is necessary in a democratic society in the interests of national security, public safety or the economic well-being of the country, for the prevention of disorder or crime, for the protection of health or morals, or for the protection of the rights and freedoms of others.&quot;

On the European stage, Malta

== 7. Right to Bodily, Spiritual, &amp; Digital Identity ==

== 8. Right to Reject Information, Clothing, &amp; Human Exhibitions ==

==== References: ====
[[Category:Malta]]
[[Category:Law in Europe]]
[[Category:Communication in Europe|Law in Malta]]</text>
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      <text bytes="35902" sha1="7ao9irtaqxqk8g7eeadaxdfmvnwi8j0" xml:space="preserve">== 1. History of Malta and Maltese Communication Law ==
{{law}}

==== History of Malta ====
The Mediterranean island of [https://www.gov.mt/en/Pages/Home.aspx Malta] is located directly in the crossroads of several formidable historical forces.  The impacts of this are clearly seen in the country's dramatic history, with effects still present in its &quot;mixed&quot; legal system.  Situated south of Sicily, Malta was [[wikipedia:Capture_of_Malta_(218_BC)#:~:text=The%20capture%20of%20Malta%20was,Punic%20War%20in%20218%20BC.|conquered by Rome]] in 218 B.C. in the Second Punic War.  This forced the small island to be placed under Roman law for the majority of time between 218 B.C. and 1530, when Holy Roman Emperor Charles V granted the island of Malta to the Order of St. John (also known as the &quot;Knights Hospitallers&quot;). The tiny island was then ruled shortly by France from 1789 - 1813 and then England from 1813 forward,&lt;ref&gt;{{Cite news|url=https://www.bbc.com/news/world-europe-17598542|title=Malta profile - Timeline|date=2012-04-03|work=BBC News|access-date=2026-02-21|language=en-GB}}&lt;/ref&gt; only gaining its sovereign independence in 1964.&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=Aquilina|first=Kevin|date=2013|title=The Nature and Sources of The Maltese Mixed Legal System: A Strange Case of Dr. Jekyll and Mr. Hyde?|url=https://heinonline.org/HOL/Page?handle=hein.journals/complv4&amp;id=&amp;collection=journals&amp;div=8|journal=Comparative law review|pages=1-38|via=HeinOnline}}&lt;/ref&gt;
[[File:Malta, 2010 - panoramio - Bengt Nyman (23).jpg|thumb]]Malta's &quot;mixed system&quot; is based most clearly on the fact that it maintains both civil and common law.&lt;ref name=&quot;:0&quot; /&gt; In practice, this means it follows a legal system predicated on both statutes and past court decisions.  Malta has been influenced greatly by international and European systems - the small country has a &quot;Westminster&quot; styled Constitution, but does not hold Constitutional Court decisions as binding on any other than the parties in dispute.&lt;ref name=&quot;:0&quot; /&gt;

==== Maltese Communication Law History ====
Malta was owned by the British as the &quot;Crown Colony of Malta&quot; from 1813 - 1964, a time period that included many critical changes in the modernization of communication.  As a colony of the British Empire, Malta was subject to British influence on both the implementation of the modern means of communication, as well as the laws surrounding the regulation of these means.&lt;ref&gt;{{Cite web|url=https://broadcasting-authority.azurewebsites.net/ourhistory?utm_source=chatgpt.com|title=Our History {{!}} Broadcasting Authority|website=broadcasting-authority.azurewebsites.net|access-date=2026-03-18}}&lt;/ref&gt; This included the creation of the Broadcast Authority, established in 1961 to regulate all broadcasting in the nation.&lt;ref&gt;{{Cite web|url=https://ba.org.mt/en/about-us|title=About Us|website=ba.org.mt|access-date=2026-03-19}}&lt;/ref&gt; 

Radio broadcasting first came to Malta in the form of cable radio through a private British company called &quot;Rediffusion&quot; in 1935.&lt;ref name=&quot;:6&quot;&gt;{{Cite web|url=https://timesofmalta.com/article/radio-broadcasting-turns-90.1119330|title=Radio broadcasting turns 90|last=Borg|first=Fr Joe|date=2025-11-11|website=Times of Malta|language=en-gb|access-date=2026-03-18}}&lt;/ref&gt;  This was especially important at the time, as Malta's cultural and physical closeness to Italy put them at risk of infiltration by the ideas of the expanding Fascist Mussolini regime through wireless radio broadcasts.&lt;ref name=&quot;:6&quot; /&gt; To combat this, British Colonial Authorities strategically placed loudspeakers with approved programs across the island, and banned the use of non-wire radio.&lt;ref name=&quot;:6&quot; /&gt;  The easy public access to radio programs, paired with the auditory nature of the news bypassing the necessity to be literate, caused radio to surpass newspapers as the main source of information to the public.&lt;ref name=&quot;:6&quot; /&gt; 

The perceived danger to Maltese citizens from Italian and Sicilian influence continued to pressure the island and those who ruled it to continually adopt updated communication systems.  In the 1950s, Maltese reception of Italian TV signals led the country to first create the television network that would become TVM, and later, the adoption of color broadcast.&lt;ref name=&quot;:6&quot; /&gt;  The geographical proximity to Italy created a situation such that, when the one-station system censored differing political ideas, the Nationalist party was able to resort to Sicilian radio and television stations to broadcast their opposing viewpoints.&lt;ref name=&quot;:6&quot; /&gt;

The small nation's reliance on Rediffusion lasted up until 1971, when the Mintoff government decided to nationalize radio and television; this was ultimately not successful until 1975.&lt;ref name=&quot;:6&quot; /&gt;  This led to the establishment of Xandir Malta, or &quot;Broadcasting Malta,&quot; a Socialist-run broadcasting network that goverened the communications sector until it was pushed out by the Nationalist party in the late 1980s.  The Nationalist party then established a public broadcasting network and issued &quot;The 1990 White Paper Commitment to Pluralism,&quot; promoting pluralism in broadcasting.&lt;ref name=&quot;:6&quot; /&gt;  The Church and the Government were the first to have their own radio stations, followed by additions in 1992.  Radio broadcasting is reportedly consumed by two-thirds of Maltese citizens to this day.&lt;ref name=&quot;:6&quot; /&gt;

When Malta joined the European Union (EU) in 2004, they were required to extend their communications legislation to meet EU standards, including promoting European and independent works.&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/legal-content/EN/TXT/?uri=legissum:e20112|website=eur-lex.europa.eu|access-date=2026-03-19}}&lt;/ref&gt;  Malta has since joined multiple international agreements, such that the Maltese lawmakers must ensure that both the law of Malta and the terms of multiple treaties and directives are followed when writing legislation.  Maltese communication law is emblematic of the legal situation of many European countries: though the country maintains its sovereign independence, the law is riddled with influence from past rulers, as well as attempts to catch up with technological progress and the ever-expanding process of globalization.  

== 2. Sources of Maltese Communications Law ==

==== National ====
[[File:Malta ali 2009224 lrg.jpg|thumb]]
The primary statute governing communications law in Malta is the [https://legislation.mt/eli/cap/399/eng?utm_source=chatgpt.com Electronic Communications (Regulation) Act].&lt;ref&gt;{{Cite web|url=https://legislation.mt/eli/cap/399/eng?utm_source=chatgpt.com|title=Electronic Communications (Regulation) Act|date=Dec. 31st, 1997|website=Legislation Malta}}&lt;/ref&gt;  This statute lays the groundwork for how communications law is to be upheld in Malta: it governs access to networks, authorizations and license requirements, radio communications, and the regulatory framework that telecommunications companies will have to adhere to.  

[https://legislation.mt/eli/cap/418/eng Shortly after, the Malta Communications Authority Act]&lt;ref&gt;{{Cite web|url=https://legislation.mt/eli/cap/418/eng|title=MALTA COMMUNICATIONS AUTHORITY ACT|date=Aug. 1, 2000}}&lt;/ref&gt; was passed and functioned to establish the [https://www.mca.org.mt/ Malta Communication Authority (MCA).]  The Act gave the MCA regulating authority to enforce Maltese communication law, established the regulatory goals of the entity, and expressed the requirements for members wishing to serve on the MCA board.  

The MCA's authority was challenged in European Court of Justice case [https://www.maltatoday.com.mt/news/court_and_police/87626/vodafone_loses_challenge_on_financing_legal_interceptions Vodafone Malta Ltd v. Malta Communication Authority.]&lt;ref name=&quot;:2&quot;&gt;{{Cite web|url=https://infocuria.curia.europa.eu/tabs/document?source=document&amp;docid=138859&amp;doclang=EN|title=Vodafone Malta Ltd and Mobisle Communications Ltd v Attorney General and Others (Case C-71/12) — Judgment of the Court (Third Chamber), 27 June 2013.|website=CURIA}}&lt;/ref&gt; Vodafone claimed that the 3% excise tax the Maltese government had levied on mobile providers was a breach of EU law.&lt;ref&gt;{{Cite web|url=https://www.mca.org.mt/articles/judgement-court-vodafone-malta-limited-et-vs-avukati-generali-et-case-c%E2%80%917112|title=Judgement of the Court - Vodafone Malta Limited et vs Avukati Generali et - Case C‑71/12 {{!}} MCA|last=Authority|first=Malta Communication|website=www.mca.org.mt|language=en|access-date=2026-02-27}}&lt;/ref&gt;  The court affirmed the MCA's authority to regulate communications law, ruled that the MCA was acting fully within their regulatory authority, and the national tax to offset interception costs was valid.&lt;ref name=&quot;:2&quot; /&gt;

Another statute governing communications law in Malta is the aforementioned [https://ba.org.mt/legislation Broadcasting Act.]&lt;ref name=&quot;:3&quot;&gt;{{Cite web|url=https://ba.org.mt/legislation|title=CHAPTER 350
BROADCASTING ACT|date=1 June 1991|website=Malta Broadcasting Authority}}&lt;/ref&gt; Where the ECR act was focused on the infrastructure of communications systems in Malta, the Broadcasting Act was directed at the content that was being pushed through television, radio, and audiovisual programming.  This Act also established the regulator the [https://ba.org.mt Malta Broadcasting Authority,] which seeks to enforce the statutes contained in the act.&lt;ref name=&quot;:3&quot; /&gt;     

==== International ====
Being an EU member state, Maltese communication law is heavily impacted on the supranational decisions of the European Union. The MCA lists 8 articles of EU legislation on their website divided into 3 subsections: e-commerce, postal service, and regulatory framework for communications.&lt;ref&gt;{{Cite web|url=https://www.mca.org.mt/general/legislation|title=Legislation {{!}} MCA|last=Authority|first=Malta Communication|website=www.mca.org.mt|language=en|access-date=2026-02-21}}&lt;/ref&gt; The MCA cites an EU directive from 2000&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/legal-content/EN/ALL/?uri=celex%3A32000L0031|title=Directive 2000/31/EC of the European Parliament and of the Council of 8 June 2000 on certain legal aspects of information society services, in particular electronic commerce, in the Internal Market ('Directive on electronic commerce')|date=8 June 2000|website=EUR-Lex}}&lt;/ref&gt; as well as a 2014 EU regulation&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/eli/reg/2014/910/oj/eng|title=Regulation (EU) No 910/2014 of the European Parliament and of the Council of 23 July 2014 on electronic identification and trust services for electronic transactions in the internal market and repealing Directive 1999/93/EC|date=23 July 2014|website=EUR-Lex}}&lt;/ref&gt; as controlling for Maltese E-commerce.  The MCA cites a 2008 EU Directive&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/eli/dir/2008/6/oj/eng|title=Directive 2008/6/EC of the European Parliament and of the Council of 20 February 2008 amending Directive 97/67/EC with regard to the full accomplishment of the internal market of Community postal services|date=20 February 2008|website=EUR-Lex}}&lt;/ref&gt; as the applicable EU law for Maltese postal services, and  a 2018 EU Directive,&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/eli/dir/2018/1972/oj/eng|title=Directive (EU) 2018/1972 of the European Parliament and of the Council of 11 December 2018 establishing the European Electronic Communications Code (Recast) (Text with EEA relevance)|date=11 December 2018|website=EUR-Lex}}&lt;/ref&gt; as well as four 2002 EU Directives&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/eli/dir/2002/22/oj/eng|title=Directive 2002/22/EC of the European Parliament and of the Council of 7 March 2002 on universal service and users' rights relating to electronic communications networks and services (Universal Service Directive)|date=7 March 2002|website=EUR-Lex}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/eli/dir/2002/20/oj/eng|title=Directive 2002/20/EC of the European Parliament and of the Council of 7 March 2002 on the authorisation of electronic communications networks and services (Authorisation Directive)|date=7 March 2002|website=EUR-Lex}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/eli/dir/2002/19/oj/eng|title=Directive 2002/19/EC of the European Parliament and of the Council of 7 March 2002 on access to, and interconnection of, electronic communications networks and associated facilities (Access Directive)|date=7 March 2002|website=EUR-Lex}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/eli/dir/2002/58/oj/eng|title=Directive 2002/58/EC of the European Parliament and of the Council of 12 July 2002 concerning the processing of personal data and the protection of privacy in the electronic communications sector (Directive on privacy and electronic communications)|website=EUR-Lex}}&lt;/ref&gt; as the applicable law for electronic communications. 

Additionally, Maltese membership in the European Union requires that the country adheres to three principles of EU membership:   

# As a member state, citizens of Malta have the ability to uphold certain EU rights in court.  This is known as the &quot;Principle of Direct Effect,&quot; and was established by the 1963 case Van Gend en Loos v Nederlandse Administratie der Belastingen.&lt;ref&gt;{{Cite web|url=https://www.lawyersnjurists.com/article/direct-effect-of-europian-union-law/|title=DIRECT EFFECT OF EUROPEAN UNION LAW|website=The Lawyers &amp; Jurists|access-date=2026-03-19}}&lt;/ref&gt;  This means that any Maltese citizen or company can invoke European Union communication rights in the Maltese domestic court system. 
# Further, Malta is obligated to interpret its domestic laws ''as much as possible'' in light of EU directives.  This is known as the &quot;Principle of Conforming Interpretation&quot; or &quot;Principle of Indirect Effect,&quot; and was established by the 1984 case of ''V''on Colson v Land Nordrhein-Westfalen.&lt;ref&gt;{{Cite web|url=https://employmentrightsireland.com/von-colson-and-kamann-establishing-the-principle-of-indirect-effect/#:~:text=The%20*Von%20Colson%20and%20Kamann*%20case%20established,which%20will%20give%20effect%20to%20EU%20directives.|title=Von Colson and Kamann-Establishing the Principle of Indirect Effect – Employment Rights Ireland|website=employmentrightsireland.com|access-date=2026-03-19}}&lt;/ref&gt;  Therefore, all of the cases relating to communications law in Malta must be decided utilizing a legislative framework that takes into account the directives of the EU. 
# Lastly, the member states of the EU must ensure that their regulatory measures are: 1) suitable 2) necessary 3) does not impose excessive burdens.  This is known as the &quot;Principle of Proportionality,&quot; and was established by the case ''I''nternationale Handelsgesellschaft mbH v Einfuhr- und Vorratsstelle für Getreide und Futtermittel.&lt;ref&gt;{{Cite web|url=https://opil.ouplaw.com/display/10.1093/law:epil/9780199231690/law-9780199231690-e1459|title=Proportionality|last=Crawford|first=Emily|date=May 2011|website=Oxford Public International Law}}&lt;/ref&gt;  This therefore requires that the Maltese Communication Authority and Broadcast Authority regulations align with the Proportionality standards. 

==== Regional ====
Malta is a member party to the [https://www.echr.coe.int/documents/d/echr/Convention_ENG European Convention on Human Rights (ECHR),]&lt;ref name=&quot;:7&quot;&gt;{{Cite web|url=https://www.echr.coe.int/documents/d/echr/Convention_ENG|title=European Convention on Human Rights|date=Nov. 4th, 1950}}&lt;/ref&gt; and therefore must abide by the ECHR's protection of freedom of expression.  This protection is enshrined in Article 10 of the convention,&lt;ref&gt;{{Cite web|url=https://fra.europa.eu/en/law-reference/european-convention-human-rights-article-10|title=European Convention on Human Rights - Article 10|website=European Union Agency for Fundamental Rights}}&lt;/ref&gt; which states the following:  

&quot;Everyone has the right to freedom of expression. This right shall include freedom to hold opinions and to receive and impart information and ideas without interference by public authority and regardless of frontiers. This Article shall not prevent States from requiring the licensing of broadcasting, television or cinema enterprises.&quot;  

Further, the ECHR established the [https://www.echr.coe.int European Court of Human Rights], which has jurisdiction over Maltese law in Malta-specific cases, and persuasive authority for cases dealing with other member states.&lt;ref name=&quot;:7&quot; /&gt;  

== 3. Censorship and Violent Content ==

==== Hate Speech ====
Malta takes a no-nonsense approach to hate speech, penalizing those convicted with 6-18 months.&lt;ref name=&quot;:1&quot;&gt;{{Cite web|url=https://legislation.mt/eli/cap/9/eng/pdf?utm_source=chatgpt.com|title=Criminal Code Chapter 9|website=LEĠIŻLAZZJONI MALTA}}&lt;/ref&gt; Hate speech is governed primarily by the [https://legislation.mt/eli/cap/9/eng/pdf?utm_source=chatgpt.com Maltese Criminal Code]&lt;ref name=&quot;:4&quot;&gt;{{Cite web|url=https://legislation.mt/eli/cap/9/eng/pdf?utm_|title=CHAPTER 9 CRIMINAL CODE To amend and consolidate the Penal Laws and the Laws of Criminal Procedure.|website=LEĠIŻLAZZJONI MALTA}}&lt;/ref&gt;, which states the following: 

# &quot;Whosoever uses any threatening, abusive or insulting words or behaviour, or displays any written or printed material which is threatening, abusive or insulting, or otherwise conducts himself in such a manner, with intent thereby to stir up violence or hatred against another person or group of persons on the grounds of gender, gender identity, sexual orientation, race, colour, language, ethnic origin, age, disability, religion or belief or political or other opinion or whereby such violence or hatred is likely, having regard to all the circumstances, to be stirred up shall, on conviction, be liable&quot;

The Code goes on in Sections 82B and 82C to crim&lt;ref&gt;{{Cite web|url=https://globalfreedomofexpression.columbia.edu/cases/garaudy-v-france/|title=Garaudy v. France|website=Global Freedom of Expression|language=en-US|access-date=2026-03-19}}&lt;/ref&gt;inalize support or denial of crimes against humanity, genocide, war crimes and crimes against peace. 

* ''(''a) likely to incite to violence or hatred against such a group or a member of such a group;

* (b) likely to disturb public order or which is threatening, abusive or insulting

==== Case law: ====
[https://fra.europa.eu/en/databases/criminal-detention/case-law/6700 Police v. Norman Lowell]&lt;ref&gt;{{Cite web|url=https://fra.europa.eu/en/databases/criminal-detention/case-law/6700|title=Malta / Criminal Court of Appeal / 98/2011 / Judgement / Police vs Norman Lowell|website=European Union Agency for Fundamental Rights}}&lt;/ref&gt; 

In this 2013 case, a political activist was convicted for the charge of &quot;incitement of racial hatred&quot; against immigrants and Jews for delivering speeches and publishing allegedly racist views including incitements to violence online. Lowell was charged under Article 82A of the Maltese Criminal Code and received a 2 year sentence (later suspended) and a €500 fine.  

[https://globalfreedomofexpression.columbia.edu/cases/garaudy-v-france/ Garaudy v. France]&lt;ref&gt;{{Cite web|url=https://globalfreedomofexpression.columbia.edu/cases/garaudy-v-france/|title=Garaudy v. France|website=Global Freedom of Expression|language=en-US|access-date=2026-03-19}}&lt;/ref&gt; 

Being bound by the European Convention of Human Rights, this case is a persuasive authority for Maltese decision making. This 2003 case ruled against French Philosopher Roger Garaudy who was convicted in France for his 1998 book that allegedly denied the Holocaust.  The Court ruled that Article 17 of the Convention preempted Article 10 - making his writing fall outside the realm of protected speech. 

==== Illegal Pornography and Child Protection ====
The Maltese government has strict laws governing pornography.  Under Section 208 of the Code Chapter 9, displays of pornographic material in public places will result in fines &quot;of not less than one thousand euro (€1,000) and not more than three thousand euro (€3,000).&quot;&lt;ref name=&quot;:4&quot; /&gt; Section 208 of the statute goes further to strictly outlaw all forms of child pornography and &quot;revenge porn&quot; (illicit photos intended for one person and then used as blackmail).&lt;ref name=&quot;:4&quot; /&gt;

==== Case Law ====
[https://hudoc.echr.coe.int/eng#{%22itemid%22:&amp;#x5B;%22001-57499%22&amp;#x5D;} Handyside v. United Kingdom]&lt;ref&gt;{{Cite web|url=https://hudoc.echr.coe.int/eng#%7B%22itemid%22:%5B%22001-57499%22%5D%7D|title=HUDOC - European Court of Human Rights|website=hudoc.echr.coe.int|access-date=2026-03-19}}&lt;/ref&gt;

This 1976 case established that while the European Union protects &quot;freedom of expression&quot; in Article 10 of the European Convention of Human Rights, the European Union recognizes the authority of countries to limit this due to protecting the morality of their citizens. The case focused on the seizure and prohibition of a pornographic schoolbook set to be disseminated throughout the United Kingdom. The court stated, 

&quot;the Court first finds that the 1959/1964 Acts have an aim that is legitimate under Article 10 para. 2 (art. 10-2), namely, the protection of morals in a democratic society. Only this latter purpose is relevant in this case since the object of the said Acts - to wage war on 'obscene' publications, defined by their tendency to 'deprave and corrupt' - is linked far more closely to the protection of morals than to any of the further purposes permitted by Article 10 para. 2 (art. 10-2).&quot; 

This decision showed to member states that while there are certain supranational requirements they must maintain, individual countries still had rights to uphold their individual public morality.  

==== Defamation and Libel Reform ====
Though previously illegal, the [https://parlament.mt/13th-leg/acts/act-xi-of-2018/ Media and Defamation] act abolished the crime of libel and relegated it to civil courts. Section 3 of the act states that &quot;defamatory words in written media&quot; constitutes libel, &lt;ref&gt;{{Cite web|url=https://parlament.mt/media/93813/act-xi-media-and-defamation-act.pdf|title=ACT No. XI of 2018 The Media and Defamation Act|date=24 April 2018|website=Parliament of Malta}}&lt;/ref&gt;  and prior to this Act, libel could be criminally prosecuted.  

==== Case law ====
[https://globalfreedomofexpression.columbia.edu/cases/falzon-v-malta/?utm_source=chatgpt.com Falzon v. Malta]&lt;ref&gt;{{Cite web|url=https://globalfreedomofexpression.columbia.edu/cases/falzon-v-malta/|title=Falzon v. Malta|website=Global Freedom of Expression|language=en-US|access-date=2026-03-14}}&lt;/ref&gt;

This 2018 ECtHR court decision directly implicated Maltese communication law, as the Maltese court system had ruled against Falzon in a civil libel case for dramatizing a negative story of the Maltese Police Force.  The ECtHR ruled that this was a violation of Article 10 of the Convention, due to the purpose of the article being political commentary based on civic engagement, rather than true defamatory allegations against individuals; the Maltese court was thus overruled.  

[https://globalfreedomofexpression.columbia.edu/cases/cuschieri-v-galizia/?utm_source=chatgpt.com Caruana Galizia v. Cuschieri] &lt;ref&gt;{{Cite web|url=https://globalfreedomofexpression.columbia.edu/cases/cuschieri-v-galizia/|title=Caruana Galizia v. Cuschieri|website=Global Freedom of Expression|language=en-US|access-date=2026-03-14}}&lt;/ref&gt;

In this case, a woman was sued for defamation for falsely alleging in an independent blog that a politician was an antisemite and had employed seditious tactics.  The court ruled that this was not protected speech, as it went beyond mere opinion or political commentary and into character attacks on the politician that were ill-founded. 

== 4. Truth, Honor &amp; Tolerance ==
Malta's membership in both the European Union and European Convention on Human Rights requires the country to provide equal protection to citizens under the law.  Malta itself has also established additional national laws instituting this. 
[[File:Basílica de San Pablo, Rabat, isla de Malta, Malta, 2021-08-25, DD 167.jpg|thumb]]
With 85% of the Maltese Population identifying as Catholic, and 2,000 years of Catholic history,&lt;ref&gt;{{Cite web|url=https://www.ewtnnews.com/world/europe/facing-declining-mass-attendance-malta-s-catholics-hope-papal-visit-will-revitalize-the-faith|title=Facing a decline in Mass attendance, Catholics in one country hope papal visit will revitalize the faith|website=Catholic News Agency|language=en|access-date=2026-03-14}}&lt;/ref&gt; it is unsurprising that Article 2 of the Maltese Constitution established Catholicism as the state religion.&lt;ref name=&quot;:5&quot;&gt;{{Cite web|url=https://legislation.mt/eli/const/eng/pdf|title=Constitution of Malta|date=21st September, 1964|website=LEGISLATION MALTA}}&lt;/ref&gt; However, under Article 40 of the Maltese Constitution,&lt;ref name=&quot;:5&quot; /&gt; as well as Article 10 of the EU Charter of Human Rights,&lt;ref&gt;{{Cite web|url=https://fra.europa.eu/en/eu-charter/article/10-freedom-thought-conscience-and-religion|title=Article 10 - Freedom of thought, conscience and religion {{!}} European Union Agency for Fundamental Rights|date=2015-04-25|website=fra.europa.eu|language=en|access-date=2026-03-19}}&lt;/ref&gt; and Article 9 of the ECtHR,&lt;ref name=&quot;:7&quot; /&gt; Maltese citizens enjoy freedom of religious practice. 

==== Constitutional Rights to Freedom of Expression ====
[https://www.constituteproject.org/constitution/Malta_2016 Article 41] of the Maltese Constitution&lt;ref name=&quot;:8&quot;&gt;{{Cite web|url=https://www.constituteproject.org/constitution/Malta_2016|title=Malta 1964 (rev. 2016) Constitution - Constitute|website=www.constituteproject.org|language=en|access-date=2026-03-16}}&lt;/ref&gt; lays out the groundwork for the parameters of freedom of expression in the country. The first section of the article grants broad &quot;freedom of expression&quot; with the exceptions of citizens consenting to be censored, and parental discipline.  

The second section of Article 41 expresses the occasions in which the government may legally impose limitations on this freedom of expression.  This includes situations in which &quot;it is reasonably required&quot; for:   

&quot;defense, public safety, public order, public morality or decency, or public health; or for the purpose of protecting the reputations, rights and freedoms of other persons, or the private lives of persons concerned in legal proceedings, preventing the disclosure of information received in confidence, maintaining the authority and independence of the courts, protecting the privileges of Parliament, or regulating telephony, telegraphy, posts, wireless broadcasting, television or other means of communication, public exhibitions or public entertainments.&quot;&lt;ref name=&quot;:8&quot; /&gt;   

The second article also explains that the government can limit the freedom of expression for public officers.&lt;ref name=&quot;:8&quot; /&gt;  

The third applicable section of this article explains the limits the government can impose on publisher's freedom of expression.  This first includes the prohibition or restriction of both editing or printing of publications by people under 21 years old; and the mandate that printers and editors inform the proper authorities of their ages and locations.&lt;ref name=&quot;:8&quot; /&gt;   

The final section of Article 41 explains the limits of seizure in response to freedom of expression.   The section first states that when the police seize a newspaper that had been used to commit a criminal offense, they are required to go to a competent court within 24 hours to see if the prima facie case of the offense was satisfied.  The section states that if the prima facie case is not satisfied, the paper must be returned to its owner.  The section ends with stating that, &quot;No person shall be deprived of his citizenship under any provisions made under article 30(1) (b) of this Constitution or of his juridical capacity by reason only of his political opinions.&quot;&lt;ref name=&quot;:8&quot; /&gt;  

==== Tolerance and Anti-Discrimination ====
The Maltese government put forth the [https://parlament.mt/en/9th-leg/acts-9th/act-no-i-of-2003/ Equality for Men and Women Act] &lt;ref&gt;{{Cite web|url=https://parlament.mt/media/1050/i-of-2003-equality-for-men-and-women-act.pdf|title=ACT No. I of 2003, AN ACT to promote equality for men and women|date=4 February 2003|website=Parliament of Malta}}&lt;/ref&gt; in order to protect the right of both sexes to be treated as equals under the law -- especially protecting women from prejudicial discrimination.  While this does not explicitly mention communications law, it does require men and women to be treated non-discriminatorily in every area, therefore including communication platforms.  Additionally, in the [https://parlament.mt/media/129409/att-xxv-gender-identity.pdf Gender Identity, Gender Expression and Sex Characteristics Act],&lt;ref&gt;{{Cite web|url=https://parlament.mt/media/129409/att-xxv-gender-identity.pdf|title=Act No. XXV of 2024 - Gender Identity, Gender Expression and Sex Characteristics (Amendment) Act|date=12th July 2024|website=Parliament of Malta}}&lt;/ref&gt; the Maltese government instituted rights of transgender individuals to legally change and be known by their preferred gender/name, and to be treated in accordance with these preferences.  This act also protects transgender individuals from discrimination for these changes. 

==== Case law: ====
[https://timesofmalta.com/article/from-bench-is-hate-speech-freedom-expression.1051613?utm_source=chatgpt.com Police v. Michael Leonard Paul Hammond]&lt;ref&gt;{{Cite web|url=https://timesofmalta.com/article/from-bench-is-hate-speech-freedom-expression.1051613|title=From the Bench: Is it hate speech or freedom of expression?|last=Contributors|date=2023-08-26|website=Times of Malta|language=en-gb|access-date=2026-03-14}}&lt;/ref&gt;

In this 2023 case, a man was convicted for posting a hateful comment on a pro-LGBT Facebook page.  The court ruled that his comment under a Gay Pride parade post of, &quot;should bomb all u (sic) freaks,&quot;&lt;ref&gt;{{Cite web|url=https://timesofmalta.com/article/from-bench-is-hate-speech-freedom-expression.1051613|title=From the Bench: Is it hate speech or freedom of expression?|last=Contributors|date=2023-08-26|website=Times of Malta|language=en-gb|access-date=2026-03-14}}&lt;/ref&gt; constituted a violation of Section 82A of the Maltese Criminal Code.  Hammond later tried to appeal this conviction, but the court upheld the lower court's decision. 

[https://www.independent.com.mt/articles/2018-05-29/local-news/Transgender-inmates-win-damages-in-constitutional-case-against-prison-authorities-6736190683?utm_source=chatgpt.com Transgender Inmates v. Prison Authorities]&lt;ref&gt;{{Cite web|url=httpss://www.independent.com.mt/articles/2018-05-29/local-news/Transgender-inmates-win-damages-in-constitutional-case-against-prison-authorities-6736190683?utm_source=chatgpt.com|title=Updated: Transgender inmates win damages in constitutional case against prison authorities - The Malta Independent|website=www.independent.com.mt|access-date=2026-03-14}}&lt;/ref&gt;

While this case does not specifically speak to communications law, it does further show the development of the law in treating transgender individuals as a protected class. In this 2018 case, 7 inmates who identified as transgender women sued for being placed in the male section of the prison in accordance with the sex listed on their official documentation. The inmates stated that being placed in this section, while living as women, created a situation in which they were placed in constant abuse and harassment.  The Maltese court ruled that the prison system was inhumane for forcing these individuals to stay in the male-section of the prison, and ruled for the inmates, granting them damages in €5,000 for their mistreatment. 

== 5.Cultural and Religious Expressions ==
- Put here how Malta is culturally Catholic, look back at the module as to what this should go into 

== 6. Privacy &amp; Data Protection ==
For purposes of this section, privacy is defined as “the quality or state of being apart from company or observation; freedom from unauthorized intrusion.&lt;ref&gt;{{Cite web|url=https://www.merriam-webster.com/dictionary/privacy|title=Definition of PRIVACY|date=2026-04-26|website=www.merriam-webster.com|language=en|access-date=2026-04-28}}&lt;/ref&gt;&quot; Many believe that there are certain aspects of life that should rightly be protected from unnecessary government intrusion. However, this creates a tension between the rights of citizens to privacy, and the rights of these same citizens to the free access of information. This tension can clearly be seen in the legislation regarding this subject in Malta. 

'''International Laws Governing Maltese Rights to Privacy:''' 

Because Malta is member of the European Union, it is under the General Data Protection Regulation (GDPR).  This is a 2018 EU regulation relating to how personal data of citizens is collected, stored, shared, and used.  Importantly, the statute not only provides the framework for how this data is treated, but provides the necessary definitions of what falls into the category of “personal data.”  The regulation states in article 4 that personal data refers to: 

''Any information'' relating to an identified or identifiable natural person (‘data subject’); an identifiable natural person is one who can be identified, directly or indirectly, in particular by reference to an identifier such as a name, an identification number, location data, an online identifier or to one or more factors specific to the physical, physiological, genetic, mental, economic, cultural or social identity of that natural person. (emphasis added)

The legislation speaks in Article 5 on the principles of how this information ought to be processed. The first line speaks to transparency, stating that the “data subject’s” (more clearly defined as a “natural” person), information should be “processed lawfully, fairly and in a transparent manner” in relation to that person.  Article 5 goes on to emphasize that all of the processing ought to be done for legitimate purposes, and only processed to the extent necessary. This “processing” is defined essentially as any use of the personal data in Article 4(2). 

The lawfulness of this use, or “processing,” of this personal data is explicated in Article 6 of the GDPR. The main points of this Ariticle can be summarized in the following: the processing of personal data must be necessary and consented to by the person. 

Significantly, Article 1 of the GDPR states that, “the free movement of personal data within the Union shall be neither restricted nor prohibited for reasons connected with the protection of natural persons with regard to the processing of personal data.” 
[[File:Surveillance video cameras, Gdynia.jpeg|thumb]]
Further, as a member of the ECHR, Malta is required to protect the rights found in Article 8, titled “Right to respect for private and family life.” These rights include the right to “respect for his private and family life, his home and his correspondence,” and grants that, &quot;there shall be no interference by a public authority with the exercise of this right except such as is in accordance with the law and is necessary in a democratic society in the interests of national security, public safety or the economic well-being of the country, for the prevention of disorder or crime, for the protection of health or morals, or for the protection of the rights and freedoms of others.&quot;

On the European stage, Malta must maintain strict protection of the privacy of its citizens. But Malta itself has a body of statutory rules governing privacy as well. 

== 7. Right to Bodily, Spiritual, &amp; Digital Identity ==

== 8. Right to Reject Information, Clothing, &amp; Human Exhibitions ==

==== References: ====
[[Category:Malta]]
[[Category:Law in Europe]]
[[Category:Communication in Europe|Law in Malta]]</text>
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      <text bytes="37032" sha1="fbz2p3xjnq7teaarpfrkciqnyejz71n" xml:space="preserve">== 1. History of Malta and Maltese Communication Law ==
{{law}}

==== History of Malta ====
The Mediterranean island of [https://www.gov.mt/en/Pages/Home.aspx Malta] is located directly in the crossroads of several formidable historical forces.  The impacts of this are clearly seen in the country's dramatic history, with effects still present in its &quot;mixed&quot; legal system.  Situated south of Sicily, Malta was [[wikipedia:Capture_of_Malta_(218_BC)#:~:text=The%20capture%20of%20Malta%20was,Punic%20War%20in%20218%20BC.|conquered by Rome]] in 218 B.C. in the Second Punic War.  This forced the small island to be placed under Roman law for the majority of time between 218 B.C. and 1530, when Holy Roman Emperor Charles V granted the island of Malta to the Order of St. John (also known as the &quot;Knights Hospitallers&quot;). The tiny island was then ruled shortly by France from 1789 - 1813 and then England from 1813 forward,&lt;ref&gt;{{Cite news|url=https://www.bbc.com/news/world-europe-17598542|title=Malta profile - Timeline|date=2012-04-03|work=BBC News|access-date=2026-02-21|language=en-GB}}&lt;/ref&gt; only gaining its sovereign independence in 1964.&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=Aquilina|first=Kevin|date=2013|title=The Nature and Sources of The Maltese Mixed Legal System: A Strange Case of Dr. Jekyll and Mr. Hyde?|url=https://heinonline.org/HOL/Page?handle=hein.journals/complv4&amp;id=&amp;collection=journals&amp;div=8|journal=Comparative law review|pages=1-38|via=HeinOnline}}&lt;/ref&gt;
[[File:Malta, 2010 - panoramio - Bengt Nyman (23).jpg|thumb]]Malta's &quot;mixed system&quot; is based most clearly on the fact that it maintains both civil and common law.&lt;ref name=&quot;:0&quot; /&gt; In practice, this means it follows a legal system predicated on both statutes and past court decisions.  Malta has been influenced greatly by international and European systems - the small country has a &quot;Westminster&quot; styled Constitution, but does not hold Constitutional Court decisions as binding on any other than the parties in dispute.&lt;ref name=&quot;:0&quot; /&gt;

==== Maltese Communication Law History ====
Malta was owned by the British as the &quot;Crown Colony of Malta&quot; from 1813 - 1964, a time period that included many critical changes in the modernization of communication.  As a colony of the British Empire, Malta was subject to British influence on both the implementation of the modern means of communication, as well as the laws surrounding the regulation of these means.&lt;ref&gt;{{Cite web|url=https://broadcasting-authority.azurewebsites.net/ourhistory?utm_source=chatgpt.com|title=Our History {{!}} Broadcasting Authority|website=broadcasting-authority.azurewebsites.net|access-date=2026-03-18}}&lt;/ref&gt; This included the creation of the Broadcast Authority, established in 1961 to regulate all broadcasting in the nation.&lt;ref&gt;{{Cite web|url=https://ba.org.mt/en/about-us|title=About Us|website=ba.org.mt|access-date=2026-03-19}}&lt;/ref&gt; 

Radio broadcasting first came to Malta in the form of cable radio through a private British company called &quot;Rediffusion&quot; in 1935.&lt;ref name=&quot;:6&quot;&gt;{{Cite web|url=https://timesofmalta.com/article/radio-broadcasting-turns-90.1119330|title=Radio broadcasting turns 90|last=Borg|first=Fr Joe|date=2025-11-11|website=Times of Malta|language=en-gb|access-date=2026-03-18}}&lt;/ref&gt;  This was especially important at the time, as Malta's cultural and physical closeness to Italy put them at risk of infiltration by the ideas of the expanding Fascist Mussolini regime through wireless radio broadcasts.&lt;ref name=&quot;:6&quot; /&gt; To combat this, British Colonial Authorities strategically placed loudspeakers with approved programs across the island, and banned the use of non-wire radio.&lt;ref name=&quot;:6&quot; /&gt;  The easy public access to radio programs, paired with the auditory nature of the news bypassing the necessity to be literate, caused radio to surpass newspapers as the main source of information to the public.&lt;ref name=&quot;:6&quot; /&gt; 

The perceived danger to Maltese citizens from Italian and Sicilian influence continued to pressure the island and those who ruled it to continually adopt updated communication systems.  In the 1950s, Maltese reception of Italian TV signals led the country to first create the television network that would become TVM, and later, the adoption of color broadcast.&lt;ref name=&quot;:6&quot; /&gt;  The geographical proximity to Italy created a situation such that, when the one-station system censored differing political ideas, the Nationalist party was able to resort to Sicilian radio and television stations to broadcast their opposing viewpoints.&lt;ref name=&quot;:6&quot; /&gt;

The small nation's reliance on Rediffusion lasted up until 1971, when the Mintoff government decided to nationalize radio and television; this was ultimately not successful until 1975.&lt;ref name=&quot;:6&quot; /&gt;  This led to the establishment of Xandir Malta, or &quot;Broadcasting Malta,&quot; a Socialist-run broadcasting network that goverened the communications sector until it was pushed out by the Nationalist party in the late 1980s.  The Nationalist party then established a public broadcasting network and issued &quot;The 1990 White Paper Commitment to Pluralism,&quot; promoting pluralism in broadcasting.&lt;ref name=&quot;:6&quot; /&gt;  The Church and the Government were the first to have their own radio stations, followed by additions in 1992.  Radio broadcasting is reportedly consumed by two-thirds of Maltese citizens to this day.&lt;ref name=&quot;:6&quot; /&gt;

When Malta joined the European Union (EU) in 2004, they were required to extend their communications legislation to meet EU standards, including promoting European and independent works.&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/legal-content/EN/TXT/?uri=legissum:e20112|website=eur-lex.europa.eu|access-date=2026-03-19}}&lt;/ref&gt;  Malta has since joined multiple international agreements, such that the Maltese lawmakers must ensure that both the law of Malta and the terms of multiple treaties and directives are followed when writing legislation.  Maltese communication law is emblematic of the legal situation of many European countries: though the country maintains its sovereign independence, the law is riddled with influence from past rulers, as well as attempts to catch up with technological progress and the ever-expanding process of globalization.  

== 2. Sources of Maltese Communications Law ==

==== National ====
[[File:Malta ali 2009224 lrg.jpg|thumb]]
The primary statute governing communications law in Malta is the [https://legislation.mt/eli/cap/399/eng?utm_source=chatgpt.com Electronic Communications (Regulation) Act].&lt;ref&gt;{{Cite web|url=https://legislation.mt/eli/cap/399/eng?utm_source=chatgpt.com|title=Electronic Communications (Regulation) Act|date=Dec. 31st, 1997|website=Legislation Malta}}&lt;/ref&gt;  This statute lays the groundwork for how communications law is to be upheld in Malta: it governs access to networks, authorizations and license requirements, radio communications, and the regulatory framework that telecommunications companies will have to adhere to.  

[https://legislation.mt/eli/cap/418/eng Shortly after, the Malta Communications Authority Act]&lt;ref&gt;{{Cite web|url=https://legislation.mt/eli/cap/418/eng|title=MALTA COMMUNICATIONS AUTHORITY ACT|date=Aug. 1, 2000}}&lt;/ref&gt; was passed and functioned to establish the [https://www.mca.org.mt/ Malta Communication Authority (MCA).]  The Act gave the MCA regulating authority to enforce Maltese communication law, established the regulatory goals of the entity, and expressed the requirements for members wishing to serve on the MCA board.  

The MCA's authority was challenged in European Court of Justice case [https://www.maltatoday.com.mt/news/court_and_police/87626/vodafone_loses_challenge_on_financing_legal_interceptions Vodafone Malta Ltd v. Malta Communication Authority.]&lt;ref name=&quot;:2&quot;&gt;{{Cite web|url=https://infocuria.curia.europa.eu/tabs/document?source=document&amp;docid=138859&amp;doclang=EN|title=Vodafone Malta Ltd and Mobisle Communications Ltd v Attorney General and Others (Case C-71/12) — Judgment of the Court (Third Chamber), 27 June 2013.|website=CURIA}}&lt;/ref&gt; Vodafone claimed that the 3% excise tax the Maltese government had levied on mobile providers was a breach of EU law.&lt;ref&gt;{{Cite web|url=https://www.mca.org.mt/articles/judgement-court-vodafone-malta-limited-et-vs-avukati-generali-et-case-c%E2%80%917112|title=Judgement of the Court - Vodafone Malta Limited et vs Avukati Generali et - Case C‑71/12 {{!}} MCA|last=Authority|first=Malta Communication|website=www.mca.org.mt|language=en|access-date=2026-02-27}}&lt;/ref&gt;  The court affirmed the MCA's authority to regulate communications law, ruled that the MCA was acting fully within their regulatory authority, and the national tax to offset interception costs was valid.&lt;ref name=&quot;:2&quot; /&gt;

Another statute governing communications law in Malta is the aforementioned [https://ba.org.mt/legislation Broadcasting Act.]&lt;ref name=&quot;:3&quot;&gt;{{Cite web|url=https://ba.org.mt/legislation|title=CHAPTER 350
BROADCASTING ACT|date=1 June 1991|website=Malta Broadcasting Authority}}&lt;/ref&gt; Where the ECR act was focused on the infrastructure of communications systems in Malta, the Broadcasting Act was directed at the content that was being pushed through television, radio, and audiovisual programming.  This Act also established the regulator the [https://ba.org.mt Malta Broadcasting Authority,] which seeks to enforce the statutes contained in the act.&lt;ref name=&quot;:3&quot; /&gt;     

==== International ====
Being an EU member state, Maltese communication law is heavily impacted on the supranational decisions of the European Union. The MCA lists 8 articles of EU legislation on their website divided into 3 subsections: e-commerce, postal service, and regulatory framework for communications.&lt;ref&gt;{{Cite web|url=https://www.mca.org.mt/general/legislation|title=Legislation {{!}} MCA|last=Authority|first=Malta Communication|website=www.mca.org.mt|language=en|access-date=2026-02-21}}&lt;/ref&gt; The MCA cites an EU directive from 2000&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/legal-content/EN/ALL/?uri=celex%3A32000L0031|title=Directive 2000/31/EC of the European Parliament and of the Council of 8 June 2000 on certain legal aspects of information society services, in particular electronic commerce, in the Internal Market ('Directive on electronic commerce')|date=8 June 2000|website=EUR-Lex}}&lt;/ref&gt; as well as a 2014 EU regulation&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/eli/reg/2014/910/oj/eng|title=Regulation (EU) No 910/2014 of the European Parliament and of the Council of 23 July 2014 on electronic identification and trust services for electronic transactions in the internal market and repealing Directive 1999/93/EC|date=23 July 2014|website=EUR-Lex}}&lt;/ref&gt; as controlling for Maltese E-commerce.  The MCA cites a 2008 EU Directive&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/eli/dir/2008/6/oj/eng|title=Directive 2008/6/EC of the European Parliament and of the Council of 20 February 2008 amending Directive 97/67/EC with regard to the full accomplishment of the internal market of Community postal services|date=20 February 2008|website=EUR-Lex}}&lt;/ref&gt; as the applicable EU law for Maltese postal services, and  a 2018 EU Directive,&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/eli/dir/2018/1972/oj/eng|title=Directive (EU) 2018/1972 of the European Parliament and of the Council of 11 December 2018 establishing the European Electronic Communications Code (Recast) (Text with EEA relevance)|date=11 December 2018|website=EUR-Lex}}&lt;/ref&gt; as well as four 2002 EU Directives&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/eli/dir/2002/22/oj/eng|title=Directive 2002/22/EC of the European Parliament and of the Council of 7 March 2002 on universal service and users' rights relating to electronic communications networks and services (Universal Service Directive)|date=7 March 2002|website=EUR-Lex}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/eli/dir/2002/20/oj/eng|title=Directive 2002/20/EC of the European Parliament and of the Council of 7 March 2002 on the authorisation of electronic communications networks and services (Authorisation Directive)|date=7 March 2002|website=EUR-Lex}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/eli/dir/2002/19/oj/eng|title=Directive 2002/19/EC of the European Parliament and of the Council of 7 March 2002 on access to, and interconnection of, electronic communications networks and associated facilities (Access Directive)|date=7 March 2002|website=EUR-Lex}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/eli/dir/2002/58/oj/eng|title=Directive 2002/58/EC of the European Parliament and of the Council of 12 July 2002 concerning the processing of personal data and the protection of privacy in the electronic communications sector (Directive on privacy and electronic communications)|website=EUR-Lex}}&lt;/ref&gt; as the applicable law for electronic communications. 

Additionally, Maltese membership in the European Union requires that the country adheres to three principles of EU membership:   

# As a member state, citizens of Malta have the ability to uphold certain EU rights in court.  This is known as the &quot;Principle of Direct Effect,&quot; and was established by the 1963 case Van Gend en Loos v Nederlandse Administratie der Belastingen.&lt;ref&gt;{{Cite web|url=https://www.lawyersnjurists.com/article/direct-effect-of-europian-union-law/|title=DIRECT EFFECT OF EUROPEAN UNION LAW|website=The Lawyers &amp; Jurists|access-date=2026-03-19}}&lt;/ref&gt;  This means that any Maltese citizen or company can invoke European Union communication rights in the Maltese domestic court system. 
# Further, Malta is obligated to interpret its domestic laws ''as much as possible'' in light of EU directives.  This is known as the &quot;Principle of Conforming Interpretation&quot; or &quot;Principle of Indirect Effect,&quot; and was established by the 1984 case of ''V''on Colson v Land Nordrhein-Westfalen.&lt;ref&gt;{{Cite web|url=https://employmentrightsireland.com/von-colson-and-kamann-establishing-the-principle-of-indirect-effect/#:~:text=The%20*Von%20Colson%20and%20Kamann*%20case%20established,which%20will%20give%20effect%20to%20EU%20directives.|title=Von Colson and Kamann-Establishing the Principle of Indirect Effect – Employment Rights Ireland|website=employmentrightsireland.com|access-date=2026-03-19}}&lt;/ref&gt;  Therefore, all of the cases relating to communications law in Malta must be decided utilizing a legislative framework that takes into account the directives of the EU. 
# Lastly, the member states of the EU must ensure that their regulatory measures are: 1) suitable 2) necessary 3) does not impose excessive burdens.  This is known as the &quot;Principle of Proportionality,&quot; and was established by the case ''I''nternationale Handelsgesellschaft mbH v Einfuhr- und Vorratsstelle für Getreide und Futtermittel.&lt;ref&gt;{{Cite web|url=https://opil.ouplaw.com/display/10.1093/law:epil/9780199231690/law-9780199231690-e1459|title=Proportionality|last=Crawford|first=Emily|date=May 2011|website=Oxford Public International Law}}&lt;/ref&gt;  This therefore requires that the Maltese Communication Authority and Broadcast Authority regulations align with the Proportionality standards. 

==== Regional ====
Malta is a member party to the [https://www.echr.coe.int/documents/d/echr/Convention_ENG European Convention on Human Rights (ECHR),]&lt;ref name=&quot;:7&quot;&gt;{{Cite web|url=https://www.echr.coe.int/documents/d/echr/Convention_ENG|title=European Convention on Human Rights|date=Nov. 4th, 1950}}&lt;/ref&gt; and therefore must abide by the ECHR's protection of freedom of expression.  This protection is enshrined in Article 10 of the convention,&lt;ref&gt;{{Cite web|url=https://fra.europa.eu/en/law-reference/european-convention-human-rights-article-10|title=European Convention on Human Rights - Article 10|website=European Union Agency for Fundamental Rights}}&lt;/ref&gt; which states the following:  

&quot;Everyone has the right to freedom of expression. This right shall include freedom to hold opinions and to receive and impart information and ideas without interference by public authority and regardless of frontiers. This Article shall not prevent States from requiring the licensing of broadcasting, television or cinema enterprises.&quot;  

Further, the ECHR established the [https://www.echr.coe.int European Court of Human Rights], which has jurisdiction over Maltese law in Malta-specific cases, and persuasive authority for cases dealing with other member states.&lt;ref name=&quot;:7&quot; /&gt;  

== 3. Censorship and Violent Content ==

==== Hate Speech ====
Malta takes a no-nonsense approach to hate speech, penalizing those convicted with 6-18 months.&lt;ref name=&quot;:1&quot;&gt;{{Cite web|url=https://legislation.mt/eli/cap/9/eng/pdf?utm_source=chatgpt.com|title=Criminal Code Chapter 9|website=LEĠIŻLAZZJONI MALTA}}&lt;/ref&gt; Hate speech is governed primarily by the [https://legislation.mt/eli/cap/9/eng/pdf?utm_source=chatgpt.com Maltese Criminal Code]&lt;ref name=&quot;:4&quot;&gt;{{Cite web|url=https://legislation.mt/eli/cap/9/eng/pdf?utm_|title=CHAPTER 9 CRIMINAL CODE To amend and consolidate the Penal Laws and the Laws of Criminal Procedure.|website=LEĠIŻLAZZJONI MALTA}}&lt;/ref&gt;, which states the following: 

# &quot;Whosoever uses any threatening, abusive or insulting words or behaviour, or displays any written or printed material which is threatening, abusive or insulting, or otherwise conducts himself in such a manner, with intent thereby to stir up violence or hatred against another person or group of persons on the grounds of gender, gender identity, sexual orientation, race, colour, language, ethnic origin, age, disability, religion or belief or political or other opinion or whereby such violence or hatred is likely, having regard to all the circumstances, to be stirred up shall, on conviction, be liable&quot;

The Code goes on in Sections 82B and 82C to crim&lt;ref&gt;{{Cite web|url=https://globalfreedomofexpression.columbia.edu/cases/garaudy-v-france/|title=Garaudy v. France|website=Global Freedom of Expression|language=en-US|access-date=2026-03-19}}&lt;/ref&gt;inalize support or denial of crimes against humanity, genocide, war crimes and crimes against peace. 

* ''(''a) likely to incite to violence or hatred against such a group or a member of such a group;

* (b) likely to disturb public order or which is threatening, abusive or insulting

==== Case law: ====
[https://fra.europa.eu/en/databases/criminal-detention/case-law/6700 Police v. Norman Lowell]&lt;ref&gt;{{Cite web|url=https://fra.europa.eu/en/databases/criminal-detention/case-law/6700|title=Malta / Criminal Court of Appeal / 98/2011 / Judgement / Police vs Norman Lowell|website=European Union Agency for Fundamental Rights}}&lt;/ref&gt; 

In this 2013 case, a political activist was convicted for the charge of &quot;incitement of racial hatred&quot; against immigrants and Jews for delivering speeches and publishing allegedly racist views including incitements to violence online. Lowell was charged under Article 82A of the Maltese Criminal Code and received a 2 year sentence (later suspended) and a €500 fine.  

[https://globalfreedomofexpression.columbia.edu/cases/garaudy-v-france/ Garaudy v. France]&lt;ref&gt;{{Cite web|url=https://globalfreedomofexpression.columbia.edu/cases/garaudy-v-france/|title=Garaudy v. France|website=Global Freedom of Expression|language=en-US|access-date=2026-03-19}}&lt;/ref&gt; 

Being bound by the European Convention of Human Rights, this case is a persuasive authority for Maltese decision making. This 2003 case ruled against French Philosopher Roger Garaudy who was convicted in France for his 1998 book that allegedly denied the Holocaust.  The Court ruled that Article 17 of the Convention preempted Article 10 - making his writing fall outside the realm of protected speech. 

==== Illegal Pornography and Child Protection ====
The Maltese government has strict laws governing pornography.  Under Section 208 of the Code Chapter 9, displays of pornographic material in public places will result in fines &quot;of not less than one thousand euro (€1,000) and not more than three thousand euro (€3,000).&quot;&lt;ref name=&quot;:4&quot; /&gt; Section 208 of the statute goes further to strictly outlaw all forms of child pornography and &quot;revenge porn&quot; (illicit photos intended for one person and then used as blackmail).&lt;ref name=&quot;:4&quot; /&gt;

==== Case Law ====
[https://hudoc.echr.coe.int/eng#{%22itemid%22:&amp;#x5B;%22001-57499%22&amp;#x5D;} Handyside v. United Kingdom]&lt;ref&gt;{{Cite web|url=https://hudoc.echr.coe.int/eng#%7B%22itemid%22:%5B%22001-57499%22%5D%7D|title=HUDOC - European Court of Human Rights|website=hudoc.echr.coe.int|access-date=2026-03-19}}&lt;/ref&gt;

This 1976 case established that while the European Union protects &quot;freedom of expression&quot; in Article 10 of the European Convention of Human Rights, the European Union recognizes the authority of countries to limit this due to protecting the morality of their citizens. The case focused on the seizure and prohibition of a pornographic schoolbook set to be disseminated throughout the United Kingdom. The court stated, 

&quot;the Court first finds that the 1959/1964 Acts have an aim that is legitimate under Article 10 para. 2 (art. 10-2), namely, the protection of morals in a democratic society. Only this latter purpose is relevant in this case since the object of the said Acts - to wage war on 'obscene' publications, defined by their tendency to 'deprave and corrupt' - is linked far more closely to the protection of morals than to any of the further purposes permitted by Article 10 para. 2 (art. 10-2).&quot; 

This decision showed to member states that while there are certain supranational requirements they must maintain, individual countries still had rights to uphold their individual public morality.  

==== Defamation and Libel Reform ====
Though previously illegal, the [https://parlament.mt/13th-leg/acts/act-xi-of-2018/ Media and Defamation] act abolished the crime of libel and relegated it to civil courts. Section 3 of the act states that &quot;defamatory words in written media&quot; constitutes libel, &lt;ref&gt;{{Cite web|url=https://parlament.mt/media/93813/act-xi-media-and-defamation-act.pdf|title=ACT No. XI of 2018 The Media and Defamation Act|date=24 April 2018|website=Parliament of Malta}}&lt;/ref&gt;  and prior to this Act, libel could be criminally prosecuted.  

==== Case law ====
[https://globalfreedomofexpression.columbia.edu/cases/falzon-v-malta/?utm_source=chatgpt.com Falzon v. Malta]&lt;ref&gt;{{Cite web|url=https://globalfreedomofexpression.columbia.edu/cases/falzon-v-malta/|title=Falzon v. Malta|website=Global Freedom of Expression|language=en-US|access-date=2026-03-14}}&lt;/ref&gt;

This 2018 ECtHR court decision directly implicated Maltese communication law, as the Maltese court system had ruled against Falzon in a civil libel case for dramatizing a negative story of the Maltese Police Force.  The ECtHR ruled that this was a violation of Article 10 of the Convention, due to the purpose of the article being political commentary based on civic engagement, rather than true defamatory allegations against individuals; the Maltese court was thus overruled.  

[https://globalfreedomofexpression.columbia.edu/cases/cuschieri-v-galizia/?utm_source=chatgpt.com Caruana Galizia v. Cuschieri] &lt;ref&gt;{{Cite web|url=https://globalfreedomofexpression.columbia.edu/cases/cuschieri-v-galizia/|title=Caruana Galizia v. Cuschieri|website=Global Freedom of Expression|language=en-US|access-date=2026-03-14}}&lt;/ref&gt;

In this case, a woman was sued for defamation for falsely alleging in an independent blog that a politician was an antisemite and had employed seditious tactics.  The court ruled that this was not protected speech, as it went beyond mere opinion or political commentary and into character attacks on the politician that were ill-founded. 

== 4. Truth, Honor &amp; Tolerance ==
Malta's membership in both the European Union and European Convention on Human Rights requires the country to provide equal protection to citizens under the law.  Malta itself has also established additional national laws instituting this. 
[[File:Basílica de San Pablo, Rabat, isla de Malta, Malta, 2021-08-25, DD 167.jpg|thumb]]
With 85% of the Maltese Population identifying as Catholic, and 2,000 years of Catholic history,&lt;ref&gt;{{Cite web|url=https://www.ewtnnews.com/world/europe/facing-declining-mass-attendance-malta-s-catholics-hope-papal-visit-will-revitalize-the-faith|title=Facing a decline in Mass attendance, Catholics in one country hope papal visit will revitalize the faith|website=Catholic News Agency|language=en|access-date=2026-03-14}}&lt;/ref&gt; it is unsurprising that Article 2 of the Maltese Constitution established Catholicism as the state religion.&lt;ref name=&quot;:5&quot;&gt;{{Cite web|url=https://legislation.mt/eli/const/eng/pdf|title=Constitution of Malta|date=21st September, 1964|website=LEGISLATION MALTA}}&lt;/ref&gt; However, under Article 40 of the Maltese Constitution,&lt;ref name=&quot;:5&quot; /&gt; as well as Article 10 of the EU Charter of Human Rights,&lt;ref&gt;{{Cite web|url=https://fra.europa.eu/en/eu-charter/article/10-freedom-thought-conscience-and-religion|title=Article 10 - Freedom of thought, conscience and religion {{!}} European Union Agency for Fundamental Rights|date=2015-04-25|website=fra.europa.eu|language=en|access-date=2026-03-19}}&lt;/ref&gt; and Article 9 of the ECtHR,&lt;ref name=&quot;:7&quot; /&gt; Maltese citizens enjoy freedom of religious practice. 

==== Constitutional Rights to Freedom of Expression ====
[https://www.constituteproject.org/constitution/Malta_2016 Article 41] of the Maltese Constitution&lt;ref name=&quot;:8&quot;&gt;{{Cite web|url=https://www.constituteproject.org/constitution/Malta_2016|title=Malta 1964 (rev. 2016) Constitution - Constitute|website=www.constituteproject.org|language=en|access-date=2026-03-16}}&lt;/ref&gt; lays out the groundwork for the parameters of freedom of expression in the country. The first section of the article grants broad &quot;freedom of expression&quot; with the exceptions of citizens consenting to be censored, and parental discipline.  

The second section of Article 41 expresses the occasions in which the government may legally impose limitations on this freedom of expression.  This includes situations in which &quot;it is reasonably required&quot; for:   

&quot;defense, public safety, public order, public morality or decency, or public health; or for the purpose of protecting the reputations, rights and freedoms of other persons, or the private lives of persons concerned in legal proceedings, preventing the disclosure of information received in confidence, maintaining the authority and independence of the courts, protecting the privileges of Parliament, or regulating telephony, telegraphy, posts, wireless broadcasting, television or other means of communication, public exhibitions or public entertainments.&quot;&lt;ref name=&quot;:8&quot; /&gt;   

The second article also explains that the government can limit the freedom of expression for public officers.&lt;ref name=&quot;:8&quot; /&gt;  

The third applicable section of this article explains the limits the government can impose on publisher's freedom of expression.  This first includes the prohibition or restriction of both editing or printing of publications by people under 21 years old; and the mandate that printers and editors inform the proper authorities of their ages and locations.&lt;ref name=&quot;:8&quot; /&gt;   

The final section of Article 41 explains the limits of seizure in response to freedom of expression.   The section first states that when the police seize a newspaper that had been used to commit a criminal offense, they are required to go to a competent court within 24 hours to see if the prima facie case of the offense was satisfied.  The section states that if the prima facie case is not satisfied, the paper must be returned to its owner.  The section ends with stating that, &quot;No person shall be deprived of his citizenship under any provisions made under article 30(1) (b) of this Constitution or of his juridical capacity by reason only of his political opinions.&quot;&lt;ref name=&quot;:8&quot; /&gt;  

==== Tolerance and Anti-Discrimination ====
The Maltese government put forth the [https://parlament.mt/en/9th-leg/acts-9th/act-no-i-of-2003/ Equality for Men and Women Act] &lt;ref&gt;{{Cite web|url=https://parlament.mt/media/1050/i-of-2003-equality-for-men-and-women-act.pdf|title=ACT No. I of 2003, AN ACT to promote equality for men and women|date=4 February 2003|website=Parliament of Malta}}&lt;/ref&gt; in order to protect the right of both sexes to be treated as equals under the law -- especially protecting women from prejudicial discrimination.  While this does not explicitly mention communications law, it does require men and women to be treated non-discriminatorily in every area, therefore including communication platforms.  Additionally, in the [https://parlament.mt/media/129409/att-xxv-gender-identity.pdf Gender Identity, Gender Expression and Sex Characteristics Act],&lt;ref&gt;{{Cite web|url=https://parlament.mt/media/129409/att-xxv-gender-identity.pdf|title=Act No. XXV of 2024 - Gender Identity, Gender Expression and Sex Characteristics (Amendment) Act|date=12th July 2024|website=Parliament of Malta}}&lt;/ref&gt; the Maltese government instituted rights of transgender individuals to legally change and be known by their preferred gender/name, and to be treated in accordance with these preferences.  This act also protects transgender individuals from discrimination for these changes. 

==== Case law: ====
[https://timesofmalta.com/article/from-bench-is-hate-speech-freedom-expression.1051613?utm_source=chatgpt.com Police v. Michael Leonard Paul Hammond]&lt;ref&gt;{{Cite web|url=https://timesofmalta.com/article/from-bench-is-hate-speech-freedom-expression.1051613|title=From the Bench: Is it hate speech or freedom of expression?|last=Contributors|date=2023-08-26|website=Times of Malta|language=en-gb|access-date=2026-03-14}}&lt;/ref&gt;

In this 2023 case, a man was convicted for posting a hateful comment on a pro-LGBT Facebook page.  The court ruled that his comment under a Gay Pride parade post of, &quot;should bomb all u (sic) freaks,&quot;&lt;ref&gt;{{Cite web|url=https://timesofmalta.com/article/from-bench-is-hate-speech-freedom-expression.1051613|title=From the Bench: Is it hate speech or freedom of expression?|last=Contributors|date=2023-08-26|website=Times of Malta|language=en-gb|access-date=2026-03-14}}&lt;/ref&gt; constituted a violation of Section 82A of the Maltese Criminal Code.  Hammond later tried to appeal this conviction, but the court upheld the lower court's decision. 

[https://www.independent.com.mt/articles/2018-05-29/local-news/Transgender-inmates-win-damages-in-constitutional-case-against-prison-authorities-6736190683?utm_source=chatgpt.com Transgender Inmates v. Prison Authorities]&lt;ref&gt;{{Cite web|url=httpss://www.independent.com.mt/articles/2018-05-29/local-news/Transgender-inmates-win-damages-in-constitutional-case-against-prison-authorities-6736190683?utm_source=chatgpt.com|title=Updated: Transgender inmates win damages in constitutional case against prison authorities - The Malta Independent|website=www.independent.com.mt|access-date=2026-03-14}}&lt;/ref&gt;

While this case does not specifically speak to communications law, it does further show the development of the law in treating transgender individuals as a protected class. In this 2018 case, 7 inmates who identified as transgender women sued for being placed in the male section of the prison in accordance with the sex listed on their official documentation. The inmates stated that being placed in this section, while living as women, created a situation in which they were placed in constant abuse and harassment.  The Maltese court ruled that the prison system was inhumane for forcing these individuals to stay in the male-section of the prison, and ruled for the inmates, granting them damages in €5,000 for their mistreatment. 

== 5.Cultural and Religious Expressions ==
- Put here how Malta is culturally Catholic, look back at the module as to what this should go into 

== 6. Privacy &amp; Data Protection ==
For purposes of this section, privacy is defined as “the quality or state of being apart from company or observation; freedom from unauthorized intrusion.&lt;ref&gt;{{Cite web|url=https://www.merriam-webster.com/dictionary/privacy|title=Definition of PRIVACY|date=2026-04-26|website=www.merriam-webster.com|language=en|access-date=2026-04-28}}&lt;/ref&gt;&quot; Many believe that there are certain aspects of life that should rightly be protected from unnecessary government intrusion. However, this creates a tension between the rights of citizens to privacy, and the rights of these same citizens to the free access of information. This tension can clearly be seen in the legislation regarding this subject in Malta. 

'''International Laws Governing Maltese Rights to Privacy:''' 

Because Malta is member of the European Union, it is under the General Data Protection Regulation (GDPR).  This is a 2018 EU regulation relating to how personal data of citizens is collected, stored, shared, and used.  Importantly, the statute not only provides the framework for how this data is treated, but provides the necessary definitions of what falls into the category of “personal data.”  The regulation states in article 4 that personal data refers to: 

''Any information'' relating to an identified or identifiable natural person (‘data subject’); an identifiable natural person is one who can be identified, directly or indirectly, in particular by reference to an identifier such as a name, an identification number, location data, an online identifier or to one or more factors specific to the physical, physiological, genetic, mental, economic, cultural or social identity of that natural person. (emphasis added)

The legislation speaks in Article 5 on the principles of how this information ought to be processed. The first line speaks to transparency, stating that the “data subject’s” (more clearly defined as a “natural” person), information should be “processed lawfully, fairly and in a transparent manner” in relation to that person.  Article 5 goes on to emphasize that all of the processing ought to be done for legitimate purposes, and only processed to the extent necessary. This “processing” is defined essentially as any use of the personal data in Article 4(2). 

The lawfulness of this use, or “processing,” of this personal data is explicated in Article 6 of the GDPR. The main points of this Ariticle can be summarized in the following: the processing of personal data must be necessary and consented to by the person. 

Significantly, Article 1 of the GDPR states that, “the free movement of personal data within the Union shall be neither restricted nor prohibited for reasons connected with the protection of natural persons with regard to the processing of personal data.” 
[[File:Surveillance video cameras, Gdynia.jpeg|thumb]]
Further, as a member of the ECHR, Malta is required to protect the rights found in Article 8, titled “Right to respect for private and family life.” These rights include the right to “respect for his private and family life, his home and his correspondence,” and grants that, &quot;there shall be no interference by a public authority with the exercise of this right except such as is in accordance with the law and is necessary in a democratic society in the interests of national security, public safety or the economic well-being of the country, for the prevention of disorder or crime, for the protection of health or morals, or for the protection of the rights and freedoms of others.&quot;

On the European stage, Malta must maintain strict protection of the privacy of its citizens. But Malta itself has a body of statutory rules governing privacy as well. 

'''Internal Maltese Law Governing Right to Privacy''' 

Under section 38, titled, “Protection for privacy of home or other property” of the Constitution of Malta first lays out the preliminary mandate that “no person shall be subjected to the search of his person or his property or the entry by others on his premises;” provided they have not consented to this, or this is not “by way of parental discipline.”  

The section then goes into situation-specific exceptions to this rule in 38(2). These include the “reasonably required” test seen in European law in the following situations: 

“in the interest of defence, public safety, public order, public morality or decency, public  health,  town  and  country  planning, the development and utilisation of mineral resources, or the  development  and  utilisation  of  any  property  in such a manner as to promote the public benefit,” as well as, “that  is  reasonably  required  for  the  purpose  of promoting the rights or freedoms of other persons.”  The statute goes on to express

== 2. Legal Framework of Privacy in Malta ==

== 7. Right to Bodily, Spiritual, &amp; Digital Identity ==

== 8. Right to Reject Information, Clothing, &amp; Human Exhibitions ==

==== References: ====
[[Category:Malta]]
[[Category:Law in Europe]]
[[Category:Communication in Europe|Law in Malta]]</text>
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      <text bytes="37119" sha1="0svbdlbn18f8le7uyrd5f4mbuc3u892" xml:space="preserve">== 1. History of Malta and Maltese Communication Law ==
{{law}}

==== History of Malta ====
The Mediterranean island of [https://www.gov.mt/en/Pages/Home.aspx Malta] is located directly in the crossroads of several formidable historical forces.  The impacts of this are clearly seen in the country's dramatic history, with effects still present in its &quot;mixed&quot; legal system.  Situated south of Sicily, Malta was [[wikipedia:Capture_of_Malta_(218_BC)#:~:text=The%20capture%20of%20Malta%20was,Punic%20War%20in%20218%20BC.|conquered by Rome]] in 218 B.C. in the Second Punic War.  This forced the small island to be placed under Roman law for the majority of time between 218 B.C. and 1530, when Holy Roman Emperor Charles V granted the island of Malta to the Order of St. John (also known as the &quot;Knights Hospitallers&quot;). The tiny island was then ruled shortly by France from 1789 - 1813 and then England from 1813 forward,&lt;ref&gt;{{Cite news|url=https://www.bbc.com/news/world-europe-17598542|title=Malta profile - Timeline|date=2012-04-03|work=BBC News|access-date=2026-02-21|language=en-GB}}&lt;/ref&gt; only gaining its sovereign independence in 1964.&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=Aquilina|first=Kevin|date=2013|title=The Nature and Sources of The Maltese Mixed Legal System: A Strange Case of Dr. Jekyll and Mr. Hyde?|url=https://heinonline.org/HOL/Page?handle=hein.journals/complv4&amp;id=&amp;collection=journals&amp;div=8|journal=Comparative law review|pages=1-38|via=HeinOnline}}&lt;/ref&gt;
[[File:Malta, 2010 - panoramio - Bengt Nyman (23).jpg|thumb]]Malta's &quot;mixed system&quot; is based most clearly on the fact that it maintains both civil and common law.&lt;ref name=&quot;:0&quot; /&gt; In practice, this means it follows a legal system predicated on both statutes and past court decisions.  Malta has been influenced greatly by international and European systems - the small country has a &quot;Westminster&quot; styled Constitution, but does not hold Constitutional Court decisions as binding on any other than the parties in dispute.&lt;ref name=&quot;:0&quot; /&gt;

==== Maltese Communication Law History ====
Malta was owned by the British as the &quot;Crown Colony of Malta&quot; from 1813 - 1964, a time period that included many critical changes in the modernization of communication.  As a colony of the British Empire, Malta was subject to British influence on both the implementation of the modern means of communication, as well as the laws surrounding the regulation of these means.&lt;ref&gt;{{Cite web|url=https://broadcasting-authority.azurewebsites.net/ourhistory?utm_source=chatgpt.com|title=Our History {{!}} Broadcasting Authority|website=broadcasting-authority.azurewebsites.net|access-date=2026-03-18}}&lt;/ref&gt; This included the creation of the Broadcast Authority, established in 1961 to regulate all broadcasting in the nation.&lt;ref&gt;{{Cite web|url=https://ba.org.mt/en/about-us|title=About Us|website=ba.org.mt|access-date=2026-03-19}}&lt;/ref&gt; 

Radio broadcasting first came to Malta in the form of cable radio through a private British company called &quot;Rediffusion&quot; in 1935.&lt;ref name=&quot;:6&quot;&gt;{{Cite web|url=https://timesofmalta.com/article/radio-broadcasting-turns-90.1119330|title=Radio broadcasting turns 90|last=Borg|first=Fr Joe|date=2025-11-11|website=Times of Malta|language=en-gb|access-date=2026-03-18}}&lt;/ref&gt;  This was especially important at the time, as Malta's cultural and physical closeness to Italy put them at risk of infiltration by the ideas of the expanding Fascist Mussolini regime through wireless radio broadcasts.&lt;ref name=&quot;:6&quot; /&gt; To combat this, British Colonial Authorities strategically placed loudspeakers with approved programs across the island, and banned the use of non-wire radio.&lt;ref name=&quot;:6&quot; /&gt;  The easy public access to radio programs, paired with the auditory nature of the news bypassing the necessity to be literate, caused radio to surpass newspapers as the main source of information to the public.&lt;ref name=&quot;:6&quot; /&gt; 

The perceived danger to Maltese citizens from Italian and Sicilian influence continued to pressure the island and those who ruled it to continually adopt updated communication systems.  In the 1950s, Maltese reception of Italian TV signals led the country to first create the television network that would become TVM, and later, the adoption of color broadcast.&lt;ref name=&quot;:6&quot; /&gt;  The geographical proximity to Italy created a situation such that, when the one-station system censored differing political ideas, the Nationalist party was able to resort to Sicilian radio and television stations to broadcast their opposing viewpoints.&lt;ref name=&quot;:6&quot; /&gt;

The small nation's reliance on Rediffusion lasted up until 1971, when the Mintoff government decided to nationalize radio and television; this was ultimately not successful until 1975.&lt;ref name=&quot;:6&quot; /&gt;  This led to the establishment of Xandir Malta, or &quot;Broadcasting Malta,&quot; a Socialist-run broadcasting network that goverened the communications sector until it was pushed out by the Nationalist party in the late 1980s.  The Nationalist party then established a public broadcasting network and issued &quot;The 1990 White Paper Commitment to Pluralism,&quot; promoting pluralism in broadcasting.&lt;ref name=&quot;:6&quot; /&gt;  The Church and the Government were the first to have their own radio stations, followed by additions in 1992.  Radio broadcasting is reportedly consumed by two-thirds of Maltese citizens to this day.&lt;ref name=&quot;:6&quot; /&gt;

When Malta joined the European Union (EU) in 2004, they were required to extend their communications legislation to meet EU standards, including promoting European and independent works.&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/legal-content/EN/TXT/?uri=legissum:e20112|website=eur-lex.europa.eu|access-date=2026-03-19}}&lt;/ref&gt;  Malta has since joined multiple international agreements, such that the Maltese lawmakers must ensure that both the law of Malta and the terms of multiple treaties and directives are followed when writing legislation.  Maltese communication law is emblematic of the legal situation of many European countries: though the country maintains its sovereign independence, the law is riddled with influence from past rulers, as well as attempts to catch up with technological progress and the ever-expanding process of globalization.  

== 2. Sources of Maltese Communications Law ==

==== National ====
[[File:Malta ali 2009224 lrg.jpg|thumb]]
The primary statute governing communications law in Malta is the [https://legislation.mt/eli/cap/399/eng?utm_source=chatgpt.com Electronic Communications (Regulation) Act].&lt;ref&gt;{{Cite web|url=https://legislation.mt/eli/cap/399/eng?utm_source=chatgpt.com|title=Electronic Communications (Regulation) Act|date=Dec. 31st, 1997|website=Legislation Malta}}&lt;/ref&gt;  This statute lays the groundwork for how communications law is to be upheld in Malta: it governs access to networks, authorizations and license requirements, radio communications, and the regulatory framework that telecommunications companies will have to adhere to.  

[https://legislation.mt/eli/cap/418/eng Shortly after, the Malta Communications Authority Act]&lt;ref&gt;{{Cite web|url=https://legislation.mt/eli/cap/418/eng|title=MALTA COMMUNICATIONS AUTHORITY ACT|date=Aug. 1, 2000}}&lt;/ref&gt; was passed and functioned to establish the [https://www.mca.org.mt/ Malta Communication Authority (MCA).]  The Act gave the MCA regulating authority to enforce Maltese communication law, established the regulatory goals of the entity, and expressed the requirements for members wishing to serve on the MCA board.  

The MCA's authority was challenged in European Court of Justice case [https://www.maltatoday.com.mt/news/court_and_police/87626/vodafone_loses_challenge_on_financing_legal_interceptions Vodafone Malta Ltd v. Malta Communication Authority.]&lt;ref name=&quot;:2&quot;&gt;{{Cite web|url=https://infocuria.curia.europa.eu/tabs/document?source=document&amp;docid=138859&amp;doclang=EN|title=Vodafone Malta Ltd and Mobisle Communications Ltd v Attorney General and Others (Case C-71/12) — Judgment of the Court (Third Chamber), 27 June 2013.|website=CURIA}}&lt;/ref&gt; Vodafone claimed that the 3% excise tax the Maltese government had levied on mobile providers was a breach of EU law.&lt;ref&gt;{{Cite web|url=https://www.mca.org.mt/articles/judgement-court-vodafone-malta-limited-et-vs-avukati-generali-et-case-c%E2%80%917112|title=Judgement of the Court - Vodafone Malta Limited et vs Avukati Generali et - Case C‑71/12 {{!}} MCA|last=Authority|first=Malta Communication|website=www.mca.org.mt|language=en|access-date=2026-02-27}}&lt;/ref&gt;  The court affirmed the MCA's authority to regulate communications law, ruled that the MCA was acting fully within their regulatory authority, and the national tax to offset interception costs was valid.&lt;ref name=&quot;:2&quot; /&gt;

Another statute governing communications law in Malta is the aforementioned [https://ba.org.mt/legislation Broadcasting Act.]&lt;ref name=&quot;:3&quot;&gt;{{Cite web|url=https://ba.org.mt/legislation|title=CHAPTER 350
BROADCASTING ACT|date=1 June 1991|website=Malta Broadcasting Authority}}&lt;/ref&gt; Where the ECR act was focused on the infrastructure of communications systems in Malta, the Broadcasting Act was directed at the content that was being pushed through television, radio, and audiovisual programming.  This Act also established the regulator the [https://ba.org.mt Malta Broadcasting Authority,] which seeks to enforce the statutes contained in the act.&lt;ref name=&quot;:3&quot; /&gt;     

==== International ====
Being an EU member state, Maltese communication law is heavily impacted on the supranational decisions of the European Union. The MCA lists 8 articles of EU legislation on their website divided into 3 subsections: e-commerce, postal service, and regulatory framework for communications.&lt;ref&gt;{{Cite web|url=https://www.mca.org.mt/general/legislation|title=Legislation {{!}} MCA|last=Authority|first=Malta Communication|website=www.mca.org.mt|language=en|access-date=2026-02-21}}&lt;/ref&gt; The MCA cites an EU directive from 2000&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/legal-content/EN/ALL/?uri=celex%3A32000L0031|title=Directive 2000/31/EC of the European Parliament and of the Council of 8 June 2000 on certain legal aspects of information society services, in particular electronic commerce, in the Internal Market ('Directive on electronic commerce')|date=8 June 2000|website=EUR-Lex}}&lt;/ref&gt; as well as a 2014 EU regulation&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/eli/reg/2014/910/oj/eng|title=Regulation (EU) No 910/2014 of the European Parliament and of the Council of 23 July 2014 on electronic identification and trust services for electronic transactions in the internal market and repealing Directive 1999/93/EC|date=23 July 2014|website=EUR-Lex}}&lt;/ref&gt; as controlling for Maltese E-commerce.  The MCA cites a 2008 EU Directive&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/eli/dir/2008/6/oj/eng|title=Directive 2008/6/EC of the European Parliament and of the Council of 20 February 2008 amending Directive 97/67/EC with regard to the full accomplishment of the internal market of Community postal services|date=20 February 2008|website=EUR-Lex}}&lt;/ref&gt; as the applicable EU law for Maltese postal services, and  a 2018 EU Directive,&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/eli/dir/2018/1972/oj/eng|title=Directive (EU) 2018/1972 of the European Parliament and of the Council of 11 December 2018 establishing the European Electronic Communications Code (Recast) (Text with EEA relevance)|date=11 December 2018|website=EUR-Lex}}&lt;/ref&gt; as well as four 2002 EU Directives&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/eli/dir/2002/22/oj/eng|title=Directive 2002/22/EC of the European Parliament and of the Council of 7 March 2002 on universal service and users' rights relating to electronic communications networks and services (Universal Service Directive)|date=7 March 2002|website=EUR-Lex}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/eli/dir/2002/20/oj/eng|title=Directive 2002/20/EC of the European Parliament and of the Council of 7 March 2002 on the authorisation of electronic communications networks and services (Authorisation Directive)|date=7 March 2002|website=EUR-Lex}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/eli/dir/2002/19/oj/eng|title=Directive 2002/19/EC of the European Parliament and of the Council of 7 March 2002 on access to, and interconnection of, electronic communications networks and associated facilities (Access Directive)|date=7 March 2002|website=EUR-Lex}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://eur-lex.europa.eu/eli/dir/2002/58/oj/eng|title=Directive 2002/58/EC of the European Parliament and of the Council of 12 July 2002 concerning the processing of personal data and the protection of privacy in the electronic communications sector (Directive on privacy and electronic communications)|website=EUR-Lex}}&lt;/ref&gt; as the applicable law for electronic communications. 

Additionally, Maltese membership in the European Union requires that the country adheres to three principles of EU membership:   

# As a member state, citizens of Malta have the ability to uphold certain EU rights in court.  This is known as the &quot;Principle of Direct Effect,&quot; and was established by the 1963 case Van Gend en Loos v Nederlandse Administratie der Belastingen.&lt;ref&gt;{{Cite web|url=https://www.lawyersnjurists.com/article/direct-effect-of-europian-union-law/|title=DIRECT EFFECT OF EUROPEAN UNION LAW|website=The Lawyers &amp; Jurists|access-date=2026-03-19}}&lt;/ref&gt;  This means that any Maltese citizen or company can invoke European Union communication rights in the Maltese domestic court system. 
# Further, Malta is obligated to interpret its domestic laws ''as much as possible'' in light of EU directives.  This is known as the &quot;Principle of Conforming Interpretation&quot; or &quot;Principle of Indirect Effect,&quot; and was established by the 1984 case of ''V''on Colson v Land Nordrhein-Westfalen.&lt;ref&gt;{{Cite web|url=https://employmentrightsireland.com/von-colson-and-kamann-establishing-the-principle-of-indirect-effect/#:~:text=The%20*Von%20Colson%20and%20Kamann*%20case%20established,which%20will%20give%20effect%20to%20EU%20directives.|title=Von Colson and Kamann-Establishing the Principle of Indirect Effect – Employment Rights Ireland|website=employmentrightsireland.com|access-date=2026-03-19}}&lt;/ref&gt;  Therefore, all of the cases relating to communications law in Malta must be decided utilizing a legislative framework that takes into account the directives of the EU. 
# Lastly, the member states of the EU must ensure that their regulatory measures are: 1) suitable 2) necessary 3) does not impose excessive burdens.  This is known as the &quot;Principle of Proportionality,&quot; and was established by the case ''I''nternationale Handelsgesellschaft mbH v Einfuhr- und Vorratsstelle für Getreide und Futtermittel.&lt;ref&gt;{{Cite web|url=https://opil.ouplaw.com/display/10.1093/law:epil/9780199231690/law-9780199231690-e1459|title=Proportionality|last=Crawford|first=Emily|date=May 2011|website=Oxford Public International Law}}&lt;/ref&gt;  This therefore requires that the Maltese Communication Authority and Broadcast Authority regulations align with the Proportionality standards. 

==== Regional ====
Malta is a member party to the [https://www.echr.coe.int/documents/d/echr/Convention_ENG European Convention on Human Rights (ECHR),]&lt;ref name=&quot;:7&quot;&gt;{{Cite web|url=https://www.echr.coe.int/documents/d/echr/Convention_ENG|title=European Convention on Human Rights|date=Nov. 4th, 1950}}&lt;/ref&gt; and therefore must abide by the ECHR's protection of freedom of expression.  This protection is enshrined in Article 10 of the convention,&lt;ref&gt;{{Cite web|url=https://fra.europa.eu/en/law-reference/european-convention-human-rights-article-10|title=European Convention on Human Rights - Article 10|website=European Union Agency for Fundamental Rights}}&lt;/ref&gt; which states the following:  

&quot;Everyone has the right to freedom of expression. This right shall include freedom to hold opinions and to receive and impart information and ideas without interference by public authority and regardless of frontiers. This Article shall not prevent States from requiring the licensing of broadcasting, television or cinema enterprises.&quot;  

Further, the ECHR established the [https://www.echr.coe.int European Court of Human Rights], which has jurisdiction over Maltese law in Malta-specific cases, and persuasive authority for cases dealing with other member states.&lt;ref name=&quot;:7&quot; /&gt;  

== 3. Censorship and Violent Content ==

==== Hate Speech ====
Malta takes a no-nonsense approach to hate speech, penalizing those convicted with 6-18 months.&lt;ref name=&quot;:1&quot;&gt;{{Cite web|url=https://legislation.mt/eli/cap/9/eng/pdf?utm_source=chatgpt.com|title=Criminal Code Chapter 9|website=LEĠIŻLAZZJONI MALTA}}&lt;/ref&gt; Hate speech is governed primarily by the [https://legislation.mt/eli/cap/9/eng/pdf?utm_source=chatgpt.com Maltese Criminal Code]&lt;ref name=&quot;:4&quot;&gt;{{Cite web|url=https://legislation.mt/eli/cap/9/eng/pdf?utm_|title=CHAPTER 9 CRIMINAL CODE To amend and consolidate the Penal Laws and the Laws of Criminal Procedure.|website=LEĠIŻLAZZJONI MALTA}}&lt;/ref&gt;, which states the following: 

# &quot;Whosoever uses any threatening, abusive or insulting words or behaviour, or displays any written or printed material which is threatening, abusive or insulting, or otherwise conducts himself in such a manner, with intent thereby to stir up violence or hatred against another person or group of persons on the grounds of gender, gender identity, sexual orientation, race, colour, language, ethnic origin, age, disability, religion or belief or political or other opinion or whereby such violence or hatred is likely, having regard to all the circumstances, to be stirred up shall, on conviction, be liable&quot;

The Code goes on in Sections 82B and 82C to crim&lt;ref&gt;{{Cite web|url=https://globalfreedomofexpression.columbia.edu/cases/garaudy-v-france/|title=Garaudy v. France|website=Global Freedom of Expression|language=en-US|access-date=2026-03-19}}&lt;/ref&gt;inalize support or denial of crimes against humanity, genocide, war crimes and crimes against peace. 

* ''(''a) likely to incite to violence or hatred against such a group or a member of such a group;

* (b) likely to disturb public order or which is threatening, abusive or insulting

==== Case law: ====
[https://fra.europa.eu/en/databases/criminal-detention/case-law/6700 Police v. Norman Lowell]&lt;ref&gt;{{Cite web|url=https://fra.europa.eu/en/databases/criminal-detention/case-law/6700|title=Malta / Criminal Court of Appeal / 98/2011 / Judgement / Police vs Norman Lowell|website=European Union Agency for Fundamental Rights}}&lt;/ref&gt; 

In this 2013 case, a political activist was convicted for the charge of &quot;incitement of racial hatred&quot; against immigrants and Jews for delivering speeches and publishing allegedly racist views including incitements to violence online. Lowell was charged under Article 82A of the Maltese Criminal Code and received a 2 year sentence (later suspended) and a €500 fine.  

[https://globalfreedomofexpression.columbia.edu/cases/garaudy-v-france/ Garaudy v. France]&lt;ref&gt;{{Cite web|url=https://globalfreedomofexpression.columbia.edu/cases/garaudy-v-france/|title=Garaudy v. France|website=Global Freedom of Expression|language=en-US|access-date=2026-03-19}}&lt;/ref&gt; 

Being bound by the European Convention of Human Rights, this case is a persuasive authority for Maltese decision making. This 2003 case ruled against French Philosopher Roger Garaudy who was convicted in France for his 1998 book that allegedly denied the Holocaust.  The Court ruled that Article 17 of the Convention preempted Article 10 - making his writing fall outside the realm of protected speech. 

==== Illegal Pornography and Child Protection ====
The Maltese government has strict laws governing pornography.  Under Section 208 of the Code Chapter 9, displays of pornographic material in public places will result in fines &quot;of not less than one thousand euro (€1,000) and not more than three thousand euro (€3,000).&quot;&lt;ref name=&quot;:4&quot; /&gt; Section 208 of the statute goes further to strictly outlaw all forms of child pornography and &quot;revenge porn&quot; (illicit photos intended for one person and then used as blackmail).&lt;ref name=&quot;:4&quot; /&gt;

==== Case Law ====
[https://hudoc.echr.coe.int/eng#{%22itemid%22:&amp;#x5B;%22001-57499%22&amp;#x5D;} Handyside v. United Kingdom]&lt;ref&gt;{{Cite web|url=https://hudoc.echr.coe.int/eng#%7B%22itemid%22:%5B%22001-57499%22%5D%7D|title=HUDOC - European Court of Human Rights|website=hudoc.echr.coe.int|access-date=2026-03-19}}&lt;/ref&gt;

This 1976 case established that while the European Union protects &quot;freedom of expression&quot; in Article 10 of the European Convention of Human Rights, the European Union recognizes the authority of countries to limit this due to protecting the morality of their citizens. The case focused on the seizure and prohibition of a pornographic schoolbook set to be disseminated throughout the United Kingdom. The court stated, 

&quot;the Court first finds that the 1959/1964 Acts have an aim that is legitimate under Article 10 para. 2 (art. 10-2), namely, the protection of morals in a democratic society. Only this latter purpose is relevant in this case since the object of the said Acts - to wage war on 'obscene' publications, defined by their tendency to 'deprave and corrupt' - is linked far more closely to the protection of morals than to any of the further purposes permitted by Article 10 para. 2 (art. 10-2).&quot; 

This decision showed to member states that while there are certain supranational requirements they must maintain, individual countries still had rights to uphold their individual public morality.  

==== Defamation and Libel Reform ====
Though previously illegal, the [https://parlament.mt/13th-leg/acts/act-xi-of-2018/ Media and Defamation] act abolished the crime of libel and relegated it to civil courts. Section 3 of the act states that &quot;defamatory words in written media&quot; constitutes libel, &lt;ref&gt;{{Cite web|url=https://parlament.mt/media/93813/act-xi-media-and-defamation-act.pdf|title=ACT No. XI of 2018 The Media and Defamation Act|date=24 April 2018|website=Parliament of Malta}}&lt;/ref&gt;  and prior to this Act, libel could be criminally prosecuted.  

==== Case law ====
[https://globalfreedomofexpression.columbia.edu/cases/falzon-v-malta/?utm_source=chatgpt.com Falzon v. Malta]&lt;ref&gt;{{Cite web|url=https://globalfreedomofexpression.columbia.edu/cases/falzon-v-malta/|title=Falzon v. Malta|website=Global Freedom of Expression|language=en-US|access-date=2026-03-14}}&lt;/ref&gt;

This 2018 ECtHR court decision directly implicated Maltese communication law, as the Maltese court system had ruled against Falzon in a civil libel case for dramatizing a negative story of the Maltese Police Force.  The ECtHR ruled that this was a violation of Article 10 of the Convention, due to the purpose of the article being political commentary based on civic engagement, rather than true defamatory allegations against individuals; the Maltese court was thus overruled.  

[https://globalfreedomofexpression.columbia.edu/cases/cuschieri-v-galizia/?utm_source=chatgpt.com Caruana Galizia v. Cuschieri] &lt;ref&gt;{{Cite web|url=https://globalfreedomofexpression.columbia.edu/cases/cuschieri-v-galizia/|title=Caruana Galizia v. Cuschieri|website=Global Freedom of Expression|language=en-US|access-date=2026-03-14}}&lt;/ref&gt;

In this case, a woman was sued for defamation for falsely alleging in an independent blog that a politician was an antisemite and had employed seditious tactics.  The court ruled that this was not protected speech, as it went beyond mere opinion or political commentary and into character attacks on the politician that were ill-founded. 

== 4. Truth, Honor &amp; Tolerance ==
Malta's membership in both the European Union and European Convention on Human Rights requires the country to provide equal protection to citizens under the law.  Malta itself has also established additional national laws instituting this. 
[[File:Basílica de San Pablo, Rabat, isla de Malta, Malta, 2021-08-25, DD 167.jpg|thumb]]
With 85% of the Maltese Population identifying as Catholic, and 2,000 years of Catholic history,&lt;ref&gt;{{Cite web|url=https://www.ewtnnews.com/world/europe/facing-declining-mass-attendance-malta-s-catholics-hope-papal-visit-will-revitalize-the-faith|title=Facing a decline in Mass attendance, Catholics in one country hope papal visit will revitalize the faith|website=Catholic News Agency|language=en|access-date=2026-03-14}}&lt;/ref&gt; it is unsurprising that Article 2 of the Maltese Constitution established Catholicism as the state religion.&lt;ref name=&quot;:5&quot;&gt;{{Cite web|url=https://legislation.mt/eli/const/eng/pdf|title=Constitution of Malta|date=21st September, 1964|website=LEGISLATION MALTA}}&lt;/ref&gt; However, under Article 40 of the Maltese Constitution,&lt;ref name=&quot;:5&quot; /&gt; as well as Article 10 of the EU Charter of Human Rights,&lt;ref&gt;{{Cite web|url=https://fra.europa.eu/en/eu-charter/article/10-freedom-thought-conscience-and-religion|title=Article 10 - Freedom of thought, conscience and religion {{!}} European Union Agency for Fundamental Rights|date=2015-04-25|website=fra.europa.eu|language=en|access-date=2026-03-19}}&lt;/ref&gt; and Article 9 of the ECtHR,&lt;ref name=&quot;:7&quot; /&gt; Maltese citizens enjoy freedom of religious practice. 

==== Constitutional Rights to Freedom of Expression ====
[https://www.constituteproject.org/constitution/Malta_2016 Article 41] of the Maltese Constitution&lt;ref name=&quot;:8&quot;&gt;{{Cite web|url=https://www.constituteproject.org/constitution/Malta_2016|title=Malta 1964 (rev. 2016) Constitution - Constitute|website=www.constituteproject.org|language=en|access-date=2026-03-16}}&lt;/ref&gt; lays out the groundwork for the parameters of freedom of expression in the country. The first section of the article grants broad &quot;freedom of expression&quot; with the exceptions of citizens consenting to be censored, and parental discipline.  

The second section of Article 41 expresses the occasions in which the government may legally impose limitations on this freedom of expression.  This includes situations in which &quot;it is reasonably required&quot; for:   

&quot;defense, public safety, public order, public morality or decency, or public health; or for the purpose of protecting the reputations, rights and freedoms of other persons, or the private lives of persons concerned in legal proceedings, preventing the disclosure of information received in confidence, maintaining the authority and independence of the courts, protecting the privileges of Parliament, or regulating telephony, telegraphy, posts, wireless broadcasting, television or other means of communication, public exhibitions or public entertainments.&quot;&lt;ref name=&quot;:8&quot; /&gt;   

The second article also explains that the government can limit the freedom of expression for public officers.&lt;ref name=&quot;:8&quot; /&gt;  

The third applicable section of this article explains the limits the government can impose on publisher's freedom of expression.  This first includes the prohibition or restriction of both editing or printing of publications by people under 21 years old; and the mandate that printers and editors inform the proper authorities of their ages and locations.&lt;ref name=&quot;:8&quot; /&gt;   

The final section of Article 41 explains the limits of seizure in response to freedom of expression.   The section first states that when the police seize a newspaper that had been used to commit a criminal offense, they are required to go to a competent court within 24 hours to see if the prima facie case of the offense was satisfied.  The section states that if the prima facie case is not satisfied, the paper must be returned to its owner.  The section ends with stating that, &quot;No person shall be deprived of his citizenship under any provisions made under article 30(1) (b) of this Constitution or of his juridical capacity by reason only of his political opinions.&quot;&lt;ref name=&quot;:8&quot; /&gt;  

==== Tolerance and Anti-Discrimination ====
The Maltese government put forth the [https://parlament.mt/en/9th-leg/acts-9th/act-no-i-of-2003/ Equality for Men and Women Act] &lt;ref&gt;{{Cite web|url=https://parlament.mt/media/1050/i-of-2003-equality-for-men-and-women-act.pdf|title=ACT No. I of 2003, AN ACT to promote equality for men and women|date=4 February 2003|website=Parliament of Malta}}&lt;/ref&gt; in order to protect the right of both sexes to be treated as equals under the law -- especially protecting women from prejudicial discrimination.  While this does not explicitly mention communications law, it does require men and women to be treated non-discriminatorily in every area, therefore including communication platforms.  Additionally, in the [https://parlament.mt/media/129409/att-xxv-gender-identity.pdf Gender Identity, Gender Expression and Sex Characteristics Act],&lt;ref&gt;{{Cite web|url=https://parlament.mt/media/129409/att-xxv-gender-identity.pdf|title=Act No. XXV of 2024 - Gender Identity, Gender Expression and Sex Characteristics (Amendment) Act|date=12th July 2024|website=Parliament of Malta}}&lt;/ref&gt; the Maltese government instituted rights of transgender individuals to legally change and be known by their preferred gender/name, and to be treated in accordance with these preferences.  This act also protects transgender individuals from discrimination for these changes. 

==== Case law: ====
[https://timesofmalta.com/article/from-bench-is-hate-speech-freedom-expression.1051613?utm_source=chatgpt.com Police v. Michael Leonard Paul Hammond]&lt;ref&gt;{{Cite web|url=https://timesofmalta.com/article/from-bench-is-hate-speech-freedom-expression.1051613|title=From the Bench: Is it hate speech or freedom of expression?|last=Contributors|date=2023-08-26|website=Times of Malta|language=en-gb|access-date=2026-03-14}}&lt;/ref&gt;

In this 2023 case, a man was convicted for posting a hateful comment on a pro-LGBT Facebook page.  The court ruled that his comment under a Gay Pride parade post of, &quot;should bomb all u (sic) freaks,&quot;&lt;ref&gt;{{Cite web|url=https://timesofmalta.com/article/from-bench-is-hate-speech-freedom-expression.1051613|title=From the Bench: Is it hate speech or freedom of expression?|last=Contributors|date=2023-08-26|website=Times of Malta|language=en-gb|access-date=2026-03-14}}&lt;/ref&gt; constituted a violation of Section 82A of the Maltese Criminal Code.  Hammond later tried to appeal this conviction, but the court upheld the lower court's decision. 

[https://www.independent.com.mt/articles/2018-05-29/local-news/Transgender-inmates-win-damages-in-constitutional-case-against-prison-authorities-6736190683?utm_source=chatgpt.com Transgender Inmates v. Prison Authorities]&lt;ref&gt;{{Cite web|url=httpss://www.independent.com.mt/articles/2018-05-29/local-news/Transgender-inmates-win-damages-in-constitutional-case-against-prison-authorities-6736190683?utm_source=chatgpt.com|title=Updated: Transgender inmates win damages in constitutional case against prison authorities - The Malta Independent|website=www.independent.com.mt|access-date=2026-03-14}}&lt;/ref&gt;

While this case does not specifically speak to communications law, it does further show the development of the law in treating transgender individuals as a protected class. In this 2018 case, 7 inmates who identified as transgender women sued for being placed in the male section of the prison in accordance with the sex listed on their official documentation. The inmates stated that being placed in this section, while living as women, created a situation in which they were placed in constant abuse and harassment.  The Maltese court ruled that the prison system was inhumane for forcing these individuals to stay in the male-section of the prison, and ruled for the inmates, granting them damages in €5,000 for their mistreatment. 

== 5.Cultural and Religious Expressions ==
- Put here how Malta is culturally Catholic, look back at the module as to what this should go into 

== 6. Privacy &amp; Data Protection ==
For purposes of this section, privacy is defined as “the quality or state of being apart from company or observation; freedom from unauthorized intrusion.&lt;ref&gt;{{Cite web|url=https://www.merriam-webster.com/dictionary/privacy|title=Definition of PRIVACY|date=2026-04-26|website=www.merriam-webster.com|language=en|access-date=2026-04-28}}&lt;/ref&gt;&quot; Many believe that there are certain aspects of life that should rightly be protected from unnecessary government intrusion. However, this creates a tension between the rights of citizens to privacy, and the rights of these same citizens to the free access of information. This tension can clearly be seen in the legislation regarding this subject in Malta. 

'''International Laws Governing Maltese Rights to Privacy:''' 

Because Malta is member of the European Union, it is under the General Data Protection Regulation (GDPR).  This is a 2018 EU regulation relating to how personal data of citizens is collected, stored, shared, and used.  Importantly, the statute not only provides the framework for how this data is treated, but provides the necessary definitions of what falls into the category of “personal data.”  The regulation states in article 4 that personal data refers to: 

''Any information'' relating to an identified or identifiable natural person (‘data subject’); an identifiable natural person is one who can be identified, directly or indirectly, in particular by reference to an identifier such as a name, an identification number, location data, an online identifier or to one or more factors specific to the physical, physiological, genetic, mental, economic, cultural or social identity of that natural person. (emphasis added)

The legislation speaks in Article 5 on the principles of how this information ought to be processed. The first line speaks to transparency, stating that the “data subject’s” (more clearly defined as a “natural” person), information should be “processed lawfully, fairly and in a transparent manner” in relation to that person.  Article 5 goes on to emphasize that all of the processing ought to be done for legitimate purposes, and only processed to the extent necessary. This “processing” is defined essentially as any use of the personal data in Article 4(2). 

The lawfulness of this use, or “processing,” of this personal data is explicated in Article 6 of the GDPR. The main points of this Ariticle can be summarized in the following: the processing of personal data must be necessary and consented to by the person. 

Significantly, Article 1 of the GDPR states that, “the free movement of personal data within the Union shall be neither restricted nor prohibited for reasons connected with the protection of natural persons with regard to the processing of personal data.” 
[[File:Surveillance video cameras, Gdynia.jpeg|thumb]]
Further, as a member of the ECHR, Malta is required to protect the rights found in Article 8, titled “Right to respect for private and family life.” These rights include the right to “respect for his private and family life, his home and his correspondence,” and grants that, &quot;there shall be no interference by a public authority with the exercise of this right except such as is in accordance with the law and is necessary in a democratic society in the interests of national security, public safety or the economic well-being of the country, for the prevention of disorder or crime, for the protection of health or morals, or for the protection of the rights and freedoms of others.&quot;

On the European stage, Malta must maintain strict protection of the privacy of its citizens. But Malta itself has a body of statutory rules governing privacy as well. 

'''Internal Maltese Law Governing Right to Privacy''' 

Maltese law tends to take on an approach similar to the American prohibition of unreasonable searches and seizures.  Under section 38, titled, “Protection for privacy of home or other property” of the Constitution of Malta first lays out the preliminary mandate that “no person shall be subjected to the search of his person or his property or the entry by others on his premises;” provided they have not consented to this, or this is not “by way of parental discipline.”  

The section then goes into situation-specific exceptions to this rule in 38(2). These include the “reasonably required” test seen in European law in the following situations: 

“in the interest of defence, public safety, public order, public morality or decency, public  health,  town  and  country  planning, the development and utilisation of mineral resources, or the  development  and  utilisation  of  any  property  in such a manner as to promote the public benefit,” as well as, “that  is  reasonably  required  for  the  purpose  of promoting the rights or freedoms of other persons.”  

== 2. Legal Framework of Privacy in Malta ==

== 7. Right to Bodily, Spiritual, &amp; Digital Identity ==

== 8. Right to Reject Information, Clothing, &amp; Human Exhibitions ==

==== References: ====
[[Category:Malta]]
[[Category:Law in Europe]]
[[Category:Communication in Europe|Law in Malta]]</text>
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  <page>
    <title>Patriarch Ages Curious Numerical Facts Response</title>
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      <text bytes="104990" sha1="hvr7ul45q9kev0nc95gj27rkhb0q1ed" xml:space="preserve">{{Original research}}
This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified.

== Summary of Main Arguments ==
The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include:

* '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or &quot;perfect&quot; numbers (such as 7, 49, and 60) rather than biological or historical reality.
* '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions.
* '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood.
* '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific &quot;years of the world&quot; (''Anno Mundi''), synchronizing human history with a divine calendar.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
[[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]]

Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE).

The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations.

* '''16 ''šūši'' (960 years)'''
** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''': Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** SKL: [[w:Atab|Atab]]
** '''Prototype 1''': Shem

* '''7 ''šūši'' (420 years)'''
** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''': Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list.

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word &quot;Adapa&quot; is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].

In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . .
*During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar.
. . .

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.

==== Mesopotamian Similarities ====
*[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions.
*[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.
*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;2&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (Armenian)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;2&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| 753 &lt;br/&gt; &lt;small&gt;(1454)&lt;br/&gt;(2207)&lt;/small&gt;
| 723 &lt;br/&gt; &lt;small&gt;(1454)&lt;br/&gt;(2177)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1642)&lt;br/&gt;(2592)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;2&quot; |&lt;small&gt;(2242)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;2&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (Armenian)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;2&quot; | 167
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth.

{{RoundBoxTop}}
====The &quot;Whoops Theory&quot;: A Digression====
Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor added 100 years to the age at which each patriarch begot his son, intending to &quot;fix&quot; the timeline, but somehow failed in the case of Methuselah. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.

==== Demetrius the Chronographer ====
Writing in the late 3rd century BC (c. 221 BC), '''Demetrius the Chronographer''' stands as the earliest known witness to biblical chronological calculations. Despite the fragmentary nature of his work, his data remains pivotal; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood, a total that inherently supports a longer chronology where Methuselah’s fathering age is '''187''' years rather than '''167'''.

In the original article's comments, a debate surfaced regarding this longer chronology and the '''187''' year fathering age for Methuselah. Paul D. defends his &quot;Whoops Theory&quot; by systematically challenging the validity of the early witnesses, particularly those that support the longer timeline:

* '''Josephus:''' Characterized as being dependent on the Masoretic tradition rather than an independent witness.
* '''Pseudo-Philo:''' Dismissed entirely due to severe textual corruption (described as &quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through a later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional two years whose precise placement remains unknown.
* '''Codex Alexandrinus:''' Identified as the lone legitimate witness to the 187-year fathering age.

The dismissal of Julius Africanus due to his survival through an intermediary, or the disqualification of Demetrius based on a two-year uncertainty, is arguably overstated. As shown in the comparative tables below, there is remarkably little variation in the &quot;LONG CHRONOLOGY&quot; begettal ages. They are identical for Adam through Enoch, with some variation in the last few patriarchs prior to the flood.

The below reconstructed Demetrius chronology employs a plausible explanation for the 2-year discrepancy: the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can account for such variances without necessitating the rejection of the witness. Given that Demetrius operates within the longest known chronological framework, the suggestion that he utilized the longer begettal ages for earlier patriarchs while failing to apply the correction for Methuselah—the very figure requiring it most in such an expanded timeline—is statistically and logically improbable.

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in patriarchs surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | TOTAL
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}
{{RoundBoxBottom}}

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all textual traditions. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors

As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;2&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (Armenian)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|130 + 800}} &lt;br/&gt;= 930
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|230 + 700}} &lt;br/&gt;= 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|105 + 807}} &lt;br/&gt;= 912
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|205 + 707}} &lt;br/&gt;= 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|90 + 815}}  &lt;br/&gt;= 905
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|190 + 715}} &lt;br/&gt;= 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|70 + 840}}  &lt;br/&gt;= 910
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|170 + 740}} &lt;br/&gt;= 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|65 + 830}}  &lt;br/&gt;= 895
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|165 + 730}} &lt;br/&gt;= 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|62 + 900}} &lt;br/&gt;= 962
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|162 + 800}} &lt;br/&gt;= 962
| {{nowrap|62 + 785}} &lt;br/&gt;= 847
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|162 + 800}} &lt;br/&gt;= 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|65 + 300}} &lt;br/&gt;= 365
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|165 + 200}} &lt;br/&gt;= 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|67 + 902}} &lt;br/&gt;= 969
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|187 + 782}} &lt;br/&gt;= 969
| {{nowrap|67 + 653}} &lt;br/&gt;= 720
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|167 + 802}} &lt;br/&gt;= 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|53 + 730}} &lt;br/&gt;= 783
| {{nowrap|182 + 595}} &lt;br/&gt;= 777
| {{nowrap|53 + 600}} &lt;br/&gt;= 653
| {{nowrap|188 + 565}} &lt;br/&gt;= 753
| {{nowrap|188 + 535}} &lt;br/&gt;= 723

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|502 + 448}} &lt;br/&gt;= 950
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|500 + 450}} &lt;br/&gt;= 950
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|502 + 448}} &lt;br/&gt;= 950
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|500 + 450}} &lt;br/&gt;= 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;5&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|100 + 500}} &lt;br/&gt;= 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 438
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|35 + 403}} &lt;br/&gt;= 438
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|135 + 303}} &lt;br/&gt;= 438
| {{nowrap|135 + 400}} &lt;br/&gt;= 535
| {{nowrap|135 + 403}} &lt;br/&gt;= 538

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| 460
| —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 433
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|30 + 403}} &lt;br/&gt;= 433
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|130 + 303}} &lt;br/&gt;= 433
| {{nowrap|130 + 330}} &lt;br/&gt;= 460
| {{nowrap|130 + 406}} &lt;br/&gt;= 536

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 464
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|34 + 430}} &lt;br/&gt;= 464
| colspan=&quot;2&quot; | {{nowrap|134 + 270}} &lt;br/&gt;= 404
| {{nowrap|134 + 433}} &lt;br/&gt;= 567

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 239
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|30 + 209}} &lt;br/&gt;= 239
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|130 + 109}} &lt;br/&gt;= 239
| colspan=&quot;2&quot; | {{nowrap|130 + 209}} &lt;br/&gt;= 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 239
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|32 + 207}} &lt;br/&gt;= 239
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|132 + 107}} &lt;br/&gt;= 239
| {{nowrap|132 + 207}} &lt;br/&gt;= 339
| {{nowrap|135 + 207}} &lt;br/&gt;= 342

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 230
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|30 + 200}} &lt;br/&gt;= 230
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|130 + 100}} &lt;br/&gt;= 230
| colspan=&quot;2&quot; | {{nowrap|130 + 200}} &lt;br/&gt;= 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 148
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|29 + 119}} &lt;br/&gt;= 148
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|79 + 69}} &lt;br/&gt;= 148
| {{nowrap|179 + 125}} &lt;br/&gt;= 304
| {{nowrap|79 + 119}} &lt;br/&gt;= 198

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|70 + 135}} &lt;br/&gt;= 205
| {{nowrap|70 + 75}} &lt;br/&gt;= 145
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|70 + 135}} &lt;br/&gt;= 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|100 + 75}} &lt;br/&gt;= 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|60 + 120}} &lt;br/&gt;= 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob..Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|345 + 323}} &lt;br/&gt;= 668
| colspan=&quot;1&quot; | {{nowrap|560 + 114}} &lt;br/&gt;= 674
| colspan=&quot;1&quot; | {{nowrap|345 + 328}} &lt;br/&gt;= 673
| colspan=&quot;2&quot; | {{nowrap|345 + 324}} &lt;br/&gt;= 669

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| colspan=&quot;1&quot; | 13,551
| colspan=&quot;1&quot; | 13,200
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

=== Samaritan Adjustment Details ===
As noted previously, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;2&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| 13,551
| 13,200
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

== The Birth of Shem (A Digression) ==
Were Noah's sons born when Noah was 500 or 502?

==== The 502 Calculation ====
While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;\begin{aligned} \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } &amp; 70 \times 40 = 2800 \end{aligned}&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.
= End TBD =

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that could result in patriarchs surviving beyond the date of the Flood.
* &lt;span style=&quot;color:#333333;&quot;&gt;'''Blank Cells'''&lt;/span&gt; indicate where primary sources do not provide specific lifespan or death data.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== The Septuagint Chronology ==

=== The Correlations ===
An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo).

The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the &quot;Universal Flood&quot; problem.


[[Category:Religion]]</text>
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      <id>2806919</id>
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      <text bytes="104986" sha1="ksbrbw8lvjbnkghtqeyjupjnfrq02pg" xml:space="preserve">{{Original research}}
This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified.

== Summary of Main Arguments ==
The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include:

* '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or &quot;perfect&quot; numbers (such as 7, 49, and 60) rather than biological or historical reality.
* '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions.
* '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood.
* '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific &quot;years of the world&quot; (''Anno Mundi''), synchronizing human history with a divine calendar.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
[[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]]

Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE).

The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations.

* '''16 ''šūši'' (960 years)'''
** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''': Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** SKL: [[w:Atab|Atab]]
** '''Prototype 1''': Shem

* '''7 ''šūši'' (420 years)'''
** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''': Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list.

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word &quot;Adapa&quot; is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].

In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . .
*During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar.
. . .

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.

==== Mesopotamian Similarities ====
*[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions.
*[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.
*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;2&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (Armenian)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;2&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| 753 &lt;br/&gt; &lt;small&gt;(1454)&lt;br/&gt;(2207)&lt;/small&gt;
| 723 &lt;br/&gt; &lt;small&gt;(1454)&lt;br/&gt;(2177)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1642)&lt;br/&gt;(2592)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;2&quot; |&lt;small&gt;(2242)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;2&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (Armenian)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;2&quot; | 167
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth.

{{RoundBoxTop}}
====The &quot;Whoops Theory&quot;: A Digression====
Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor added 100 years to the age at which each patriarch begot his son, intending to &quot;fix&quot; the timeline, but somehow failed in the case of Methuselah. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.

==== Demetrius the Chronographer ====
Writing in the late 3rd century BC (c. 221 BC), '''Demetrius the Chronographer''' stands as the earliest known witness to biblical chronological calculations. Despite the fragmentary nature of his work, his data remains pivotal; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood, a total that inherently supports a longer chronology where Methuselah’s fathering age is '''187''' years rather than '''167'''.

In the original article's comments, a debate surfaced regarding this longer chronology and the '''187''' year fathering age for Methuselah. Paul D. defends his &quot;Whoops Theory&quot; by systematically challenging the validity of the early witnesses, particularly those that support the longer timeline:

* '''Josephus:''' Characterized as being dependent on the Masoretic tradition rather than an independent witness.
* '''Pseudo-Philo:''' Dismissed entirely due to severe textual corruption (described as &quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through a later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional two years whose precise placement remains unknown.
* '''Codex Alexandrinus:''' Identified as the lone legitimate witness to the 187-year fathering age.

The dismissal of Julius Africanus due to his survival through an intermediary, or the disqualification of Demetrius based on a two-year uncertainty, is arguably overstated. As shown in the comparative tables below, there is remarkably little variation in the &quot;LONG CHRONOLOGY&quot; begettal ages. They are identical for Adam through Enoch, with some variation in the last few patriarchs prior to the flood.

The below reconstructed Demetrius chronology employs a plausible explanation for the 2-year discrepancy: the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can account for such variances without necessitating the rejection of the witness. Given that Demetrius operates within the longest known chronological framework, the suggestion that he utilized longer begettal ages for earlier patriarchs while failing to apply the correction for Methuselah—the very figure requiring it most in such an expanded timeline—is statistically and logically improbable.

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in patriarchs surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | TOTAL
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}
{{RoundBoxBottom}}

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all textual traditions. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors

As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;2&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (Armenian)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|130 + 800}} &lt;br/&gt;= 930
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|230 + 700}} &lt;br/&gt;= 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|105 + 807}} &lt;br/&gt;= 912
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|205 + 707}} &lt;br/&gt;= 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|90 + 815}}  &lt;br/&gt;= 905
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|190 + 715}} &lt;br/&gt;= 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|70 + 840}}  &lt;br/&gt;= 910
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|170 + 740}} &lt;br/&gt;= 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|65 + 830}}  &lt;br/&gt;= 895
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|165 + 730}} &lt;br/&gt;= 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|62 + 900}} &lt;br/&gt;= 962
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|162 + 800}} &lt;br/&gt;= 962
| {{nowrap|62 + 785}} &lt;br/&gt;= 847
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|162 + 800}} &lt;br/&gt;= 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|65 + 300}} &lt;br/&gt;= 365
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|165 + 200}} &lt;br/&gt;= 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|67 + 902}} &lt;br/&gt;= 969
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|187 + 782}} &lt;br/&gt;= 969
| {{nowrap|67 + 653}} &lt;br/&gt;= 720
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|167 + 802}} &lt;br/&gt;= 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|53 + 730}} &lt;br/&gt;= 783
| {{nowrap|182 + 595}} &lt;br/&gt;= 777
| {{nowrap|53 + 600}} &lt;br/&gt;= 653
| {{nowrap|188 + 565}} &lt;br/&gt;= 753
| {{nowrap|188 + 535}} &lt;br/&gt;= 723

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|502 + 448}} &lt;br/&gt;= 950
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|500 + 450}} &lt;br/&gt;= 950
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|502 + 448}} &lt;br/&gt;= 950
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|500 + 450}} &lt;br/&gt;= 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;5&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|100 + 500}} &lt;br/&gt;= 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 438
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|35 + 403}} &lt;br/&gt;= 438
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|135 + 303}} &lt;br/&gt;= 438
| {{nowrap|135 + 400}} &lt;br/&gt;= 535
| {{nowrap|135 + 403}} &lt;br/&gt;= 538

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| 460
| —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 433
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|30 + 403}} &lt;br/&gt;= 433
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|130 + 303}} &lt;br/&gt;= 433
| {{nowrap|130 + 330}} &lt;br/&gt;= 460
| {{nowrap|130 + 406}} &lt;br/&gt;= 536

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 464
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|34 + 430}} &lt;br/&gt;= 464
| colspan=&quot;2&quot; | {{nowrap|134 + 270}} &lt;br/&gt;= 404
| {{nowrap|134 + 433}} &lt;br/&gt;= 567

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 239
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|30 + 209}} &lt;br/&gt;= 239
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|130 + 109}} &lt;br/&gt;= 239
| colspan=&quot;2&quot; | {{nowrap|130 + 209}} &lt;br/&gt;= 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 239
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|32 + 207}} &lt;br/&gt;= 239
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|132 + 107}} &lt;br/&gt;= 239
| {{nowrap|132 + 207}} &lt;br/&gt;= 339
| {{nowrap|135 + 207}} &lt;br/&gt;= 342

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 230
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|30 + 200}} &lt;br/&gt;= 230
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|130 + 100}} &lt;br/&gt;= 230
| colspan=&quot;2&quot; | {{nowrap|130 + 200}} &lt;br/&gt;= 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 148
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|29 + 119}} &lt;br/&gt;= 148
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|79 + 69}} &lt;br/&gt;= 148
| {{nowrap|179 + 125}} &lt;br/&gt;= 304
| {{nowrap|79 + 119}} &lt;br/&gt;= 198

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|70 + 135}} &lt;br/&gt;= 205
| {{nowrap|70 + 75}} &lt;br/&gt;= 145
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|70 + 135}} &lt;br/&gt;= 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|100 + 75}} &lt;br/&gt;= 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|60 + 120}} &lt;br/&gt;= 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob..Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|345 + 323}} &lt;br/&gt;= 668
| colspan=&quot;1&quot; | {{nowrap|560 + 114}} &lt;br/&gt;= 674
| colspan=&quot;1&quot; | {{nowrap|345 + 328}} &lt;br/&gt;= 673
| colspan=&quot;2&quot; | {{nowrap|345 + 324}} &lt;br/&gt;= 669

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| colspan=&quot;1&quot; | 13,551
| colspan=&quot;1&quot; | 13,200
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

=== Samaritan Adjustment Details ===
As noted previously, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;2&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| 13,551
| 13,200
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

== The Birth of Shem (A Digression) ==
Were Noah's sons born when Noah was 500 or 502?

==== The 502 Calculation ====
While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;\begin{aligned} \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } &amp; 70 \times 40 = 2800 \end{aligned}&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.
= End TBD =

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that could result in patriarchs surviving beyond the date of the Flood.
* &lt;span style=&quot;color:#333333;&quot;&gt;'''Blank Cells'''&lt;/span&gt; indicate where primary sources do not provide specific lifespan or death data.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== The Septuagint Chronology ==

=== The Correlations ===
An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo).

The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the &quot;Universal Flood&quot; problem.


[[Category:Religion]]</text>
      <sha1>ksbrbw8lvjbnkghtqeyjupjnfrq02pg</sha1>
    </revision>
    <revision>
      <id>2806921</id>
      <parentid>2806919</parentid>
      <timestamp>2026-04-28T22:33:30Z</timestamp>
      <contributor>
        <username>~2026-25504-03</username>
        <id>3068785</id>
      </contributor>
      <comment>/* Masoretic Adjustments */</comment>
      <origin>2806921</origin>
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      <text bytes="104992" sha1="qjrls7bgngaac7y5at41h391tm3ekd5" xml:space="preserve">{{Original research}}
This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified.

== Summary of Main Arguments ==
The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include:

* '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or &quot;perfect&quot; numbers (such as 7, 49, and 60) rather than biological or historical reality.
* '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions.
* '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood.
* '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific &quot;years of the world&quot; (''Anno Mundi''), synchronizing human history with a divine calendar.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
[[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]]

Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE).

The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations.

* '''16 ''šūši'' (960 years)'''
** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''': Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** SKL: [[w:Atab|Atab]]
** '''Prototype 1''': Shem

* '''7 ''šūši'' (420 years)'''
** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''': Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list.

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word &quot;Adapa&quot; is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].

In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . .
*During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar.
. . .

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.

==== Mesopotamian Similarities ====
*[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions.
*[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.
*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;2&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (Armenian)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;2&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| 753 &lt;br/&gt; &lt;small&gt;(1454)&lt;br/&gt;(2207)&lt;/small&gt;
| 723 &lt;br/&gt; &lt;small&gt;(1454)&lt;br/&gt;(2177)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1642)&lt;br/&gt;(2592)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;2&quot; |&lt;small&gt;(2242)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;2&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (Armenian)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;2&quot; | 167 / 187
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth.

{{RoundBoxTop}}
====The &quot;Whoops Theory&quot;: A Digression====
Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor added 100 years to the age at which each patriarch begot his son, intending to &quot;fix&quot; the timeline, but somehow failed in the case of Methuselah. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.

==== Demetrius the Chronographer ====
Writing in the late 3rd century BC (c. 221 BC), '''Demetrius the Chronographer''' stands as the earliest known witness to biblical chronological calculations. Despite the fragmentary nature of his work, his data remains pivotal; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood, a total that inherently supports a longer chronology where Methuselah’s fathering age is '''187''' years rather than '''167'''.

In the original article's comments, a debate surfaced regarding this longer chronology and the '''187''' year fathering age for Methuselah. Paul D. defends his &quot;Whoops Theory&quot; by systematically challenging the validity of the early witnesses, particularly those that support the longer timeline:

* '''Josephus:''' Characterized as being dependent on the Masoretic tradition rather than an independent witness.
* '''Pseudo-Philo:''' Dismissed entirely due to severe textual corruption (described as &quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through a later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional two years whose precise placement remains unknown.
* '''Codex Alexandrinus:''' Identified as the lone legitimate witness to the 187-year fathering age.

The dismissal of Julius Africanus due to his survival through an intermediary, or the disqualification of Demetrius based on a two-year uncertainty, is arguably overstated. As shown in the comparative tables below, there is remarkably little variation in the &quot;LONG CHRONOLOGY&quot; begettal ages. They are identical for Adam through Enoch, with some variation in the last few patriarchs prior to the flood.

The below reconstructed Demetrius chronology employs a plausible explanation for the 2-year discrepancy: the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can account for such variances without necessitating the rejection of the witness. Given that Demetrius operates within the longest known chronological framework, the suggestion that he utilized longer begettal ages for earlier patriarchs while failing to apply the correction for Methuselah—the very figure requiring it most in such an expanded timeline—is statistically and logically improbable.

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in patriarchs surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | TOTAL
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}
{{RoundBoxBottom}}

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all textual traditions. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors

As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;2&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (Armenian)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|130 + 800}} &lt;br/&gt;= 930
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|230 + 700}} &lt;br/&gt;= 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|105 + 807}} &lt;br/&gt;= 912
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|205 + 707}} &lt;br/&gt;= 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|90 + 815}}  &lt;br/&gt;= 905
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|190 + 715}} &lt;br/&gt;= 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|70 + 840}}  &lt;br/&gt;= 910
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|170 + 740}} &lt;br/&gt;= 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|65 + 830}}  &lt;br/&gt;= 895
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|165 + 730}} &lt;br/&gt;= 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|62 + 900}} &lt;br/&gt;= 962
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|162 + 800}} &lt;br/&gt;= 962
| {{nowrap|62 + 785}} &lt;br/&gt;= 847
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|162 + 800}} &lt;br/&gt;= 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|65 + 300}} &lt;br/&gt;= 365
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|165 + 200}} &lt;br/&gt;= 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|67 + 902}} &lt;br/&gt;= 969
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|187 + 782}} &lt;br/&gt;= 969
| {{nowrap|67 + 653}} &lt;br/&gt;= 720
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|167 + 802}} &lt;br/&gt;= 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|53 + 730}} &lt;br/&gt;= 783
| {{nowrap|182 + 595}} &lt;br/&gt;= 777
| {{nowrap|53 + 600}} &lt;br/&gt;= 653
| {{nowrap|188 + 565}} &lt;br/&gt;= 753
| {{nowrap|188 + 535}} &lt;br/&gt;= 723

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|502 + 448}} &lt;br/&gt;= 950
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|500 + 450}} &lt;br/&gt;= 950
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|502 + 448}} &lt;br/&gt;= 950
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|500 + 450}} &lt;br/&gt;= 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;5&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|100 + 500}} &lt;br/&gt;= 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 438
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|35 + 403}} &lt;br/&gt;= 438
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|135 + 303}} &lt;br/&gt;= 438
| {{nowrap|135 + 400}} &lt;br/&gt;= 535
| {{nowrap|135 + 403}} &lt;br/&gt;= 538

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| 460
| —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 433
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|30 + 403}} &lt;br/&gt;= 433
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|130 + 303}} &lt;br/&gt;= 433
| {{nowrap|130 + 330}} &lt;br/&gt;= 460
| {{nowrap|130 + 406}} &lt;br/&gt;= 536

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 464
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|34 + 430}} &lt;br/&gt;= 464
| colspan=&quot;2&quot; | {{nowrap|134 + 270}} &lt;br/&gt;= 404
| {{nowrap|134 + 433}} &lt;br/&gt;= 567

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 239
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|30 + 209}} &lt;br/&gt;= 239
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|130 + 109}} &lt;br/&gt;= 239
| colspan=&quot;2&quot; | {{nowrap|130 + 209}} &lt;br/&gt;= 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 239
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|32 + 207}} &lt;br/&gt;= 239
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|132 + 107}} &lt;br/&gt;= 239
| {{nowrap|132 + 207}} &lt;br/&gt;= 339
| {{nowrap|135 + 207}} &lt;br/&gt;= 342

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 230
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|30 + 200}} &lt;br/&gt;= 230
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|130 + 100}} &lt;br/&gt;= 230
| colspan=&quot;2&quot; | {{nowrap|130 + 200}} &lt;br/&gt;= 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 148
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|29 + 119}} &lt;br/&gt;= 148
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|79 + 69}} &lt;br/&gt;= 148
| {{nowrap|179 + 125}} &lt;br/&gt;= 304
| {{nowrap|79 + 119}} &lt;br/&gt;= 198

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|70 + 135}} &lt;br/&gt;= 205
| {{nowrap|70 + 75}} &lt;br/&gt;= 145
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|70 + 135}} &lt;br/&gt;= 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|100 + 75}} &lt;br/&gt;= 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|60 + 120}} &lt;br/&gt;= 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob..Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|345 + 323}} &lt;br/&gt;= 668
| colspan=&quot;1&quot; | {{nowrap|560 + 114}} &lt;br/&gt;= 674
| colspan=&quot;1&quot; | {{nowrap|345 + 328}} &lt;br/&gt;= 673
| colspan=&quot;2&quot; | {{nowrap|345 + 324}} &lt;br/&gt;= 669

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| colspan=&quot;1&quot; | 13,551
| colspan=&quot;1&quot; | 13,200
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

=== Samaritan Adjustment Details ===
As noted previously, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;2&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| 13,551
| 13,200
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

== The Birth of Shem (A Digression) ==
Were Noah's sons born when Noah was 500 or 502?

==== The 502 Calculation ====
While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;\begin{aligned} \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } &amp; 70 \times 40 = 2800 \end{aligned}&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.
= End TBD =

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that could result in patriarchs surviving beyond the date of the Flood.
* &lt;span style=&quot;color:#333333;&quot;&gt;'''Blank Cells'''&lt;/span&gt; indicate where primary sources do not provide specific lifespan or death data.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== The Septuagint Chronology ==

=== The Correlations ===
An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo).

The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the &quot;Universal Flood&quot; problem.


[[Category:Religion]]</text>
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    </revision>
    <revision>
      <id>2806922</id>
      <parentid>2806921</parentid>
      <timestamp>2026-04-28T22:38:56Z</timestamp>
      <contributor>
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        <id>2646631</id>
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      <comment>/* Demetrius the Chronographer */</comment>
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      <text bytes="104988" sha1="e0aikrfqmt09w0y44naodij6vqiialt" xml:space="preserve">{{Original research}}
This page extends the mathematical insights presented in the 2017 article, [https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ ''Some Curious Numerical Facts about the Ages of the Patriarchs''] by Paul D. While the original article offers compelling arguments, this analysis provides additional evidence and demonstrates that the underlying numerical data is even more robust and systematic than initially identified.

== Summary of Main Arguments ==
The ages of the patriarchs in Genesis are not historical records, but are a symbolic mathematical structure. Key points include:

* '''Artificial Mathematical Design:''' Patriarchal lifespans and event years are based on symbolic or &quot;perfect&quot; numbers (such as 7, 49, and 60) rather than biological or historical reality.
* '''The Universal Flood as a Later Insertion:''' Evidence suggests the universal scope of Noah's Flood was a later addition to a patriarchal foundation story. This insertion disrupted the original timelines, forcing recalibrations in the Masoretic Text (MT), Samaritan Pentateuch (SP), and Septuagint (LXX) to avoid chronological contradictions.
* '''Chronological Overlaps:''' In the original numerical framework (prior to recalibration for a universal flood), four patriarchs survived beyond the date of the Flood.
* '''Alignment with Sacred Cycles:''' The chronologies are designed to align significant events—such as the Exodus and the dedication of Solomon’s Temple—with specific &quot;years of the world&quot; (''Anno Mundi''), synchronizing human history with a divine calendar.

= ''Arichat Yamim'' (Long Life) =

Most of the patriarchs' lifespans in the Hebrew Bible exceed typical human demographics, and many appear to be based on rounded multiples of 101 years. For example, the combined lifespans of Seth, Enosh, and Kenan total '''2,727 years''' (27 × 101). Likewise, the sum for Mahalalel, Jared, and Enoch is '''2,222 years''' (22 × 101), and for Methuselah and Noah, it is '''1,919 years''' (19 × 101).

This phenomenon is difficult to explain, as no known ancient number system features &quot;101&quot; as a significant unit. However, a possible explanation emerges if we assume the original chronographer arrived at these figures through a two-stage process: an initial prototype relying on Mesopotamian sexagesimal numbers, followed by a refined prototype rounded to the nearest Jubilee cycle.
 
In his 1989 London Bible College thesis, ''The Genealogies of Genesis: A Study of Their Structure and Function'', Richard I. Johnson argues that the cumulative lifespans of the patriarchs from Adam to Moses derive from a &quot;perfect&quot; Mesopotamian value: seven ''šar'' (7 × 3,600) or 420 ''šūši'' (420 × 60), divided by two. Using the sexagesimal (base-60) system, the calculation is structured as follows:

*:&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
\frac{7\,\text{šar}}{2} &amp;= \frac{420\,\text{šūši}}{2} \\ &amp;= 210\,\,\text{šūši}  \\ 
&amp;= \left(210 \times 60 \,\text{years} \right) \\ 
&amp;= 12,600 \, \text{years}
\end{aligned}
&lt;/math&gt;

This 12,600-year total was partitioned into three allotments, each based on a 100-Jubilee cycle (4,900 years) but rounded to the nearest Mesopotamian ''šūši'' (multiples of 60).

==== Prototype 1: Initial &quot;Mesopotamian&quot; Allocation ====

----

&lt;div style=&quot;background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;&quot;&gt;
The initial &quot;PT1&quot; framework partitioned the 12,600-year total into three allotments based on 100-Jubilee cycles (rounded to the nearest ''šūši''):

* '''Group 1 (Seth to Enoch):''' Six patriarchs allotted a combined sum of '''82 ''šūši'' (4,920 years)'''. This approximates 100 Jubilees (82 × 60 ≈ 100 × 49).
* '''Group 2 (Adam, plus Shem to Moses):''' These 17 patriarchs were also allotted a combined sum of '''82 ''šūši'' (4,920 years)'''.
* '''Group 3 (The Remainder):''' Methuselah, Lamech, and Noah were allotted the remaining '''46 ''šūši'' (2,760 years)''' (12,600 − 4,920 − 4,920).
&lt;/div&gt;

----

==== Prototype 2: Refined &quot;Jubilee&quot; Allocation ====

----

&lt;div style=&quot;background-color: #fdf7ff; padding: 15px; border-left: 5px solid #9c27b0;&quot;&gt;
Because the rounded Mesopotamian sums in Prototype 1 were not exact Jubilee multiples, the framework was refined by shifting 29 years from the &quot;Remainder&quot; to each of the two primary groups. This resulted in the &quot;PT2&quot; figures as follows:

* '''Group 1 (Seth to Enoch):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 2 (Adam, plus Shem to Moses):''' Increased to '''4,949 years''' (101 × 49-year Jubilees).
* '''Group 3 (The Remainder):''' Decreased by 58 years to '''2,702 years''' (12,600 − 4,949 − 4,949).
&lt;/div&gt;

----

'''Table Legend:'''
* &lt;span style=&quot;width:100%; color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in a patriarch surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Prototype Chronologies (Age at death)

|-
! colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e0f2f1; border-bottom:2px solid #009688;&quot; | PROTOTYPE 1&lt;br/&gt;(PT1)
! colspan=&quot;3&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PROTOTYPE 2&lt;br/&gt;(PT2)

|-
| rowspan=&quot;6&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 1}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|45 šūši}}&lt;br/&gt;&lt;small&gt;(2700)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2727&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|37 šūši}}&lt;br/&gt;&lt;small&gt;(2220)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 15 &lt;small&gt;(900)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2222&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 6 &lt;small&gt;(360)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 365

|-
| rowspan=&quot;3&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 3}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot;  | Methuselah
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|46 šūši}}&lt;br/&gt;&lt;small&gt;(2760)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|32 šūši}}&lt;br/&gt;&lt;small&gt;(1920)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2702&lt;/div&gt;
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1919&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 14 &lt;small&gt;(840)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 783
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783

|-
| rowspan=&quot;18&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|Group 2}}&lt;/div&gt;
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|82 šūši}}&lt;br/&gt;&lt;small&gt;(4920)&lt;/small&gt;&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|40 šūši}}&lt;br/&gt;&lt;small&gt;(2400)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 16 &lt;small&gt;(960)&lt;/small&gt;
| rowspan=&quot;18&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;4949&lt;/div&gt;
| rowspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;2401&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 10 &lt;small&gt;(600)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 438

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 7 &lt;small&gt;(420)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 433

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|25 šūši}}&lt;br/&gt;&lt;small&gt;(1500)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 8 &lt;small&gt;(480)&lt;/small&gt;
| rowspan=&quot;6&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1525&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 464

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 4 &lt;small&gt;(240)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 239

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 230

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 148

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;{{nowrap|17 šūši}}&lt;br/&gt;&lt;small&gt;(1020)&lt;/small&gt;&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| rowspan=&quot;7&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | &lt;div style=&quot;display:inline-block; transform:rotate(270deg);&quot;&gt;1023&lt;/div&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 3 &lt;small&gt;(180)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 147

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Levi
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 137

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kohath
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 133

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Amram
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 131

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 2 &lt;small&gt;(120)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | 120

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;6&quot; | 210 šūši&lt;br/&gt;&lt;small&gt;(12,600 years)&lt;/small&gt;
|}

==Mesopotamian Derived Lifespans==
[[File:Diagram of the Supplementary Hypothesis.jpg|thumb|upright=0.8|Diagram of the [[w:supplementary_hypothesis|supplementary hypothesis]], a popular model of the [[w:composition_of_the_Torah|composition of the Torah]]. The Priestly source is shown as '''P'''.]]

Many scholars believe the biblical chronology was developed by an individual or school of scribes known as the [[w:Priestly_source|Priestly source]] (see diagram). Deprived of a physical Temple, the Judean elite focused on transforming oral traditions into a permanent, 'portable' written Law. To do so, scribes likely adopted the prestigious sexagesimal (base-60) mathematical system of their captors, codifying a history that would command respect within a Mesopotamian intellectual context. The presence of these mathematical structures provides strong evidence that these lifespans were integrated into the biblical narrative during or shortly after the Babylonian captivity (c. 586–538 BCE).

The following comparison illustrates how the '''Prototype 1''' chronology utilized timespans found in the [[w:Sumerian_King_List|Sumerian King List (SKL)]]. The longest lifespans in this chronology—960 and 900 years—are figures well-represented as Sumerian kingship durations.

* '''16 ''šūši'' (960 years)'''
** SKL: [[w:Kullassina-bel|Kullassina-bel]], [[w:Kalibum|Kalibum]]
** '''Prototype 1''': Adam, Jared, Methuselah, Noah

* '''15 ''šūši'' (900 years)'''
** SKL: [[w:Zuqaqip|Zuqaqip]], [[w:Melem-Kish|Melem-Kish]], [[w:Ilku|Ilku]], [[w:Enmebaragesi|Enmebaragesi]]
** '''Prototype 1''': Seth, Enosh, Kenan, Mahalalel

* '''10 ''šūši'' (600 years)'''
** SKL: [[w:Atab|Atab]]
** '''Prototype 1''': Shem

* '''7 ''šūši'' (420 years)'''
** SKL: [[w:En-tarah-ana|En-tarah-ana]], [[w:Enmerkar|Enmerkar]]
** '''Prototype 1''': Arpachshad, Shelah

The precise alignment of these four distinct groupings suggests that the Prototype 1 Chronology was not merely inspired by Mesopotamian traditions, but was mathematically calibrated to synchronize with them. Notably, in his work ''[[w:Antiquities_of_the_Jews|Antiquities of the Jews]]'', [[w:Josephus|Flavius Josephus]] characterizes several pre-flood (antediluvian) patriarchs as having explicit leadership or ruling roles, further mirroring the regal nature of the Sumerian list.

==The Grouping of Adam==
The placement of Adam in Group 2 for lifespan allotments is surprising given his role as the first human male in the Genesis narrative. Interestingly, Mesopotamian mythology faces a similar ambiguity regarding the figure Adapa. In [[w:Apkallu#Uanna_(Oannes)_or_Adapa?|some inscriptions (click here)]], the word &quot;Adapa&quot; is linked to the first sage and associated with the first pre-flood king, Ayalu (often identified as [[w:Alulim|Alulim]]). In [[w:Adapa#Other_myths|other myths (click here)]], Adapa is associated with the post-flood king, [[w:Enmerkar|Enmerkar]].

In the [[w:Apkallu#Uruk_List_of_Kings_and_Sages|&quot;Uruk List of Kings and Sages&quot;]] (165 BC), discovered in 1959/60 in the Seleucid-era temple of Anu in Bīt Rēš, the text documents a clear succession of divine and human wisdom. It consists of a list of seven antediluvian kings and their associated semi-divine sages (apkallū), followed by a note on the 'Deluge' (see [[w:Gilgamesh_flood_myth|Gilgamesh flood myth]]). After this break, the list continues with eight more king-sage pairs representing the post-flood era, where the &quot;sages&quot; eventually transition into human scholars.

A tentative translation reads:

*During the reign of [[w:Alulim|'''Ayalu''', the king, '''Adapa''' was sage]].
*During the reign of [[w:Alalngar|'''Alalgar''', the king, '''Uanduga''' was sage]].
*During the reign of '''Ameluana''', the king, '''Enmeduga''' was sage.
*During the reign of '''Amegalana''', the king, '''Enmegalama''' was sage.
*During the reign of '''Enmeusumgalana''', the king, '''Enmebuluga''' was sage.
*During the reign of '''Dumuzi''', the shepherd, the king, '''Anenlilda''' was sage.
*During the reign of '''Enmeduranki''', the king, '''Utuabzu''' was sage.
*After the flood, during the reign of '''Enmerkar''', the king, '''Nungalpirigal''' was sage . . .
*During the reign of '''Gilgamesh''', the king, '''Sin-leqi-unnini''' was scholar.
. . .

This list illustrates the traditional sequence of sages that parallels the biblical patriarchs, leading to several specific similarities in their roles and narratives.

==== Mesopotamian Similarities ====
*[[w:Adapa#As_Adam|Adam as Adapa]]: Possible parallels include the similarity in names (potentially sharing the same linguistic root) and thematic overlaps. Both accounts feature a trial involving the consumption of purportedly deadly food, and both figures are summoned before a deity to answer for their transgressions.
*[[w:En-men-dur-ana#Myth|Enoch as Enmeduranki]]: Enoch appears in the biblical chronology as the seventh pre-flood patriarch, while Enmeduranki is listed as the seventh pre-flood king in the Sumerian King List. The Hebrew [[w:Book of Enoch|Book of Enoch]] describes Enoch’s divine revelations and heavenly travels. Similarly, the Akkadian text ''Pirišti Šamê u Erṣeti'' (Secrets of Heaven and Earth) recounts Enmeduranki being taken to heaven by the gods Shamash and Adad to be taught the secrets of the cosmos.
*[[w:Utnapishtim|Noah as Utnapishtim]]: Similar to Noah, Utnapishtim is warned by a deity (Enki) of an impending flood and tasked with abandoning his possessions to build a massive vessel, the Preserver of Life. Both narratives emphasize the preservation of the protagonist's family, various animals, and seeds to repopulate the world. Utnapishtim is the son of [[w:Ubara-Tutu|Ubara-Tutu]], who in broader Mesopotamian tradition was understood to be the son of En-men-dur-ana, who traveled to heaven. Similarly, Noah is a descendant (the great-grandson) of Enoch, who was also taken to heaven.

==== Conclusion ====
The dual association of Adapa—as both the first antediluvian sage and a figure linked to the post-flood king Enmerkar—provides a compelling mythological parallel to the numerical &quot;surprise&quot; of Adam’s grouping. Just as Adapa bridges the divide between the primordial era and the post-flood world, Adam’s placement in Group 2 suggests a similar thematic ambiguity. This pattern is further reinforced by the figure of Enoch, whose role as the seventh patriarch mirrors Enmeduranki, the seventh king; both serve as pivotal links between humanity and the divine realm. Together, these overlaps imply that the biblical lifespan allotments were influenced by ancient conventions that viewed the progression of kingship and wisdom as a fluid, structured tradition rather than a strictly linear history.

==The Universal Flood==
In the '''Prototype 2''' chronology, four pre-flood patriarchs—[[w:Jared (biblical figure)|Jared]], [[w:Methuselah|Methuselah]], [[w:Lamech (Genesis)|Lamech]], and [[w:Noah|Noah]]—are attributed with exceptionally long lifespans, and late enough in the chronology that their lives overlap with the Deluge. This creates a significant anomaly where these figures survive the [[w:Genesis flood narrative|Universal Flood]], despite not being named among those saved on the Ark in the biblical narrative.

It is possible that the survival of these patriarchs was initially not a theological problem. For example, in the eleventh tablet of the ''[[w:Epic of Gilgamesh|Epic of Gilgamesh]]'', the hero [[w:Utnapishtim|Utnapishtim]] is instructed to preserve civilization by loading his vessel not only with kin, but with &quot;all the craftsmen.&quot; 

Given the evident Mesopotamian influences on early biblical narratives, it is possible that the original author of the biblical chronology may have operated within a similar conceptual framework—one in which Noah preserved certain forefathers alongside his immediate family, thereby bypassing the necessity of their death prior to the deluge. Alternatively, the author may have envisioned the flood as a localized event rather than a universal cataclysm, which would not have required the total destruction of human life outside the Ark.

Whatever the intentions of the original author, later chronographers were clearly concerned with the universality of the Flood. Consequently, chronological &quot;corrections&quot; were implemented to ensure the deaths of these patriarchs prior to the deluge. The lifespans of the problematic patriarchs are detailed in the table below. Each entry includes the total lifespan with the corresponding birth and death years (Anno Mundi, or years after creation) provided in parentheses.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Bible Chronologies: Lifespan (Birth year) (Death year)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;2&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (Armenian)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| 847 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1307)&lt;/small&gt;
| 962 &lt;br/&gt;&lt;small&gt;(460)&lt;br/&gt;(1422)&lt;/small&gt;
| colspan=&quot;2&quot; | 962 &lt;br/&gt;&lt;small&gt;(960)&lt;br/&gt;(1922)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1556)&lt;/small&gt;
| 720 &lt;br/&gt; &lt;small&gt;(587)&lt;br/&gt;(1307)&lt;/small&gt;
| 969 &lt;br/&gt; &lt;small&gt;(687)&lt;br/&gt;(1656)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 969 &lt;br/&gt;&lt;small&gt;(1287)&lt;br/&gt;(2256)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 783 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1437)&lt;/small&gt;
| 653 &lt;br/&gt; &lt;small&gt;(654)&lt;br/&gt;(1307)&lt;/small&gt;
| 777 &lt;br/&gt; &lt;small&gt;(874)&lt;br/&gt;(1651)&lt;/small&gt;
| 753 &lt;br/&gt; &lt;small&gt;(1454)&lt;br/&gt;(2207)&lt;/small&gt;
| 723 &lt;br/&gt; &lt;small&gt;(1454)&lt;br/&gt;(2177)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(707)&lt;br/&gt;(1657)&lt;/small&gt;
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1056)&lt;br/&gt;(2006)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 950 &lt;br/&gt; &lt;small&gt;(1642)&lt;br/&gt;(2592)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | The Flood
| colspan=&quot;2&quot; | &lt;small&gt;(1307)&lt;/small&gt;
| &lt;small&gt;(1656)&lt;/small&gt;
| colspan=&quot;2&quot; |&lt;small&gt;(2242)&lt;/small&gt;
|}

=== Samaritan Adjustments ===

As shown in the table above, the '''Samaritan Pentateuch''' (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech while leaving their birth years unchanged (460 AM, 587 AM, and 654 AM respectively). This adjustment ensures that all three patriarchs die precisely in the year of the Flood (1307 AM), leaving Noah as the sole survivor. 

While this mathematically resolves the overlap, the solution is less than ideal from a theological perspective; it suggests that these presumably righteous forefathers were swept away in the same judgment as the wicked generation, perishing in the same year as the Deluge.

=== Masoretic Adjustments ===

The '''Masoretic Text''' (MT) maintains the original lifespan and birth year for Jared, but implements specific shifts for his successors. It moves Methuselah's birth and death years forward by exactly '''one hundred years'''; he is born in year 687 AM (rather than 587 AM) and dies in year 1656 AM (rather than 1556 AM). 

Lamech's birth year is moved forward by '''two hundred and twenty years''', and his lifespan is reduced by six years, resulting in a birth in year 874 AM (as opposed to 654 AM) and a death in year 1651 AM (as opposed to 1437 AM). Finally, Noah's birth year and the year of the flood are moved forward by '''three hundred and forty-nine years''', while his original lifespan remains unchanged.

These adjustments shift the timeline of the Flood forward sufficiently so that Methuselah's death occurs in the year of the Flood and Lamech's death occurs five years prior, effectively resolving the overlap. However, this solution is less than ideal because it creates significant irregularities in the ages of the fathers at the birth of their successors (see table below). In particular, Jared, Methuselah, and Lamech are respectively '''162''', '''187''', and '''182''' years old at the births of their successors—ages that are notably higher than the preceding patriarchs Adam, Seth, Enosh, Kenan, Mahalalel, and Enoch, who are respectively '''130''', '''105''', '''90''', '''70''', '''65''', and '''65''' in the Masoretic Text.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;2&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (Armenian)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;3&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; | 67
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;2&quot; | 167 / 187
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; | 53
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 188

|}

=== Septuagint Adjustments ===
In his article ''[https://isthatinthebible.wordpress.com/2017/08/24/some-curious-numerical-facts-about-the-ages-of-the-patriarchs/ Some Curious Numerical Facts about the Ages of the Patriarchs]'', the author Paul D makes the following statement regarding the Septuagint (LXX):

&lt;blockquote&gt;“The LXX’s editor methodically added 100 years to the age at which each patriarch begat his son. Adam begat Seth at age 230 instead of 130, and so on. This had the result of postponing the date of the Flood by 900 years without affecting the patriarchs’ lifespans, which he possibly felt were too important to alter. Remarkably, however, the editor failed to account for Methuselah’s exceptional longevity, so old Methuselah still ends up dying 14 years after the Flood in the LXX. (Whoops!)”&lt;/blockquote&gt;

The Septuagint solution avoids the Samaritan issue where multiple righteous forefathers were swept away in the same year as the wicked. It also avoids the Masoretic issue of having disparate fathering ages. 

However, the Septuagint solution of adding hundreds of years to the chronology subverts mathematical motifs upon which the chronology was originally built. Abraham fathering Isaac at the age of 100 is presented as a miraculous event within the post-flood Abraham narrative; yet, having a long line of ancestors who begat sons when well over a hundred and fifty significantly dilutes the miraculous nature of Isaac's birth.

{{RoundBoxTop}}
====The &quot;Whoops Theory&quot;: A Digression====
Paul D.’s &quot;Whoops Theory&quot; suggests the LXX editor added 100 years to the age at which each patriarch begot his son, intending to &quot;fix&quot; the timeline, but somehow failed in the case of Methuselah. Fortunately, in addition to the biblical text traditions themselves, the writings of early chronographers provide insight into how these histories were developed. The LXX was the favored source for most Christian scholars during the early church period. Consider the following statement by Eusebius in his ''Chronicon'': 

&lt;blockquote&gt;&quot;Methusaleh fathered Lamech when he was 167 years of age. He lived an additional 802 years. Thus he would have survived the flood by 22 years.&quot;&lt;/blockquote&gt; 

This statement illustrates that Eusebius, as early as 325 AD, was aware of these chronological tensions. If he recognized the discrepancy, it is highly probable that earlier chronographers would also have been conscious of the overlap, suggesting it was not part of the earliest traditions but was a later development.

==== Demetrius the Chronographer ====
Writing in the late 3rd century BC (c. 221 BC), '''Demetrius the Chronographer''' stands as the earliest known witness to biblical chronological calculations. Despite the fragmentary nature of his work, his data remains pivotal; Demetrius explicitly calculated 2,264 years between the creation of Adam and the Flood, a total that inherently supports a longer chronology where Methuselah’s fathering age is '''187''' years rather than '''167'''.

In the original article's comments, a debate surfaced regarding this longer chronology and the '''187''' year fathering age for Methuselah. Paul D. defends his &quot;Whoops Theory&quot; by systematically challenging the validity of early witnesses, particularly those that support the longer timeline:

* '''Josephus:''' Characterized as being dependent on the Masoretic tradition rather than an independent witness.
* '''Pseudo-Philo:''' Dismissed entirely due to severe textual corruption (described as &quot;a real mess&quot;).
* '''Julius Africanus:''' Questioned because his records survive only through a later intermediary, Syncellus.
* '''Demetrius:''' Rejected as a witness because his chronology contains an additional two years whose precise placement remains unknown.
* '''Codex Alexandrinus:''' Identified as the lone legitimate witness to the 187-year fathering age.

The dismissal of Julius Africanus due to his survival through an intermediary, or the disqualification of Demetrius based on a two-year uncertainty, is arguably overstated. As shown in the comparative tables below, there is remarkably little variation in the &quot;LONG CHRONOLOGY&quot; begettal ages. They are identical for Adam through Enoch, with some variation in the last few patriarchs prior to the flood.

The below reconstructed Demetrius chronology employs a plausible explanation for the 2-year discrepancy: the ambiguity surrounding the precise timing of the Flood in relation to the births of Shem and Arphaxad. As explored later in this resource, chronographers frequently differ on whether Arphaxad was born two years after the Flood (Gen 11:10) or in the same year—a nuance that can account for such variances without necessitating the rejection of the witness. Given that Demetrius operates within the longest known chronological framework, the suggestion that he utilized longer begettal ages for earlier patriarchs while failing to apply the correction for Methuselah—the very figure requiring it most in such an expanded timeline—is statistically and logically improbable.

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that result in patriarchs surviving beyond the date of the Flood.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | TOTAL
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}
{{RoundBoxBottom}}

=== Flood Adjustment Summary ===

In summary, there was no ideal methodology for accommodating a universal flood within the various textual traditions. 

* In the '''Prototype 2''' chronology, multiple ancestors survive the flood alongside Noah. This dilutes Noah's status as the sole surviving patriarch, which in turn weakens the legitimacy of the [[w:Covenant_(biblical)#Noahic|Noahic covenant]]—a covenant predicated on the premise that God had destroyed all humanity in a universal reset, making Noah a fresh start in God's relationship with humanity.

* The '''Samaritan''' solution was less than ideal because Noah's presumably righteous ancestors perish in the same year as the wicked, which appears to undermine the discernment of God's judgments. 

* The '''Masoretic''' and '''Septuagint''' solutions, by adding hundreds of years to begettal ages, normalize what is intended to be the miraculous birth of Isaac when Abraham was an hundred years old.

== Additional Textual Evidence ==
Because no single surviving manuscript preserves the original PT2 in its entirety, it must be reconstructed using internal textual evidence. As described previously, a primary anchor for this reconstruction is the '''4,949-year sum''' for the Seth-to-Enoch group (Group 1), which is preserved across nearly all textual traditions. Also, where traditions diverge from this sum, they do so in patterns that preserve underlying symmetries and reveal the editorial intent of later redactors

As shown in the following tables (While most values are obtained directly from the primary source texts listed in the header, the '''Armenian Eusebius''' chronology does not explicitly record lifespans for Levi, Kohath, and Amram. These specific values are assumed to be shared across other known ''Long Chronology'' traditions.)

=== Lifespan Adjustments by Individual Patriarch ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;
|+ Comparison of Chronological Traditions (Individual Patriarch Lifespans)
|-
! rowspan = &quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | Patriarch
! colspan=&quot;1&quot; rowspan = &quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;2&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (Armenian)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|130 + 800}} &lt;br/&gt;= 930
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|230 + 700}} &lt;br/&gt;= 930

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|105 + 807}} &lt;br/&gt;= 912
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|205 + 707}} &lt;br/&gt;= 912

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|90 + 815}}  &lt;br/&gt;= 905
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|190 + 715}} &lt;br/&gt;= 905

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|70 + 840}}  &lt;br/&gt;= 910
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|170 + 740}} &lt;br/&gt;= 910

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|65 + 830}}  &lt;br/&gt;= 895
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|165 + 730}} &lt;br/&gt;= 895

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|62 + 900}} &lt;br/&gt;= 962
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|162 + 800}} &lt;br/&gt;= 962
| {{nowrap|62 + 785}} &lt;br/&gt;= 847
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|162 + 800}} &lt;br/&gt;= 962

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|65 + 300}} &lt;br/&gt;= 365
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|165 + 200}} &lt;br/&gt;= 365

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|67 + 902}} &lt;br/&gt;= 969
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|187 + 782}} &lt;br/&gt;= 969
| {{nowrap|67 + 653}} &lt;br/&gt;= 720
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|167 + 802}} &lt;br/&gt;= 969

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|53 + 730}} &lt;br/&gt;= 783
| {{nowrap|182 + 595}} &lt;br/&gt;= 777
| {{nowrap|53 + 600}} &lt;br/&gt;= 653
| {{nowrap|188 + 565}} &lt;br/&gt;= 753
| {{nowrap|188 + 535}} &lt;br/&gt;= 723

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Noah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|502 + 448}} &lt;br/&gt;= 950
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|500 + 450}} &lt;br/&gt;= 950
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|502 + 448}} &lt;br/&gt;= 950
| colspan=&quot;2&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|500 + 450}} &lt;br/&gt;= 950

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shem
| colspan=&quot;5&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | {{nowrap|100 + 500}} &lt;br/&gt;= 600

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Arpachshad
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 438
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|35 + 403}} &lt;br/&gt;= 438
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|135 + 303}} &lt;br/&gt;= 438
| {{nowrap|135 + 400}} &lt;br/&gt;= 535
| {{nowrap|135 + 403}} &lt;br/&gt;= 538

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Cainan (II)
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | —
| 460
| —

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 433
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|30 + 403}} &lt;br/&gt;= 433
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|130 + 303}} &lt;br/&gt;= 433
| {{nowrap|130 + 330}} &lt;br/&gt;= 460
| {{nowrap|130 + 406}} &lt;br/&gt;= 536

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 464
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|34 + 430}} &lt;br/&gt;= 464
| colspan=&quot;2&quot; | {{nowrap|134 + 270}} &lt;br/&gt;= 404
| {{nowrap|134 + 433}} &lt;br/&gt;= 567

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 239
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|30 + 209}} &lt;br/&gt;= 239
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|130 + 109}} &lt;br/&gt;= 239
| colspan=&quot;2&quot; | {{nowrap|130 + 209}} &lt;br/&gt;= 339

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 239
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|32 + 207}} &lt;br/&gt;= 239
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|132 + 107}} &lt;br/&gt;= 239
| {{nowrap|132 + 207}} &lt;br/&gt;= 339
| {{nowrap|135 + 207}} &lt;br/&gt;= 342

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Serug
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 230
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|30 + 200}} &lt;br/&gt;= 230
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|130 + 100}} &lt;br/&gt;= 230
| colspan=&quot;2&quot; | {{nowrap|130 + 200}} &lt;br/&gt;= 330

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|? + ?}} &lt;br/&gt;= 148
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|29 + 119}} &lt;br/&gt;= 148
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|79 + 69}} &lt;br/&gt;= 148
| {{nowrap|179 + 125}} &lt;br/&gt;= 304
| {{nowrap|79 + 119}} &lt;br/&gt;= 198

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|70 + 135}} &lt;br/&gt;= 205
| {{nowrap|70 + 75}} &lt;br/&gt;= 145
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|70 + 135}} &lt;br/&gt;= 205

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Abraham
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|100 + 75}} &lt;br/&gt;= 175

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Isaac
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|60 + 120}} &lt;br/&gt;= 180

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Jacob..Moses
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8; font-weight:bold; color:#555;&quot; | {{nowrap|345 + 323}} &lt;br/&gt;= 668
| colspan=&quot;1&quot; | {{nowrap|560 + 114}} &lt;br/&gt;= 674
| colspan=&quot;1&quot; | {{nowrap|345 + 328}} &lt;br/&gt;= 673
| colspan=&quot;2&quot; | {{nowrap|345 + 324}} &lt;br/&gt;= 669

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! style=&quot;text-align:left; color:black;&quot; | LIFESPAN&lt;br/&gt;DURATION&lt;br/&gt;SUM
| colspan=&quot;2&quot; | 12,600
| colspan=&quot;1&quot; | 11,991
| colspan=&quot;1&quot; | 13,551
| colspan=&quot;1&quot; | 13,200
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

=== Samaritan Adjustment Details ===
As noted previously, the Samaritan Pentateuch (SP) systematically reduces the total lifespans of Jared, Methuselah, and Lamech so that all three die precisely in the year of the Flood, leaving Noah as the sole survivor. The required reduction in Jared's lifespan was '''115 years'''. Interestingly, the Samaritan tradition also reduces the lifespans of later patriarchs by a combined total of 115 years, seemingly to maintain a numerical balance between the &quot;Group 1&quot; and &quot;Group 2&quot; patriarchs.

Specifically, this balance was achieved through the following adjustments:
* '''Eber''' and '''Terah''' each had their lifespans reduced by 60 years (one ''šūši'' each).
* '''Amram's''' lifespan was increased by five years.

This net adjustment of 115 years (60 + 60 - 5) suggests a deliberate schematic balancing.

=== Masoretic Adjustment Details ===

In the 2017 article, &quot;[https://wordpress.com Some Curious Numerical Facts about the Ages of the Patriarchs],&quot; Paul D. describes a specific shift in Lamech's death age in the Masoretic tradition:

&lt;blockquote&gt;&quot;The original age of Lamech was 753, and a late editor of the MT changed it to the schematic 777 (inspired by Gen 4:24, it seems, even though that is supposed to be a different Lamech: If Cain is avenged sevenfold, truly Lamech seventy-sevenfold). (Hendel 2012: 8; Northcote 251)&quot;&lt;/blockquote&gt;

While Paul D. accepts 753 as the original age, this conclusion creates significant tension within his own numerical analysis. A central pillar of his article is the discovery that the sum of all patriarchal ages from Adam to Moses totals exactly '''12,600 years'''—a result that relies specifically on Lamech living 777 years. To dismiss 777 as a late &quot;tweak&quot; in favor of 753 potentially overlooks the intentional mathematical architecture that defines the Masoretic tradition. As Paul D. acknowledges:

&lt;blockquote&gt;&quot;Alas, it appears that the lifespan of Lamech was changed from 753 to 777. Additionally, the age of Eber was apparently changed from 404 (as it is in the LXX) to 464... Presumably, these tweaks were made after the MT diverged from other versions of the text, in order to obtain the magic number 12,600 described above.&quot;&lt;/blockquote&gt;

==== ''Lectio Difficilior Potior'' ====
The principle of ''[[Wikipedia:Lectio difficilior potior|Lectio Difficilior Potior]]'' (the &quot;harder reading is stronger&quot;) suggests that scribes tend to simplify or &quot;smooth&quot; texts by introducing patterns. Therefore, when reconstructing an earlier tradition, the critic should often favor the reading with the least amount of artificial internal structure. This concept is particularly useful in evaluating major events in Noah's life.

In the Samaritan Pentateuch (SP) tradition, Noah is born in [https://www.stepbible.org/?q=reference=Gen.5:28,31%7Cversion=SPE Lamech’s 53rd year and Lamech dies when he is 653]. In the Septuagint tradition Lamech dies [https://www.stepbible.org/?q=reference=Gen.5:31%7Cversion=AB when he is 753, exactly one hundred years later than the Samaritan tradition]. If we combine that with the 500-year figure for Noah's age at the birth of his sons, a suspiciously neat pattern emerges:

* '''Year 500 (of Noah):''' Shem is born.
* '''Year 600 (of Noah):''' The Flood occurs.
* '''Year 700 (of Noah):''' Lamech dies.

This creates a perfectly intervalic 200-year span (500–700) between the birth of the heir and the death of the father. Such a &quot;compressed chronology&quot; (500–600–700) is a hallmark of editorial smoothing. Applying ''Lectio Difficilior'', one might conclude that these specific figures (53, 653, and 753) are secondary schematic developments rather than original data. In the reconstructed prototype chronology (PT2), it is proposed that Lamech's original lifespan was '''783 years'''—a value not preserved in any surviving tradition. Under this theory, Lamech's lifespan was reduced by six years in the Masoretic tradition to reach the '''777''' figure described previously, while Amram's was increased by six years in a deliberate &quot;balancing&quot; of total chronological years.

=== Armenian Eusebius Adjustments ===
Perhaps the most surprising adjustments of all are those found in the Armenian recension of Eusebius's Long Chronology. Eusebius's original work is dated to 325 AD, and the Armenian recension is presumed to have diverged from the Greek text approximately a hundred years later. It is not anticipated that the Armenian recension would retain Persian-era mathematical motifs; however, when the lifespan durations for all of the patriarchs are added up, the resulting figure is 13,200 years, which is exactly 600 years (or 10 ''šūši'') more than the Masoretic Text. Also, the specific adjustments to lifespans between the Prototype 2 (PT2) chronology and the Armenian recension of Eusebius's Long Chronology appear to be formulated using the Persian 60-based system.

Specifically, the following adjustments appear to have occurred for Group 2 patriarchs:
* '''Arpachshad''', '''Peleg''', and '''Serug''' each had their lifespans increased by 100 years.
* '''Shelah''', '''Eber''', and '''Reu''' each had their lifespans increased by 103 years.
* '''Nahor''' had his lifespan increased by 50 years.
* '''Amram''' had his lifespan increased by 1 year.

=== Lifespan Adjustments by Group ===

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center;&quot;

|+ Comparison of Chronological Traditions (Patriarch Group Lifespan Duration Sum)
|-
! rowspan=&quot;2&quot; | Patriarch Groups
! rowspan=&quot;2&quot; style=&quot;background-color:#f3e5f5; border-bottom:2px solid #9c27b0;&quot; | PT2
! colspan=&quot;2&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;2&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! style=&quot;background-color:#e3f2fd;&quot; | Masoretic&lt;br/&gt;(MT)
! style=&quot;background-color:#e3f2fd;&quot; | Samaritan&lt;br/&gt;(SP)
! style=&quot;background-color:#fff3e0;&quot; | Septuagint&lt;br/&gt;(LXX)
! style=&quot;background-color:#fff3e0;&quot; | Eusebius&lt;br/&gt;(325 AD)

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 1: Seth to Enoch&lt;br/&gt;&lt;small&gt;(6 Patriarchs)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 3: Methuselah, Lamech, Noah&lt;br/&gt;&lt;small&gt;(The Remainder)&lt;/small&gt;
| style=&quot;font-weight:bold; background-color:#f9f9f9;&quot; | 2702
| style=&quot;background-color:#f9f9f9;&quot; | 2696&lt;br/&gt;&lt;small&gt;(2702 - 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2323&lt;br/&gt;&lt;small&gt;(2702 - 379)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2672&lt;br/&gt;&lt;small&gt;(2702 - 30)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 2642&lt;br/&gt;&lt;small&gt;(2702 - 60)&lt;/small&gt;

|-
| style=&quot;font-weight:bold; text-align:left; background-color:#f9f9f9;&quot; | Group 2: Adam &amp; Shem to Moses&lt;br/&gt;&lt;small&gt;(The &quot;Second Half&quot;)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9; font-weight:bold;&quot; | 4949
| style=&quot;background-color:#f9f9f9;&quot; | 4955&lt;br/&gt;&lt;small&gt;(4949 + 6)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 4834&lt;br/&gt;&lt;small&gt;(4949 - 115)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5930&lt;br/&gt;&lt;small&gt;(4949 + 981)&lt;/small&gt;
| style=&quot;background-color:#f9f9f9;&quot; | 5609&lt;br/&gt;&lt;small&gt;(4949 + 660)&lt;/small&gt;

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:14px;&quot;
! LIFESPAN DURATION SUM
| colspan=&quot;2&quot; | 12,600
| 11,991
| 13,551
| 13,200
|}
&lt;small&gt;* '''Dash (—)''' indicates where primary sources do not provide complete death data.&lt;/small&gt;

* '''The Masoretic Text (MT):''' This tradition shifted 6 years from the &quot;Remainder&quot; to Group 2. This move broke the original symmetry but preserved the '''4,949-year sum''' for the Group 1 block.
* '''The Samaritan Pentateuch (SP):''' This tradition reduced both Group 1 and Group 2 by exactly 115 years each. While this maintained the underlying symmetry between the two primary blocks, the 101-Jubilee connection was lost.
* '''The Septuagint (LXX):''' This tradition adds 981 years to Group 2 while subtracting 30 years from the Remainder. This breaks the symmetry of the primary blocks and subverts any obvious connection to sexagesimal (base-60) influence.
* '''The Armenian Eusebius Chronology:''' This tradition reduced the Remainder by 60 years while increasing Group 2 by 660 years. This resulted in a net increase of exactly 600 years, or '''10 ''šūši''''' (base-60 units).

The use of rounded Mesopotamian figures in the '''Armenian Eusebius Chronology''' suggests it likely emerged prior to the Hellenistic conquest of Persia. Conversely, the '''Septuagint's''' divergence indicates a later development—likely in [[w:Alexandria|Alexandria]]—where Hellenized Jews were more focused on correlating Hebrew history with Greek and Egyptian chronologies than on maintaining Persian-era mathematical motifs.

The sum total of the above adjustments amounts to 660 years, or 11 ''šūši''. When combined with the 60-year reduction in Lamech's life (from 783 years to 723 years), the combined final adjustment is 10 ''šūši''.

= It All Started With Grain =

[[File:Centres_of_origin_and_spread_of_agriculture_labelled.svg|thumb|500px|Centres of origin of agriculture in the Neolithic revolution]]

The chronology found in the ''Book of Jubilees'' has deep roots in the Neolithic Revolution, stretching back roughly 14,400 years to the [https://www.biblicalarchaeology.org/daily/news/ancient-bread-jordan/ Black Desert of Jordan]. There, Natufian hunter-gatherers first produced flatbread by grinding wild cereals and tubers into flour, mixing them with water, and baking the dough on hot stones. This original flour contained a mix of wild wheat, wild barley, and tubers like club-rush (''Bolboschoenus glaucus''). Over millennia, these wild plants transformed into domesticated crops.

The first grains to be domesticated in the Fertile Crescent, appearing around 10,000–12,000 years ago, were emmer wheat (''Triticum dicoccum''), einkorn wheat (''Triticum monococcum''), and hulled barley (''Hordeum vulgare''). Early farmers discovered that barley was essential for its early harvest, while wheat was superior for making bread. The relative qualities of these two grains became a focus of early biblical religion, as recorded in [https://www.stepbible.org/?q=reference=Lev.23:10-21 Leviticus 23:10-21], where the people were commanded to bring the &quot;firstfruits of your harvest&quot; (referring to barley) before the Lord:

&lt;blockquote&gt;&quot;then ye shall bring a sheaf of the firstfruits of your harvest unto the priest: And he shall wave the sheaf before the Lord&quot;&lt;/blockquote&gt;

To early farmers, for whom hunger was a constant reality and winter survival uncertain, that first barley harvest was a profound sign of divine deliverance from the hardships of the season. The commandment in Leviticus 23 continues:

&lt;blockquote&gt;&quot;And ye shall count unto you from the morrow after the sabbath, from the day that ye brought the sheaf of the wave offering; seven sabbaths shall be complete: Even unto the morrow after the seventh sabbath shall ye number fifty days; and ye shall offer a new meat offering unto the Lord. Ye shall bring out of your habitations two wave loaves of two tenth deals; they shall be of fine flour; they shall be baken with leaven; they are the firstfruits unto the Lord.&quot;&lt;/blockquote&gt;

[[File:Ghandum_ki_katai_-punjab.jpg|thumb|500px|[https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.]]

These seven sabbaths amount to forty-nine days. The number 49 is significant because wheat typically reaches harvest roughly 49 days after barley. This grain carried a different symbolism: while barley represented survival and deliverance from winter, wheat represented the &quot;better things&quot; and the abundance provided to the faithful. [https://www.stepbible.org/?q=reference=Deu.16:9-10 Deuteronomy 16:9-10] similarly commands the people to count seven weeks from the time the sickle is first put to the standing grain, celebrating the feast on the fiftieth day.

This 49-day interval between the barley and wheat harvests was so integral to ancient worship that it informed the timeline of the Exodus. Among the plagues of Egypt, [https://www.stepbible.org/?q=reference=Exo.9:31-32 Exodus 9:31-32] describes the destruction of crops: 

&lt;blockquote&gt;&quot;And the flax and the barley was smitten: for the barley was in the ear, and the flax was bolled. But the wheat and the rye &lt;small&gt;(likely emmer wheat or spelt)&lt;/small&gt; were not smitten: for they were not grown up.&quot;&lt;/blockquote&gt;

This text establishes that the Exodus—God's deliverance from slavery—began during the barley harvest. Just as the barley harvest signaled the end of winter’s hardship, it symbolized Israel's release from bondage. 

The Israelites left Egypt on the 15th of Nisan (the first month) and arrived at the Wilderness of Sinai on the 1st of Sivan (the third month), 45 days later. In Jewish tradition, the giving of the Ten Commandments is identified with the 6th or 7th of Sivan—exactly 50 days after the Exodus. Thus, the Exodus (deliverance) corresponds to the barley harvest and is celebrated as the [[wikipedia:Passover|Passover]] holiday, while the Law (the life of God’s subjects) corresponds to the wheat harvest and is celebrated as [[wikipedia:Shavuot|Shavuot]]. This pattern carries into Christianity: Jesus was crucified during Passover (barley harvest), celebrated as [[wikipedia:Easter|Easter]], and fifty days later, the Holy Spirit was sent at [[wikipedia:Pentecost|Pentecost]] (wheat harvest).

=== The Mathematical Structure of Jubilees ===

The chronology of the ''Book of Jubilees'' is built upon this base-7 agricultural cycle, expanded into a fractal system of &quot;weeks&quot;:

* '''Week of Years:''' 7&lt;sup&gt;1&lt;/sup&gt; = 7 years
* '''Jubilee of Years:''' 7&lt;sup&gt;2&lt;/sup&gt; = 49 years
* '''Week of Jubilees:''' 7&lt;sup&gt;3&lt;/sup&gt; = 343 years
* '''Jubilee of Jubilees:''' 7&lt;sup&gt;4&lt;/sup&gt; = 2,401 years

The author of the ''Jubilees'' chronology envisions the entirety of early Hebraic history, from the creation of Adam to the entry into Canaan, as occurring within a Jubilee of Jubilees, concluding with a fiftieth Jubilee of years. In this framework, the 2,450-year span (2,401 + 49 = 2,450) serves as a grand-scale reflection of the agricultural transition from the barley of deliverance to the wheat of the Promised Land.

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs.png|thumb|center|500px|Early Hebraic history as envisioned by the author of the ''Jubilees'' chronology]]

The above diagram illustrates the reconstructed Jubilee of Jubilees fractal chronology. The first twenty rows in the left column respectively list 20 individual patriarchs, with parentheses indicating their age at the birth of their successor. Shem, the 11th patriarch and son of Noah, is born in reconstructed year 1209, which is roughly halfway through the 2,401-year structure. Abram is listed in the 21st position with a 77 in parentheses, indicating that Abram entered Canaan when he was 77 years old. The final three rows represent the Canaan, Egypt, and 40-year Sinai eras. Chronological time flows from the upper left to the lower right, utilizing 7x7 grids to represent 49-year Jubilees within a larger, nested &quot;Jubilee of Jubilees&quot; (49x49). Note that the two black squares at the start of the Sinai era mark the two-year interval between the Exodus and the completion of the Tabernacle.

* The '''first Jubilee''' (top-left 7x7 grid) covers the era from Adam's creation through his 49th year.
* The '''second Jubilee''' (the adjacent 7x7 grid to the right) spans Adam's 50th through 98th years.
* The '''third Jubilee''' marks the birth of Seth in the year 130, indicated by a color transition within the grid.
* The '''twenty-fifth Jubilee''' occupies the center of the 49x49 structure; it depicts Shem's birth and the chronological transition from pre-flood to post-flood patriarchs.

== The Birth of Shem (A Digression) ==
Were Noah's sons born when Noah was 500 or 502?

==== The 502 Calculation ====
While [https://www.stepbible.org/?q=reference=Gen.5:32 Genesis 5:32] states that &quot;Noah was 500 years old, and Noah begat Shem, Ham, and Japheth,&quot; this likely indicates the year Noah ''began'' having children rather than the year all three were born. Shem’s specific age can be deduced by comparing other verses: 
# Noah was 600 years old when the floodwaters came ([https://www.stepbible.org/?q=reference=Gen.7:6 Genesis 7:6]). 
# Shem was 100 years old when he fathered Arpachshad, two years after the flood ([https://www.stepbible.org/?q=reference=Gen.11:10 Genesis 11:10])

'''The Calculation:''' If Shem was 100 years old two years after the flood, he was 98 when the flood began. Subtracting 98 from Noah’s 600th year (600 - 98) results in '''502'''. This indicates that either Japheth or Ham was the eldest son, born when Noah was 500, followed by Shem two years later. Shem is likely listed first in the biblical text due to his status as the ancestor of the Semitic peoples.

== The Mathematical relationship between 40 and 49 ==

As noted previously, the ''Jubilees'' author envisions early Hebraic history within a &quot;Jubilee of Jubilees&quot; fractal chronology (2,401 years). Shem is born in year 1209, which is a nine-year offset from the exact mathematical center of 1200. To understand this shift, one must look at a mathematical relationship that exists between the foundational numbers 40 and 49. Specifically, 40 can be expressed as a difference of squares derived from 7; using the distributive property, the relationship is demonstrated as follows:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 (7-3)(7+3) &amp;=  7^2 - 3^2 \\
 &amp;= 49 - 9 \\
 &amp;= 40
\end{aligned}
&lt;/math&gt;

The following diagram graphically represents the above mathematical relationship. A Jubilee may be divided into two unequal portions of 9 and 40.

[[File:Jubilee_to_Generation_Division.png|thumb|center|500px|Diagram illustrating the division of a Jubilee into unequal portions of 9 and 40.]] 

Shem's placement within the structure can be understood mathematically as the first half of the fractal plus nine pre-flood years, followed by the second half of the fractal plus forty post-flood years, totaling the entire fractal plus one Jubilee (49 years):

[[File:Schematic_Diagram_Book_of_Jubilees_Early_Patriarchs_split.png|thumb|center|500px|Diagram of early Hebraic history as envisioned by the author of the ''Jubilees'' chronology with a split fractal framework]]

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan)'''
** Pre-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 - 1}{2} + 3^2 = 1200 + 9 = 1209&lt;/math&gt;
** Post-Flood Patriarch years:
*:&lt;math display=&quot;block&quot;&gt;\frac{7^4 + 1}{2} + (7^2 - 3^2) = 1201 + 40 = 1241&lt;/math&gt;
** Total Years:
*:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450&lt;/math&gt;
&lt;/div&gt;

== The Samaritan Pentateuch Connection ==
Of all biblical chronologies, the ''Book of Jubilees'' and the ''Samaritan Pentateuch'' share the closest affinity during the pre-flood era, suggesting that the Jubilee system may be a key to unlocking the SP’s internal logic. The diagram below illustrates the structural organization of the patriarchs within the Samaritan tradition.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Jubilees mathematical framework]]

=== Determining Chronological Priority ===
A comparison of the begettal ages in the above Samaritan diagram with the Jubilees diagram reveals a deep alignment between these systems. From Adam to Shem, the chronologies are nearly identical, with minor discrepancies likely resulting from scribal transmission. In the Samaritan Pentateuch, Shem, Ham, and Japheth are born in year 1207 (with Shem's reconstructed birth year as 1209), maintaining a birth position within the 25th Jubilee—the approximate center of the 49x49 &quot;Jubilee of Jubilees.&quot;

This raises a vital question of chronological priority: which system came first? Shem’s placement at the center of the 49x49 grid suggests that the schematic framework of the Book of Jubilees may have influenced the Samaritan Pentateuch's chronology, even if the latter's narratives are older. It is highly probable that Shem's &quot;pivot&quot; position was an intentional design feature inherited or shared by the Samaritan tradition, rather than a coincidental alignment.

=== The 350-Year Symmetrical Extension ===
Post-flood begettal ages differ significantly between these two chronologies. In the Samaritan Pentateuch, the ages of six patriarchs at the birth of their successors are significantly higher than those in the ''Book of Jubilees'', extending the timeline by exactly 350 years (assuming the inclusion of a six-year conquest under Joshua, represented by the black-outlined squares in the SP diagram). This extension appears to be a deliberate, symmetrical addition: a &quot;week of Jubilees&quot; (343 years) plus a &quot;week of years&quot; (7 years).

&lt;div style=&quot;line-height: 1.5;&quot;&gt;
* '''Book of Jubilees (Adam to Canaan):'''
:&lt;math display=&quot;block&quot;&gt;7^4 + 7^2 = 2401 + 49 = 2450 \text{ years}&lt;/math&gt;

* '''Samaritan Pentateuch (Adam to Conquest):'''
:&lt;math display=&quot;block&quot;&gt;\begin{aligned} \text{(Base 49): } &amp; 7^4 + 7^3 + 7^2 + 7^1 = 2401 + 343 + 49 + 7 = 2800 \\ \text{(Base 40): } &amp; 70 \times 40 = 2800 \end{aligned}&lt;/math&gt;
&lt;/div&gt;

=== Mathematical Structure of the Early Samaritan Chronology ===
To understand the motivation for the 350-year variation between the ''Book of Jubilees'' and the SP, a specific mathematical framework must be considered. The following diagram illustrates the Samaritan tradition using a '''40-year grid''' (4x10 year blocks) organized into 5x5 clusters (25 blocks each):

* '''The first cluster''' (outlined in dark grey) contains 25 blocks, representing exactly '''1,000 years'''.
* '''The second cluster''' represents a second millennium.
* '''The final set''' contains 20 blocks (4x5), representing '''800 years'''.

Notably, when the SP chronology is mapped to this 70-unit format, the conquest of Canaan aligns precisely with the end of the 70th block. This suggests a deliberate structural design—totaling 2,800 years—rather than a literal historical record.


[[File:Schematic_Diagram_Samaritan_Pentateuch_Early_Patriarchs_40.png|thumb|center|500px|Diagram of Hebraic history as presented in ''the Samaritan Pentateuch'' chronology, organized into a Generational (4x10 year blocks) mathematical framework]]

== Living in the Rough ==

[[File:Samaritan Passover sacrifice IMG 1988.JPG|thumb|350px|A Samaritan Passover Sacrifice 1988]]

As explained previously, 49 (a Jubilee) is closely associated with agriculture and the 49-day interval between the barley and wheat harvests. The symbolic origins of the number '''40''' (often representing a &quot;generation&quot;) are less clear, but the number is consistently associated with &quot;living in the rough&quot;—periods of trial, transition, or exile away from the comforts of civilization. 

Examples of this pattern include:
* '''Noah''' lived within the ark for 40 days while the rain fell; 
* '''Israel''' wandered in the wilderness for 40 years; 
* '''Moses''' stayed on Mount Sinai for 40 days and nights without food or water. 

Several other prophets followed this pattern, most notably '''Jesus''' in the New Testament, who fasted in the wilderness for 40 days before beginning his ministry. In each case, the number 40 marks a period of testing that precedes a new spiritual or national era.

Another recurring theme in the [[w:Pentateuch|Pentateuch]] is the tension between settled farmers and mobile pastoralists. This friction is first exhibited between Cain and Abel: Cain, a farmer, offered grain as a sacrifice to God, while Abel, a pastoralist, offered meat. When Cain’s offering was rejected, he slew Abel in a fit of envy. The narrative portrays Cain as clever and deceptive, whereas Abel is presented as honest and earnest—a precursor to the broader biblical preference for the wilderness over the &quot;civilized&quot; city.

In a later narrative, Isaac’s twin sons, Jacob and Esau, further exemplify this dichotomy. Jacob—whose name means &quot;supplanter&quot;—is characterized as clever and potentially deceptive, while Esau is depicted as a rough, hairy, and uncivilized man, who simply says what he feels, lacking the calculated restraint of his brother. Esau is described as a &quot;skillful hunter&quot; and a &quot;man of the field,&quot; while Jacob is &quot;dwelling in tents&quot; and cooking &quot;lentil stew.&quot;

The text draws a clear parallel between these two sets of brothers:
* In the '''Cain and Abel''' narrative, the plant-based sacrifice of Cain is rejected in favor of the meat-based one. 
* In the '''Jacob and Esau''' story, Jacob’s mother intervenes to ensure he offers meat (disguised as game) to secure his father's blessing. Through this &quot;clever&quot; intervention, Jacob successfully secures the favor that Cain could not.

Jacob’s life trajectory progresses from the pastoralist childhood he inherited from Isaac toward the most urbanized lifestyle of the era. His son, Joseph, ultimately becomes the vizier of Egypt, tasked with overseeing the nation's grain supply—the ultimate symbol of settled, agricultural civilization. 

This path is juxtaposed against the life of Moses: while Moses begins life in the Egyptian court, he is forced into the wilderness after killing a taskmaster. Ultimately, Moses leads all of Israel back into the wilderness, contrasting with Jacob, who led them into Egypt. While Jacob’s family found a home within civilization, Moses was forbidden to enter the Promised Land, eventually dying in the &quot;rough&quot; of the wilderness.

Given the contrast between the lives of Jacob and Moses—and the established associations of 49 with grain and 40 with the wilderness—it is likely no coincidence that their lifespans follow these exact mathematical patterns. Jacob is recorded as living 147 years, precisely three Jubilees (3 x 49). In contrast, Moses lived exactly 120 years, representing three &quot;generations&quot; (3 x 40).

The relationship between these two &quot;three-fold&quot; lifespans can be expressed by the same nine-year offset identified in the Shem chronology:

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 3(49 - 9) &amp;= 3(40) \\
 147 - 27 &amp;= 120
\end{aligned}
&lt;/math&gt;

[[File:Three_Jubilees_vs_Three_Generations.png|thumb|center|500px|Jacob lived for 147 years, or three Jubilees of 49 years each as illustrated by the above 7 x 7 squares. Jacob's life is juxtaposed against the life of Moses, who lived 120 years, or three generations of 40 years each as illustrated by the above 4 x 10 rectangles.]] 

Samaritan tradition maintains a unique cultural link to the &quot;pastoralist&quot; ideal: unlike mainstream Judaism, Samaritans still practice animal sacrifice on Mount Gerizim to this day. This enduring ritual focus on meat offerings, rather than the &quot;grain-based&quot; agricultural system symbolized by the 49-year Jubilee, further aligns the Samaritan identity with the symbolic number 40. Building on this connection to &quot;wilderness living,&quot; the Samaritan chronology appears to structure the era prior to the conquest of Canaan using the number 40 as its primary mathematical unit.

=== A narrative foil for Joshua ===
As noted in the previous section, the ''Samaritan Pentateuch'' structures the era prior to Joshua using 40 years as a fundamental unit; in this system, Joshua completes his six-year conquest of Canaan exactly 70 units of 40 years (2,800 years) after the creation of Adam. It was also observed that the Bible positions Moses as a &quot;foil&quot; for Jacob: Moses lived exactly three &quot;generations&quot; (3x40) and died in the wilderness, whereas Jacob lived three Jubilees (3x49) and died in civilization.

This symmetry suggests an intriguing possibility: if Joshua conquered Canaan exactly 70 units of 40 years (2,800 years) after creation, is there a corresponding &quot;foil&quot; to Joshua—a significant event occurring exactly 70 Jubilees (3,430 years) after the creation of Adam?

&lt;math display=&quot;block&quot;&gt;
\begin{aligned}
 49 - 9 &amp;= 40 \\
 70(49 - 9) &amp;= 70(40) \\
 3,430 - 630 &amp;= 2,800
\end{aligned}
&lt;/math&gt;

Unfortunately, unlike mainstream Judaism, the Samaritans do not grant post-conquest writings the same scriptural status as the Five Books of Moses. While the Samaritans maintain various historical records, these were likely not preserved with the same mathematical rigor as the ''Samaritan Pentateuch'' itself. Consequently, it remains difficult to determine with certainty if a specific &quot;foil&quot; to Joshua existed in the original architect's mind.

The Samaritans do maintain a continuous, running calendar. However, this system uses a &quot;Conquest Era&quot; epoch—calculated by adding 1,638 years to the Gregorian date—which creates a 1639 BC (there is no year 0 AD) conquest that is historically irreconcilable. For instance, at that time, the [[w:Hyksos|Hyksos]] were only beginning to establish control over Lower Egypt. Furthermore, the [[w:Amarna letters|Amarna Letters]] (c. 1360–1330 BC) describe a Canaan still governed by local city-states under Egyptian influence. If the Samaritan chronology were a literal historical record, the Israelite conquest would have occurred centuries before these letters; yet, neither archaeological nor epistolary evidence supports such a massive geopolitical shift in the mid-17th century BC.

There is, however, one more possibility to consider: what if the &quot;irreconcilable&quot; nature of this running calendar is actually the key? What if the Samaritan chronographers specifically altered their tradition to ensure that the Conquest occurred exactly 2,800 years after Creation, and the subsequent &quot;foil&quot; event occurred exactly 3,430 years after Creation? 

As it turns out, this is precisely what occurred. The evidence for this intentional mathematical recalibration was recorded by none other than a Samaritan High Priest, providing a rare &quot;smoking gun&quot; for the artificial design of the chronology.

=== A Mystery Solved ===
In 1864, the Rev. John Mills published ''Three Months' Residence at Nablus'', documenting his time spent with the Samaritans in 1855 and 1860. During this period, he consulted regularly with the High Priest Amram. In Chapter XIII, Mills records a specific chronology provided by the priest.

The significant milestones in this timeline include:
* '''Year 1''': &quot;This year the world and Adam were created.&quot; 
* '''Year 2801''': &quot;The first year of Israel's rule in the land of Canaan.&quot; 
* '''Year 3423''': &quot;The commencement of the kingdom of Solomon.&quot;

According to 1 Kings 6:37–38, Solomon began the Temple in his fourth year and completed it in his eleventh, having labored for seven years. This reveals that the '''3,430-year milestone'''—representing exactly 70 Jubilees (70 × 49) after Creation—corresponds precisely to the midpoint of the Temple’s construction. This chronological &quot;anchor&quot; was not merely a foil for Joshua; it served as a mathematical foil for the Divine Presence itself.

In Creation Year 2800—marking exactly 70 &quot;generations&quot; of 40 years—God entered Canaan in a tent, embodying the &quot;living rough&quot; wilderness tradition symbolized by the number 40. Later, in Creation Year 3430—marking 70 &quot;Jubilees&quot; of 49 years—God moved into the permanent Temple built by Solomon, the ultimate archetype of settled, agricultural civilization. Under this schema, the 630 years spanning Joshua's conquest to Solomon's temple are not intended as literal history; rather, they represent the 70 units of 9 years required to transition mathematically from the 70&lt;sup&gt;th&lt;/sup&gt; generation to the 70&lt;sup&gt;th&lt;/sup&gt; Jubilee:

:&lt;math&gt;70 \times 40 + (70 \times 9) = 70 \times 49&lt;/math&gt;

=== Mathematical Structure of the Later Samaritan Chronology ===
The following diagram illustrates 2,400 years of reconstructed chronology, based on historical data provided by the Samaritan High Priest Amram. This system utilizes a '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 10 individual blocks representing the period from '''4,000 to 4,400''' after Creation.

The 70th generation and 70th Jubilee are both marked with callouts in this diagram. There is a '''676-year &quot;Tabernacle&quot; era''', which is composed of:
* The 40 years of wandering in the wilderness;
* The 6 years of the initial conquest;
* The 630 years between the conquest and the completion of Solomon’s Temple.

Following the '''676-year &quot;Tabernacle&quot; era''' is a '''400-year &quot;First Temple&quot; era''' and a '''70-year &quot;Exile&quot; era''' as detailed in the historical breakdown below.

[[File:Schematic_Diagram_Samaritan_Pentateuch_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Samaritan chronology, demonstrating a generational mathematical framework.]]

The Book of Daniel states: &quot;In the third year of the reign of Jehoiakim king of Judah, Nebuchadnezzar king of Babylon came to Jerusalem and besieged it&quot; (Daniel 1:1). While scholarly consensus varies regarding the historicity of this first deportation, if historical, it occurred in approximately '''606 BC'''—ten years prior to the second deportation of '''597 BC''', and twenty years prior to the final deportation and destruction of Solomon’s Temple in '''586 BC'''.

The '''539 BC''' fall of Babylon to the Persian armies opened the way for captive Judeans to return to their homeland. By '''536 BC''', a significant wave of exiles had returned to Jerusalem—marking fifty years since the Temple's destruction and seventy years since the first recorded deportation in 606 BC. A Second Temple (to replace Solomon's) was completed by '''516 BC''', seventy years after the destruction of the original structure.

High Priest Amram places the fall of Babylon in year '''3877 after Creation'''. If synchronized with the 539 BC calculation of modern historians, then year '''3880''' (three years after the defeat of Babylon) corresponds with '''536 BC''' and the initial return of the Judeans. 

Using this synchronization, other significant milestones are mapped as follows:
* '''The Exile Period (Years 3810–3830):''' The deportations occurred during this 20-year window, represented in the diagram by '''yellow squares outlined in red'''.
* '''The Desolation (Years 3830–3880):''' The fifty years between the destruction of the Temple and the initial return of the exiles are represented by '''solid red squares'''.
* '''Temple Completion (Years 3880–3900):''' The twenty years between the return of the exiles and the completion of the Second Temple are marked with '''light blue squares outlined in red'''.

High Priest Amram places the founding of Alexandria in the year '''4100 after Creation'''. This implies a 200-year &quot;Second Temple Persian Era&quot; (spanning years 3900 to 4100). While this duration is not strictly historical—modern historians date the founding of Alexandria to 331 BC, only 185 years after the completion of the Second Temple in 516 BC—it remains remarkably close to the scholarly timeline. 

The remainder of the diagram represents a 300-year &quot;Second Temple Hellenistic Era,&quot; which concludes in '''Creation Year 4400''' (30 BC).

=== Competing Temples ===
There is one further significant aspect of the Samaritan tradition to consider. In High Priest Amram's reconstructed chronology, the year '''4000 after Creation'''—representing exactly 100 generations of 40 years—falls precisely in the middle of the 200-year &quot;Second Temple Persian Era&quot; (spanning creation years 3900 to 4100, or approximately 516 BC to 331 BC). This alignment suggests that the 4000-year milestone may have been significant within the Samaritan historical framework.

According to the Book of Ezra, the Samaritans were excluded from participating in the reconstruction of the Jerusalem Temple:

&lt;blockquote&gt;&quot;But Zerubbabel, and Jeshua, and the rest of the chief of the fathers of Israel, said unto them, Ye have nothing to do with us to build an house unto our God; but we ourselves together will build unto the Lord God of Israel&quot; (Ezra 4:3).&lt;/blockquote&gt;

After rejection in Jerusalem, the Samaritans established a rival sanctuary on '''[[w:Mount Gerizim|Mount Gerizim]]'''. [[w:Mount Gerizim Temple|Archaeological evidence]] suggests the original temple and its sacred precinct were built around the mid-5th century BC (c. 450 BC). For nearly 250 years, this modest 96-by-98-meter site served as the community's religious center. However, the site was transformed in the early 2nd century BC during the reign of '''Antiochus III'''. This massive expansion replaced the older structures with white ashlar stone, a grand entrance staircase, and a fortified priestly city capable of housing a substantial population.

[[File:Archaeological_site_Mount_Gerizim_IMG_2176.JPG|thumb|center|500px|Mount Gerizim Archaeological site, Mount Gerizim.]]

This era of prosperity provides a plausible window for dating the final '''[[w:Samaritan Pentateuch|Samaritan Pentateuch]]''' chronological tradition. If the chronology was intentionally structured to mark a milestone with the year 4000—perhaps the Temple's construction or other significant event—then the final form likely developed during this period. However, this Samaritan golden age had ended by 111 BC when the Hasmonean ruler '''[[w:John Hyrcanus|John Hyrcanus I]]''' destroyed both the temple and the adjacent city. The destruction was so complete that the site remained largely desolate for centuries; consequently, the Samaritan chronological tradition likely reached its definitive form sometime after 450 BC but prior to 111 BC.

= The Rise of Zadok =
The following diagram illustrates 2,200 years of reconstructed Masoretic chronology. This diagram utilizes the same system as the previous Samaritan diagram, '''40-year grid''' (modeled on 4x10 year blocks) organized into 5x5 clusters (25 blocks per cluster), where each cluster represents exactly 1,000 years:

* '''The first cluster''' (outlined in dark grey) spans years '''2,000 to 3,000''' after Creation.
* '''The second cluster''' spans years '''3,000 to 4,000''' after Creation.
* '''The final set''' contains 5 individual blocks representing the period from '''4,000 to 4,200''' after Creation.

The Masoretic chronology has many notable distinctions from the Samaritan chronology described in the previous section.  Most notable is the absence of important events tied to siginificant dates. There was nothing of significance that happened on the 70th generation or 70th Jubilee in the Masoretic chronology. The 40 years of wandering in the wilderness and Conquest of Canaan are shown in the diagram, but the only significant date associated with these events in the exodus falling on year 2666 after creation. The Samaritan chronology was a collage of spiritual history. The Masoretic chronology is a barren wilderness.  To understand why the Masoretic chronology is so devoid of featured dates, it is important to understand the two important dates that are featured, the exodus at 2666 years after creation, and the 4000 year event.

[[File:Schematic_Diagram_Masoretic_Text_Late_Era_40.png|thumb|center|500px|Schematic of later Hebraic history based on Masoretic chronology, demonstrating a generational mathematical framework.]]

The Maccabean Revolt (167–160 BCE) was a successful Jewish rebellion against the Seleucid Empire that regained religious freedom and eventually established an independent Jewish kingdom in Judea. Triggered by the oppressive policies of King Antiochus IV Epiphanes, the uprising is the historical basis for the holiday of Hanukkah, which commemorates the rededication of the Second Temple in Jerusalem after its liberation. In particular, Hanukkah celebrates the rededication of the Second Temple in Jerusalem, which took place in 164 BC, cooresponding to creation year 4000.

= Hellenized Jews =
Hellenized Jews were
ancient Jewish individuals, primarily in the Diaspora (like Alexandria) and some in Judea, who adopted Greek language, education, and cultural customs after Alexander the Great's conquests, particularly between the 3rd century BCE and 1st century CE. While integrating Hellenistic culture—such as literature, philosophy, and naming conventions—most maintained core religious monotheism, avoiding polytheism while producing unique literature like the Septuagint.
= End TBD =

'''Table Legend:'''
* &lt;span style=&quot;color:#b71c1c;&quot;&gt;'''Red Cells'''&lt;/span&gt; indicate figures that could result in patriarchs surviving beyond the date of the Flood.
* &lt;span style=&quot;color:#333333;&quot;&gt;'''Blank Cells'''&lt;/span&gt; indicate where primary sources do not provide specific lifespan or death data.

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;
|+ Comparison of Pre-Flood Chronological Traditions (Age at birth of son)

|-
! rowspan=&quot;2&quot; colspan=&quot;1&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Adam
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 230

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Seth
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 105
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 205

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enosh
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 90
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 190

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Kenan
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 70
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 170

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Mahalalel
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 66
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Jared
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 162
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 62
| colspan=&quot;6&quot; style=&quot;background-color:#f9f9f9;&quot; | 162

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Enoch
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 65
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 165

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Methuselah
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 65
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 67
| colspan=&quot;2&quot; style=&quot;background-color:#f9f9f9;&quot; | 187
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167
| colspan=&quot;3&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 167 / 187

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Lamech
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 55
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182
| colspan=&quot;1&quot; style=&quot;background-color:#ffcdd2; color:#b71c1c; font-weight:bold; border:2px solid #ef5350;&quot; | 53
| colspan=&quot;5&quot; style=&quot;background-color:#f9f9f9;&quot; | 188
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 182 / 188

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Noah
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 602
| rowspan=&quot;2&quot; colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | 600
| rowspan=&quot;2&quot; colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shem

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | Pre-Flood TOTAL
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied
|}

{| class=&quot;wikitable&quot; style=&quot;width:100%; font-family:sans-serif; font-size:13px; text-align:center; table-layout:fixed;&quot;

|+ Comparison of Post-Flood Chronological Traditions (Age at birth of son)
|-
! colspan=&quot;1&quot; rowspan=&quot;2&quot; | Patriarch
! colspan=&quot;3&quot; style=&quot;background-color:#e3f2fd; border-bottom:2px solid #2196f3;&quot; | SHORT CHRONOLOGY
! colspan=&quot;6&quot; style=&quot;background-color:#fff3e0; border-bottom:2px solid #ff9800;&quot; | LONG CHRONOLOGY

|-
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Jubilees &lt;br/&gt; (Jub)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Masoretic &lt;br/&gt; (MT)
! colspan=&quot;1&quot; style=&quot;background-color:#e3f2fd;&quot; | Samaritan &lt;br/&gt; (SP)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Demetrius &lt;br/&gt; (204 BC)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Africanus &lt;br/&gt; (221 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Theophilus &lt;br/&gt; (192 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Septuagint &lt;br/&gt; (LXX)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Eusebius &lt;br/&gt; (325 AD)
! colspan=&quot;1&quot; style=&quot;background-color:#fff3e0;&quot; | Josephus &lt;br/&gt; (94 AD)

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot;  style=&quot;text-align:left; color:black;&quot; | Pre-Flood
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 1656
| colspan=&quot;1&quot; | 1309
| colspan=&quot;1&quot; | 2264
| colspan=&quot;1&quot; | 2262
| colspan=&quot;1&quot; | 2242
| colspan=&quot;3&quot; | Varied

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Arphaxad
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 66
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 35
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 135

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Cainan II
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 130
| colspan=&quot;2&quot; style=&quot;background-color:#e8e8e8;&quot; | -

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Shelah
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 71
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Eber
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 64
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 34
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 134

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Peleg
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 61
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;7&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Reu
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 59
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 32
| colspan=&quot;5&quot; style=&quot;background-color:#e8e8e8;&quot; | 132
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 135
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 130

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Serug	
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 57
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 30
| colspan=&quot;6&quot; style=&quot;background-color:#e8e8e8;&quot; | 130
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 132

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Nahor
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 62
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 29
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 75
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79 / 179
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 79
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 120

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Terah
| colspan=&quot;9&quot; style=&quot;background-color:#e8e8e8;&quot; | 70

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Abram
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 78
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 75

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Canaan
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 218
| colspan=&quot;8&quot; style=&quot;background-color:#e8e8e8;&quot; | 215

|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Egypt
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 238
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
| colspan=&quot;1&quot; style=&quot;background-color:#e8e8e8;&quot; | 430
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 215
|-
! colspan=&quot;1&quot; style=&quot;text-align:left; background-color:#f9f9f9;&quot; | Sinai +/-
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | 40
| colspan=&quot;1&quot; style=&quot;background-color:#f9f9f9;&quot; | -
| colspan=&quot;3&quot; style=&quot;background-color:#e8e8e8;&quot; | 46
| colspan=&quot;4&quot; style=&quot;background-color:#e8e8e8;&quot; | 40

|- style=&quot;background-color:#333; color:white; font-weight:bold; font-size:15px;&quot;
! colspan=&quot;1&quot; style=&quot;text-align:left; color:black;&quot; | GRAND TOTAL
| colspan=&quot;1&quot; | 2450
| colspan=&quot;1&quot; | 2666
| colspan=&quot;1&quot; | 2800
| colspan=&quot;1&quot; | 3885
| colspan=&quot;1&quot; | 3754
| colspan=&quot;1&quot; | 3938
| colspan=&quot;3&quot; | Varied
|}

== The Septuagint Chronology ==

=== The Correlations ===
An interesting piece of corroborating evidence exists in the previously mentioned 1864 publication by Rev. John Mills, ''Three Months' Residence at Nablus'', where High Priest Amram records his own chronological dates based on the Samaritan Pentateuch. Priest Amram lists the Flood date as 1307 years after creation, but then lists the birth of Arphaxad as 1309 years—exactly two years after the Flood—which presumably places Shem's birth in year 502 of Noah's life (though Shem's actual birth date in the text is obscured by a typo).

The internal tension in Priest Amram's calculations likely reflects the same two-year variance seen between Demetrius and Africanus. Priest Amram lists the birth years of Shelah, Eber, and Peleg as 1444, 1574, and 1708, respectively. Africanus lists those same birth years as 2397, 2527, and 2661. In each case, the Priest Amram figure differs from the Africanus value by exactly 953 years. While the chronology of Africanus may reach us through an intermediary, as Paul D. notes, the values provided by both Demetrius and Africanus are precisely what one would anticipate to resolve the &quot;Universal Flood&quot; problem.


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      <text bytes="29919" sha1="b1064ow57vtlqfx56b1zr6y8rossed8" xml:space="preserve">:''I am entitled to my [[Wiktionary:cockamamie|cockamamie]] ideas, and you are entitled to yours.'' [Humor is important but must be offered in a way that does not offend others. If others are offended, they may be less interested in dialogue.]

:This book is a combination instruction manual on [[w:Media literacy|media literacy]] and an invitation to you to support collaborative / crowd-sourced research on how to improve the world's understanding of media literacy and how to accelerate its understanding and use globally for the betterment of humanity. 

The previous chapter on [[Media Literacy and You/The impact of the media on political economy since the time of the Pharaohs|The impact of the media on political economy since the time of the Pharaohs]] describes how hierarchical societies prior to King James of the King James bible were divided between those who fought, prayed, and worked. It was the responsibility of those who prayed to convince those who worked to live in poverty while giving increasing shares of what they produced so those who fought and prayed could live lives of leisure and opulence. During the reign of King James, pamphlets and newspapers began to compete with the church for helping commoners understand their roles in society. This produced the Industrial Revolution and modern democracies. Media consolidation since World War II gradually slowed and then reversed this trend.

Then a chapter on [[Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future|Fox, the Great Depression, the Great Recession, and our future]] describes how Fox convinced its audience and a critical mass of the electorate that President Franklin Roosevelt (FDR)'s New Deal made the Great Depression ''worse'', not better. That chapter provides plots that seem to contradict the &quot;experts&quot; featured by Fox. The existence of that contrary evidence suggests that by talking politics calmly and respectfully with others with whom we may vehemently disagree, we may be able to build a culture in which we can agree to disagree agreeably and collaborate effectively to study the available evidence and make progress on issues of common concern.  

This chapter discusses the relative importance of the US [[w:Postal Service Act|Postal Service Act]] of 1792, the consolidation of ownership of the major media since the end of World War II, the rise of social media, in recent decades, and the evolution of political polarization in the history of the [[w:United States Congress|US Congress]]. 

== American Revolution and the US Postal Service Act of 1792== 

A major concern of the [[w:Continental Congress|Continental Congress]] meeting in Philadelphia between 1774 and 1789 was censorship of the press by King [[w:George III|George]]'s postal service. The revolutionaries agreed that the survival of their experiment in Republican government depended on an electorate with access to diverse views. Those discussions led to the US [[w:Postal Service Act|Postal Service Act]] of 1792, under which newspapers were delivered up to 100 miles for a penny when first class postage was between 6 and 25 cents. Those subsidies gave the young United States of American more independent newspaper publishers per million population in the early nineteenth century than at any other time or place in human history.&lt;ref&gt;John and Graves (2025). John (1995).&lt;/ref&gt; These subsidies amounted to roughly 0.21 percent of GDP in 1841, according to McChesney and Nichols.&lt;ref&gt;McChesney and Nichols (2010, pp. 310-311, note 88).&lt;/ref&gt; [[w:Alexis de Tocqueville|Alexis de Tocqueville]], who visited the relatively young United States in 1831, wrote, &quot;There is scarcely a hamlet that does not have its own newspaper.&quot;&lt;ref&gt;Tocqueville (1835, p. 93).&lt;/ref&gt; 

That diversity of independent newspaper publishers per million population began to decline in the 1850s and 1860s as high speed rotary presses reduced the per issue cost of printing a newspaper while increasing the capital required to start a newspaper.&lt;ref&gt;John and Silberstein-Loeb (2015, p. 80).&lt;/ref&gt; However, that diversity did not disappear overnight. It apparently continued to encourage literacy and limit political corruption, both of which helped the US stay together and grow in land area and average annual income while expanding the right to vote as contemporary New Spain, then Mexico, fractured, shrank and stagnated economically. The Wikiversity article on &quot;[[The Great American Paradox]]&quot; suggests three reasons why the young US fared better than Mexico and other countries in that &quot;[[w:Age of Revolution|Age of Revolution]]&quot; from late late-18th to the mid-19th centuries:&lt;ref&gt;&quot;[[The Great American Paradox]]&quot; accessed 2026-02-12.&lt;/ref&gt; 

# '''Advanced democracy before the revolution''': The British colonies that rebelled in 1776 already had possibly the most advanced democratic culture on the planet at that time. Acemoglu and Robinson document how the British colonies that became the US all initially ''failed'' until they switched from autocratic, extractive governance to giving property to substantial portions of the population and a voice in a colonial legislature to all adult male property owners.&lt;ref&gt;Acemoglu and Robinson (2012, pp. 26-27).&lt;/ref&gt; By 1776 almost 60 percent of adult white males could vote, and the violence of the Revolution did not change that.&lt;ref&gt;Keyssar (2000, pp. 7, 24); Graves (2005).&lt;/ref&gt; 
# '''Citizen-directed subsidies for news''': Citizen-directed subsidies for newspapers provided by the [[w:Postal Service Act|US Postal Service Act of 1792]] helped limit political corruption and encourage literacy, as previously mentioned. 
# '''Washington did not win the Revolution''': Some historians insist that Washington did not win many battles. His success came from attracting enough humans to join the [[w:Continental Army|Continental Army]] and keep it in the field for the 6.5 years between the &quot;[[w:Shot heard round the world|Shot heard round the world]]&quot; at [[w:Battles of Lexington and Concord|Lexington and Concord]], 1775-04-19 to the surrender of [[w:Charles Cornwallis, 1st Marquess Cornwallis|General Cornwallis]] at [[w:Siege of Yorktown|Yorktown]] 1781-10-19&lt;ref&gt;e.g., Phelps (2001). Washington was perpetually short of supplies. His own state of Virginia often could not send their allotment of troops, because they were needed for slave patrols, as reported by Nash (2005, pp. 344-345). Many in Washington's army often did not have shoes.  This made it hard for them to move, especially in the winter, because the British could easily follow the blood in the snow, as mentioned by Raphael (2001, p. 89).&lt;/ref&gt; and keep from getting captured.&lt;ref&gt;Ferling (1988).&lt;/ref&gt; He was active enough to provoking the British to do stupid things and manufacture enemies for themselves without getting captured or otherwise defeated during that period. Standard histories of the American Revolution begin with the [[w:Battles of Lexington and Concord|Battles of Lexington and Concord]], 1775-04-19. This in essence initiated the [[w:Siege of Boston|Siege of Boston]]. [[w:Siege of Boston#Stalemate|General Washington arrived]] 3.5 months later on 1775-07-02. During the following 8 months, the revolutions managed to install heavy artillery on [[w:Fortification of Dorchester Heights|Dorchester Heights]], which overlooked Boston harbor in a way that could prevent the British from getting more supplies. The British retreated. During the next five years, the revolutionaries won enough battles to encourage the French to provide more support [[w:Southern theater of the American Revolutionary War|until the British general Cornwallis]] constructed a fortified naval port at Yorktown, where he could be resupplied or evacuated. Then [[w:Battle of the Chesapeake|the French navy bottled up the Chesapeake]] while French military units helped Washington convince Cornwallis to surrender.&lt;ref&gt;Graves (2005).&lt;/ref&gt; Historians describe the &quot;[[w:Age of Revolution|Age of Revolution]] as starting with the [[w:Age of Revolution#American Revolution (1765–1783)|American Revolution (1765–1783)]], encompassing revolutionary violence in many countries from Europe to Latin America, the vast majority of which arguably replaced one brutal repressive system with another. Many got improvements in the rule of law through adoption of procedures like the [[w:Napoleonic Code|Napoleonic Code]], but few got substantive improvements in democracy. [[The Great American Paradox|The US is ''not'' an exception]]: Almost 60 percent of adult white males could vote before the revolution, and the violence of the revolution did not change that. This is consistent with the [[Freedom and abundance|twenty-first century research by Chenoweth and Stephan (2011), which found no change in the average level of democratization in the over 200 violent revolutions of the twentieth century. By contrast, win or lose, major nonviolent governmental change efforts increased the average level of democratization.  

Between 1792 and 1920 the right to vote was gradually extended to all adult males then to females, supported by political agitation and a media system whose ownership was still quite broadly held. 

During the 1930s with over 20 percent of the workforce unemployed, newspapers could not retain an audience by blaming the unemployed for their plight. That environment helped the Franklin Roosevelt administration get popular support to tax the ultra-wealthy like they had never been taxed before or since. And during world War II, FDR got the support needed to impose wage and price controls to limit price gouging that had stifled economic growth and generated inflation in previous major wars, like the War of 1812, the Civil War and World War I. The result was unprecedented economic growth with only nominal inflation, as documented in the chapter on [[Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future|Fox, the Great Depression, the Great Recession, and our future]].

== The consolidation of ownership of the major media since the end of World War II == 

The development of broadcasting differed from that of newspapers in at least two ways that may have had a substantial impact on concentration of ownership: 
# It is not easy to charge for what is broadcasted without government intervention, unlike newspapers, which can charge for a copy of the paper. 
# Content for broadcasting has been easily and quickly shared between different broadcasters by electronic means. That increases the incentives for organizational integration among broadcasters over what was available between the time that [[w:List of the oldest newspapers|newspapers started appearing in the 1600s]] and two centuries later as steam and later electronics began to dramatically increase the speed of communications. 

[[w:History of broadcasting|As radio technology was being developed, many governments established state-sponsored networks]], sometimes with competing commercial systems. Developments in intellectual property law since the 1600s also made it easier for major corporations to dominate the development of broadcasting. [[w:Guglielmo Marconi|Guglielmo Marconi]] was a leader among many attempting to develop wireless telegraphy in the late nineteenth century. [[w:Guglielmo Marconi#Transmission breakthrough|In 1895 he demonstrated a transmitter and receiver]] able to communicate up to two miles. [[w:Invention of radio|A few years later it was used for ship-to-shore]] communications. In the US, Marconi [[w:Broadcasting in the United States#Commercial development|Marconi]] created the [[w:Marconi Wireless Telegraph Company of America|Marconi Wireless Telegraph Company of America]] in 1899, which was nationalized by the US government during [[w:World War I|World War I]]. After the war, US government officials did ''not'' want to return control of the company to its British owners, so it was sold in 1919 as a [[w:patent trust|patent trust]] called [[w:Radio Corporation of America|Radio Corporation of America]], owned by a partnership of [[w:General Electric| General Electric]] (GE), [[w:Westinghouse|Westinghouse]], [[w:AT&amp;T Corporation|AT&amp;T]] and [[w:United Fruit Company|United Fruit]]. Their mission was to make and sell radio equipment. In 1926 they created the [[w:NBC#History|National Broadcasting Company (NBC)]] to produce content. [[w:Competition law|Antitrust action]] in 1932 forced GE to sell RCA and NBC. 

The [[w:Columbia Broadcasting System|Columbia Broadcasting System]] (CBS) began in 1927. NBC split into &quot;Red&quot; and &quot;Blue&quot; networks. In 1939 antitrust action forced NBC to divest its Blue network, which became the [[w:American Broadcasting Company|American Broadcasting Company]] (ABC). A decade later, in 1949 ABC merged with United Paramount Theatres after antitrust action forced the latter to divest itself from its movie production business.  

There were other, smaller networks, but as television became feasible, NBC, CBS, and ABC became the &quot;[[w:Big Three (American television)|Big Three]]&quot; dominating television in the US after [[w:World War II|World War II]]. Biased reporting by that [[w:Oligopoly|oligopoly]] facilitated the rise of [[w:McCarthyism|McCarthyism]] and have made politicians unelectable unless they support policies contrary to the best interests of the vast majority of their constituents, as discussed in the rest of this book. 

The decline in inequality documented in Figures 4 and 5 of the chapter on &quot;[[Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future|Fox, the Great Depression, the Great Recession, and our future]]&quot; meant that the ultra-wealth could not afford as many personal servants, and the success of iconoclasts like [[w:Ralph Nader|Ralph Nader]] threatened their control of the political economy. 

Meanwhile the media coverage given to the [[w:Civil rights movement|civil rights movement]] of the 1950s and 1960s generated a backlash among those who resented the loss of their previous racial-based social superiority and complained that the mainstream media had a liberal bias. These claims were already well established by the time [[w:Richard Nixon|Richard Nixon]] became US president in 1969.&lt;ref&gt;Major (2019).&lt;/ref&gt;

In this environment, [[w:Lewis F. Powell Jr.|Lewis Powell]] wrote a confidential memorandum titled, &quot;Attack on American Free Enterprise System&quot;&lt;ref&gt;Powell (1971).&lt;/ref&gt; for the [[w:United States Chamber of Commerce|US Chamber of Commerce]] after he had agreed to serve on the [[w:Supreme Court of the United States|US Supreme Court]] but before he was officially nominated. This memo recommended that business elites fund a conservative research organizations and &quot;monitor&quot; the media and activities on universities and become more politically active in lobbying to support the interests of big business. That inspired of multiple right-wing think tanks and lobbying organizations that have since had a major influence on the US political economy. This includes the creation of the [[w:Business Roundtable|Business Roundtable]] on 1972-10-13, [[w:The Heritage Foundation|The Heritage Foundation]], 1973-02-16, the [[w:American Legislative Exchange Council|American Legislative Exchange Council]] (ALEC), September 1973, the [[w:Cato Institute|Cato Institute]], January 1977, and the [[w:Manhattan Institute for Policy Research|Manhattan Institute for Policy Research]], in 1978. 

As documented in the chapter on [[Media Literacy and You/Criminal justice|Criminal justice]], beginning around 1975 the major media fired nearly all their investigative journalists and replaced them with the police blotter. This replaced reporting on questionable business practices with individual crime. The public thought that crime was out of control, when there had been no substantive change in crime. They voted in a generation of politicians promising to get tough on crime, and the portion of the US population in state and federal prisons shot up from roughly 0.1 percent, where it had been over the previous 50 years, to closer to 0.5 percent in the next 25 years. 

Beginning in the 1980s major corporations strengthened their control of the &quot;Big Three&quot; broadcast networks. [[w:NBC|GE bought RCA including NBC]] in late 1986 and immediately divested itself of RCA's other assets, retaining only NBC. In 2004 they merged with the French media company [[w:Vivendi|Vivendi]], forming [[w:NBCUniversal|NBCUniversal]]. In 2011 Comcast purchased a controlling interest in NBCUniversal. And [[w:Westinghouse Electric Corporation|Westinghouse Electric Corporation]] bought CBS in 1994.

==Fairness Doctrine==  

In 1987 the FCC ended the [[w:Fairness doctrine|fairness doctrine]], with conservatives claiming it was biased against them. 

A cynic might argue that honest conservatives should be able to negotiate a fairness doctrine that was more equitable, unless, of course, conservative media is of its essence in herently unfair. This view is supported by research by Reece Peck, Anthony Nadler, and others, and by the settlement in ''[[w:Dominion Voting Systems v. Fox News Network|Dominion v. Fox]]'', discussed further below. 

[[w:Project 2025|Project 2025]] recommended stripping public funds from the [[w:Corporation for Public Broadcasting|Corporation for Public Broadcasting]], which included the [[w:PBS|Public Broadcasting Service (PBS)]], i.e., public television, and [[w:NPR|National Public Radio (NPR)]]. They added, &quot;Stripping public funding would, of course, mean that NPR, PBS, Pacifica Radio, and the other leftist broadcasters would be shorn of the presumption that they act in the public interest ... . They should no longer, for example, be qualified as noncommercial education stations (NCE stations), which they clearly no longer are.&quot;&lt;ref&gt;Project 2025 (pp. 246-251).&lt;/ref&gt; 

[[Fighting back against the campaign of censorship and control|On 2025-09-12 FCC Commissioner Anna Gomez agreed that broadcasters have]] &quot;public interest obligations&quot;. However, the &quot;FCC is being weaponized by using an undefined and distorted public interest justification for targeting government critics and censoring disfavored content. ... [R]egardless of how often or how loudly the Commission asserts that it has the right to police media bias because broadcasters have a public interest obligation, that is simply not true. The long defunct fairness doctrine is the closest thing to the vague standard being asserted. ... [B]efore initiating any more retaliatory investigations or politically motivated actions, the FCC must initiate a proceeding to define what the public interest means.&quot; But as long as the FCC can get what it wants from broadcasters ''without issuing an official ruling'', those actions are not reviewable by courts, where the FCC would likely lose.&lt;ref&gt;Gomez (2025).&lt;/ref&gt; 

== Liberal Judges == 

Conservatives have long complained about &quot;liberal&quot; judges. In the 1966 ''[[w:Miranda v. Arizona|Miranda]]'' ruling, the US Supreme Court said that confessions cannot be used unless law enforcement first advised suspects of their constitutional rights. Without that, confessions cannot be used in court. Conservatives complained that the decision undermined the efficiency of the police and argued that it would contribute to an increase in crime. 

Complaints like this combined with the Powell memorandum mentioned above helped inspire the creation of the [[w:Federalist Society|Federalist Society]] 1982-04-03 to challeng [[w:Liberalism in the United States|liberal]] or [[w:Left-wing politics|left-wing]] ideology within American law schools and universities. They have since played a major role in networking and mentoring young conservative attorneys and recommending conservatives for judicial appointments. In that regard, they have had considerable success, contributing to US Supreme Court decisions like ''[[w:Citizens United v. FEC|Citizens United]]'', which cynic describe as saying that corporations are people, money is speech, and humans are second-class citizens. The major media have a conflict of interest in honestly reporting on the impact of such rulings. 
  
== Election denialism == 

:''The rules of evidence in the court of public opinion is whatever will most please those who control most of the money for the media.''
:''The rules of evidence in courts of law in the US tend to be more equitable.''

In 1980 Republican Christian Conservative [[w:Paul Weyrich|Paul Weyrich]] said, [[q:Paul Weyrich|I don't want I don't want everybody to vote. ... [O]ur leverage in the elections ... goes up as the voting populace goes down.]]&lt;ref&gt;Weyrich (1980).&lt;/ref&gt; In 1981, the [[w:Republican National Committee|Republican National Committee]] (RNC) created the [[w:Ballot Security Task Force|National Ballot Security Task Force]] to discourage voter turnout among likely Democratic voters in the gubernatorial election. The [[w:Democratic National Committee|Democratic National Committee]] (DNC) sued claiming a violation of the [[w:Voting Rights Act of 1965|Voting Rights Act of 1965]], illegal harassment, and voter intimidation. The RNC and New Jersey Republican State Committee entered into a consent decree in 1982 agreeing to stop doing it. That consent decree was repeatedly extended with Democrats arguing that recent campaigns showed it was still unnecessary until a different judge decided not to extend it in 2018.  

The decision in ''[[w:Fish v. Kobach|Fish v. Kobach]]'' provides some of the most compelling documentation of Republican attempts to disfranchise likely Democratic voters. Judge [[w:Julie A. Robinson|Julie Robinson]], who had been appointed to the bench by president [[w:George W. Bush|George W. Bush]], a Republican, found that &quot;31,089 total applicants ... were denied registration for failure to provide DPOC, ... [which] represented 12.4% of new voter registrations between January 1, 2013 and December 11, 2015&quot;.&lt;ref&gt;Robinson (2018, pp. 26-28).&lt;/ref&gt; Meanwhile, Kansas Secretary of State Kobach, who claimed this was a massive problem, provided evidence of only 39 cases of non-citizens having registered to vote in Kansas, which represented only &quot;0.002% of all registered voters&quot;.&lt;ref&gt;Robinson (2018, pp. 87-89).&lt;/ref&gt; In that case, [[w:Hans von Spakovsky|Hans von Spakovsky]], a senior fellow at [[w:The Heritage Foundation|The Heritage Foundation]], testified that a [[w:United States Government Accountability Office|U.S. GAO]] 'found that up to 3 percent of the 30,000 individuals called for jury duty from voter registration roles over a two-year period in just one U.S. district court were not U.S. citizens.' On cross-examination, however, he acknowledged that he had misleadingly failed to mention that the GAO study in question contained information from 8 district courts. Four of the 8 reported that there was not a single non-citizen who had been called for jury duty, and the 3 remaining district courts reported that less than 1% of those called for jury duty from voter rolls were noncitizens. Judge Robinson concluded that &quot;his clear agenda and misleading statements ... render his opinions unpersuasive.&quot;

[[w:Election denial movement in the United States|Regarding the specific claim that the 2020 election was stolen from Trump]], in March 2021, [[w:Dominion Voting Systems|Dominion Voting Systems]] sued Fox for defamation, because Fox repeatedly broadcasted claims that the [2020 election had been stolen from Trump. During [[w:Discovery (law)|pre-trial discovery]], Dominion obtained numerous internal Fox communications that documented that Fox had initially called the [[w:2020 United States presidential election|2020 United States presidential election]] for Biden. Then Fox executives saw that they were losing audience to other conservative outlets that were claiming that the election had been stolen from Trump, so they switched to repeating that fraudulent claim to protect their audience share. On 2023-04-18 Fox agreed to pay Dominion $787.5 million -- over three quarters of a billion dollars -- while agreeing that they had lied to their audience; the settlement did ''not'' require Fox to admit to their audience that they had lied to them. If Fox had 6 percent higher audience for a year, they made money lying to their audience, even after paying Dominion $787.5 million. 

Over 1.5 years later, a survey conducted the month after the 2024 US presidential election found that 63 percent of Republicans and 31 percent of voters overall still believed that 2020 election had been stolen from Trump.&lt;ref&gt;Public Religion Research Institute (2024).&lt;/ref&gt;

== Evolution of political polarization in the US Congress == 

[Coming soon, summarizing an update to &quot;[[Evolution of political polarization in the US Congress]]&quot;.]

== Exercises ==

1. Create a list of your greatest concerns, identify sources for contrary information about those issues, and spend some of your free time with them and less with for-profit media media like television and commercial social media. 

2. Get on email lists and / or social media of your elected officials to see what they are telling their constituents. US citizens and residents can contact representatives in the US Congress. You do not have to be eligible to vote in the US to contact them or to talk politics with others. However, elected officials may not respond if you cannot claim residence in their jurisdiction. At [https://www.house.gov house.gov] you are asked to &quot;Enter your zip code&quot; under &quot;FIND YOUR REPRESENTATIVE&quot;. At [https://www.senate.gov senate.gov], they want the your state. 

3. Pick an issue that most concerns you and review relevant research literature on appropriate Wikipedia articles -- and maybe in this book -- and maybe also [https://scholar.google.com Google Scholar]. 

3.1. Send emails to appropriate elected officials asking their thoughts and their response to claims you found in the literature. 

3.2. Call their office phone number and ask to speak with someone about the issue outlined in your email. 

4. Discuss your experience with others. Invite them to join you in meeting(s) (via Zoom?) with staff(s) of elected official(s) if you can arrange such. If others agree, then go to web site(s) of your elected representative(s) and request a meeting. 

5. Share your experience in some appropriate place in this book, e.g., a &quot;Discuss&quot; page associated with a chapter relative to the issue of your concern or to the main ''[[Media Literacy and You]]'' page.

== See also ==


== Notes == 
{{reflist}}

== Bibliography == 

* &lt;!--Daron Acemoğlu and James A. Robinson (2012-03-20). Why Nations Fail: The Origins of Power, Prosperity, and Poverty (in en). Crown Publishing Group--&gt;{{cite Q|Q7997840}}
*  &lt;!--Chenoweth and Stephan (2011) Why Civil Resistance Works: The Strategic Logic of Nonviolent Conflict--&gt;{{cite Q|Q88725216}}
* &lt;!--John E. Ferling (1988) The First of Men--&gt;{{cite Q|Q59297825}}
* &lt;!-- Bernhard Fulda (2009) Press and Politics in the Weimar Republic (Oxford U. Pr)--&gt;{{cite Q|Q123561961}}
* &lt;!--Anna Gomez (2025-09-12) &quot;Fighting back against the campaign of censorship and control&quot;, presentation to the Grassroots Radio Conference--&gt;{{cite Q|Q138499094}}
* &lt;!--Spencer Graves (2005-02-26) &quot;Violence, Nonviolence, and the American Revolution&quot;--&gt;{{cite Q|Q58635938}}
* &lt;!--Don Higginbotham (2001) George Washington Reconsidered--&gt;{{cite Q|Q59362787|author=Don Higginbotham, ed.}}
* &lt;!--Richard R. John (1995) Spreading the News: The American Postal System from Franklin to Morse--&gt;{{cite Q|Q54641943}}
* &lt;!--Richard R. John and Jonathan Silberstein-Loeb (eds.; 2015) Making News: The Political Economy of Journalism in Britain and America from the Glorious Revolution to the Internet (Oxford University Press)--&gt;{{cite Q|Q131468166}}
* &lt;!--Alexander Keyssar (2000) The Right to Vote: The Contested History of Democracy in the United States--&gt;{{cite Q|Q97621556}}
* &lt;!--Mark Major (2019) &quot;Bridging the Marginal and the Mainstream: Methodological Considerations for Conservative News as a Subfield&quot;, ch. 12 in Nadler and Bauer--&gt;{{cite Q|Q138497692}}
* &lt;!-- Robert W. McChesney; John Nichols (2010). The Death and Life of American Journalism (Bold Type Books) --&gt;{{cite Q|Q104888067}}.
* &lt;!--Gary B. Nash (2005) The Unknown American Revolution--&gt;{{cite Q|Q59319726}}
* &lt;!--Glenn A. Phelps (2001) The Republican General, ch. 7 in Higginbotham--&gt;{{cite Q|Q138211200}}
* &lt;!--Lewis F. Powell Jr. (1971) Attack on American Free Enterprise System--&gt;{{cite Q|Q16577699}}
* &lt;!--  Project 2025--&gt;{{cite Q|Q122382481}} 
* &lt;!--Public Religion Research Institute (2024-12-13) &quot; Analyzing the 2024 Presidential Vote: PRRI’s Post-Election Survey&quot;--&gt;{{cite Q|Q136415370}}
* {{Citation | last = Robinson | first = Julie A. 
| date = 2018-06-18 
| title = Findings of fact and conclusions of law in Fish v. Kobach, Case No. 16-2105-JAR-JPO, and Bednasek and Kobach, Case No. 15-9300-JAR-JPO (published 2018-06-18 with corrections 2018-06-19)
| publisher = US District Court for the District of Kansas 
| url = https://www.courtlistener.com/recap/gov.uscourts.ksd.110435/gov.uscourts.ksd.110435.542.0_3.pdf
| accessdate = 2018-06-28}}
* &lt;!-- Alexis de Tocqueville (1835, 1840; trad. 2001) Democracy in America (trans. by Richard Heffner, 2001; New America Library) --&gt;{{cite Q|Q112166602|publication-date=unset|author=Alexis de Tocqueville (1835, 1840; trad. 2001)}}
* &lt;!--Paul Weyrich (1980-08) &quot;I don't want everybody to vote&quot;--&gt;{{cite Q|Q98749513}}

[[Category:Media literacy]]
[[Category:Communication]]
[[Category:Political science]]
[[Category:Law]]
[[Category:Psychology]]
[[Category:Sociology]]
[[Category:Education]]
[[Category:Economics]]
[[Category:Armed forces and combat]]
[[Category:Immigration]]
[[Category:Self improvement]]
[[Category:Media Literacy and You]]
[[Category:Social media]]
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      <comment>/* w 17, Saturday, April 25 */ PTSD-like experience related to studying?</comment>
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      <text bytes="17132" sha1="r4zddvf1s8og9bm4yvbz25xwwgzrhgi" xml:space="preserve">All times noted down are UTC, Universal Time Coordinated.

This is a continuation of [[User:Dekatriofovia/Draftspace/Coursera]].

I'm continuing to make notes on a computer with a screen that allows me to see text more clearly.

{{tl|AI-generated}}, adding this template link to make clear that I am going to use &quot;AI Mode&quot; by Google to help me in prioritizing what I should do based on my motivation for independent study of the Coursera material and luckily some Wikimedia content, perhaps going completely to the linked Wikibooks even though that may sound a bit extreme.

== w 13, Thursday, March 26, 2026 ==

March 26: We are still at the transcripts and I'm supposed to read from page 4 to page 16...which I doubt I'll be able to do since I have much less time right now. Tomorrow or later in the future I can reflect on how this went if I don't gain any more insights today.
Me and my mother made this more organized this time. We decided that during certain days and hours of the week we will 'study together' but that also includes homework. My homework was page 4-16. We'll see how that goes...

Today was a day we had scheduled. 2 hours studying, taking turns reading paragraphs.

We began at page 16, which is 2 pages after a chapter about Bernard Mandeville and Adam Smith begins, and went until page 25 of the transcript where the chapter about Auguste Comte begins.

Now we got a new day scheduled along with a suitable time and then we'll read another 2 hours I guess.

== w 13, Saturday, March 28 ==

This is the 2nd day we studied for 2 hours at a specific day and a specific hour. With other words before March 26 we managed to make a schedule and now the schedule is complete. 6 hours per week, 3 days a week and 2 hours per day.

We read from page 16 and until page 33 where &quot;3.6 Comte’s Religion of Humanity&quot; begins.

== w 14, Monday, March 30 ==

Our study technique has changed now as we have started to view the videos instead of only focusing on the transcript. The videos contain the transcript so it makes sense and the videos automatically pause when there's a question to ask the student whether they've understood the material so far.

The two previous days, Thursday and Saturday our &quot;study technique&quot; was to read the transcript, one paragraph each person in our own pace and if we asked the other we needed more time we gave that time to the other person. You don't need a family relationship for this. Any two persons can agree to do that as long as there is mutual respect.

My mother was better at listening with focus on her end to what our teacher &quot;Dr. Bart van Heerikhuizen&quot; said and showed on his media slides.

== w 14, Thursday and Friday, April 2 and 3 ==

I'm still getting used to learning our new way which I'm trying to get used to. My mother did not find that watching the video helped very much with understanding and she opted to read the text first then watch the video.

Myself I try to do the same. So sad that what I'm studying isn't on Wikiversity. Maybe what I learn I can later use here on Wikiversity? ie. if I learn about Adam Smith on my own using my own sources I could create a &quot;course&quot; too but on Wikiversity with knowledge for everyone. Though with my attention span who knows if that will ever be completed haha. Dreams to dream.

Though let's talk tech and problems(both in a good way problems and in a bad way problems and perhaps even neutral way problems) on Coursera. If I gather enough interesting problems maybe that might lead to some sort of &quot;portal&quot; of information for a particular subject, ie. Coursera. What studying methods do they use? What studying methods do the universities and other institutions use through them?

[[User:ThinkingScience/Draftspace/CourseraHeadscratchers]] - This is for personal notes that are about problems or &quot;bugs&quot; on the site or me making notes of things I found helpful or less helpful in regards to the &quot;system&quot;. If I gather enough notes that might make it worth to make a page on Wikiversity all for the better. If not, at least I tried and I told you what I find interesting and what I find less interesting and more toward 'what can be improved'.

== w 15(easter caused disruptions in the schedule), Monday, April 6 ==

It did not go very well as I fell into my &quot;old habits&quot;, so for a future &quot;studying together&quot; I will now propose to my mother that we can study together but text only, no voice.

On Monday my mother was &quot;confused&quot; regarding how we learn best and I've not figured that out for myself yet. Part of it though is probably reading with another student/human being and taking turns, I read one paragraph, my studying partner reads the next and so forth rinse and repeat. Trying something new to discover if perhaps I'm better at studying reading alone but actively collaborating with another student might be the way to go.

She used to read alone and that was good for her but this Monday when she was extra stressed due to her personal life with scheduled stuff, I felt her stress and then the rest of that day was &quot;destroyed&quot; for me.

I don't think I learnt anything yesterday on Monday April 6. Thus I think it is important to try new things and &quot;text only collaborative studying&quot; might do the trick. Perhaps combining this with an Etherpad session might help, for note-taking.

Last time I hesitated showing my schedule but this time I think it would just be common sense so here it is:

UTC TIME: 07:30 - 09:30 (2 hours a day, 6 hours a week)

* Monday
* Thursday
* Saturday

== w 15, Thursday, April 9 ==

I insisted on trying the new &quot;study technique&quot; where we do not talk at each other but my mother had a greater need in talking. So it happened all through a conversation: 

* She: &quot;Lets talk for a while&quot;
* Me: &quot;But do you think it will help you learn?&quot; and &quot;If you believe it helps you sure let's talk&quot;
* She: &quot;Ok&quot;
* Me: &quot;but I will connect because of your learning&quot;

That's what happened, then we talked using 'live' voice chat.(not through Coursera)
It did not go as I had planned with other words but since I'm not only here for me but I am mainly a &quot;group person&quot; and if I'm alone I'll need &quot;hand-holding&quot; even if I'm an adult cause I feel like I have a &quot;similar to PTSD&quot; background that somehow made it very hard for me to study in my earlier years and I wouldn't be able to take this course solo because of that, I need a &quot;study buddy&quot; cause otherwise I easily isolate myself from the outside world.

She learnt with my help that the &quot;Discussion Forums&quot; posts she made were &quot;breaking a silence&quot; of &quot;a year ago&quot; and she laughed and I intended it as a joke to lighten up the mood after we found it difficult to reach fellow classmates through the forums.
The most recent post I can find now is from 4 months ago from another user. That probably means there are probably other classmates too but the forum inactivity(not exactly &quot;alive&quot; with activity at 2026-04-09 UTC) might mean there is nobody to collaborate with right now.
Also I have noticed users on the forums with the label &quot;unenrolled&quot;. Perhaps the change to charge for every lesson happened within last year?

Today we started with 1.8 and read until the end but then we had an argument about continuing to module 2 or first doing the Readings and Quiz before we go on to module 2 and that took some time. That's where we stopped. It took us our usual 2 hours. As another Wikiversity user told me( this is my interpretation of what they told me: ) Coursera's user interface may be difficult to work with, it may take some time to learn how to use it.

Tomorrow we will study together instead of on Saturday because &quot;real life&quot; caught up with at least me regarding that day.

== w 15, Friday, April 10 ==

I made a mistake. When my mother found something on Wikibooks it was actually referenced in the Coursera course.
I'm happy though it was Wikibooks...any Wikimedia project that we get involved in I think is a good thing!

We read on Wikibooks for the full 2 hours we had. [[wikibooks:Introduction to Sociology|Introduction to Sociology]].

The PDF file is called &quot;Introduction_to_Sociology-v3.0.pdf&quot;. It is version 3 of the Introduction to Sociology.

We gotta read from page 5-15 but these are not &quot;technical page numbers&quot;. The actual page numbers are 11-21 of 443.

Next we're gonna read &quot;2.2.4 Quantitative vs. Qualitative Sociology&quot;. That we might do on Sunday instead of Monday.

== w 16, Tuesday, April 14 - Learn button super important on the app! ==

'''Learn''' on Coursera is a button on the app.

Super important. It took us almost an hour to figure out it was that button.

My mother made an observation: if she reads the text out loud it may be harder to understand it so she reads silently after the reading out loud part. I'm not sure I learn well while reading out loud but it creates a sort of &quot;human connection with the material&quot; that I guess is a &quot;placeholder&quot; for me until I figure out how I learn best.

We studied for 2 hours. My mother continues cause she dislikes &quot;leaving something in the middle&quot;.

Tomorrow we study at 7:30.

Next time we/I read from: 

&quot;2.4 Sociology Today&quot; page 14 (20 of 443 &quot;A4 paper&quot;)

== w 16, Wednesday, April 15 - Coursera, fix my mother's problems in your app ==

April 14 notes were *SUPER IMPORTANT*. I got help from it today and it helped me a lot! Otherwise who knows how the next hour will be lost and the next and the next?

My mother is very upset with Coursera not being helpful. Too much &quot;ChatGPT coach&quot; and not enough &quot;tech support&quot; I guess cause, we are talking right now and she is clearly upset but learning how to make notes while listening to somebody/my mother being upset is a good challenge, for me to stay calm while this is happening.

08:15 Coursera is causing us problems. Discrepancy in what is shown to a user.
The discrepancy is specifically about the &quot;ChatGPT coach&quot; appearing in one of our apps and not in the other. So we are not synced.

08:59 My mother *ACTUALLY* got a bug on her version of Coursera. She presses on Start on the quiz and nothing happens. All she got was a &quot;ChatGPT coach&quot;.
So:
1. Find coursera support and attach the image of nothing happening
2. Email the university that put the course on coursera

My mother doesn't want to take a video of her pushing the &quot;Start&quot; button. So we'll just tell them what we did and that nothing happened. She has internet because she could open Wikibooks moments before which requires internet.

09:31 we are done. We have studied what we could. Our next thing was watching the video &quot;2.1 The Fable of the Bees&quot;.
I could not concentrate based on what happened today. I'm still in &quot;distress&quot; but I did what I could on my end.

Also that the person in the video is ... I need to research this. Maybe I can help Wikidata in adding accurate data about whatever that was about 

In Memoriam: bartvanheerikhuizen.nl/2023/02/in-memoriam/ I read that today. That may be the reference I add if it needs to be added on Wikidata.

I'm just happy with whatever opportunity I have to stay on Wikimedia projects, that is my ultimate goal.

== w 17, Monday, April 20 ==

Module 1, &quot;Classical Sociological Theory - An Introduction&quot;, the quiz part which is the final part of Module 1. My mother had a bug on her end so I suggested today that we do it together from my end by me telling her loudly. This is what I'll try to do now.

My mother's relation to technical problems and the digital divide:
* No, she doesn't need to input her email. She only needs to login with Google. It looks by what she tells me that she must have done that. I'm assuming by what she tells me that she has logged in.
* She can see the questions, with other words she can at least read the Quiz now. Before she couldn't.
* She can also choose the answers she now tells me. Great progress! We have worked through the Coursera bug she got!
* I'm missing a vital detail: I sent her the exact link to the quiz from the web and '''that helped her'''! I missed that vital part! I need to always send her the permalink of every module.

A healthy way to see how we study together: See us as &quot;one unit&quot; rather than separate. Pooling our resources. That we are &quot;one being&quot; conceptually and that this will help us in the long run based on what we learn.

We are probably not aware how powerful of an influence Google has over Android and the older generation to what is known as the &quot;Digital Divide&quot;. The older generation may see computer technology as a &quot;diety&quot; that cannot be &quot;comprehended&quot;. This could be a great danger to our society that may be actively destabilizing it.(fear/feeling)

&quot;Turning on app archiving on the tablet is a GREAT choice!&quot; I think I wanted to say to my mother but I don't feel like I get many answers or feedback to what I'm saying to her but that's ok. Neurodiversity is extremely complex!

My mother's comments: Coursera is a very isolated place. There is nobody to communicate with. Coursera isolates people? How can Coursera be improved or what can we do on Wikiversity so that students who are stuck on Coursera can find help? Here on Wikiversity?

Now we are updating Google Chrome on the tablet which is great news!

Etherpad has helped me immensely in giving her links and making sure we are opening the right apps and knowing what apps we are using on the tablet. I don't have physical access to the tablet so I need to give links to make sure she is opening the right apps.

I got a &quot;crazy idea&quot; that might work or maybe won't work but maybe will work if I work hard enough on it. Social networking with her contacts. Asking her social contacts if they would find &quot;Classical Sociological Theory&quot; interesting and if they'd like to study it with my mother because two minds are better than 1 when tackling difficult technical challenges like it is for people who have to deal with the &quot;Digital Divide&quot;.

Despite the distress that I caused for myself I feel good about the time we spent on this today and the feeling was mutual! 

Next time we begin on chapter 2.1. So basically module 2.

== w 17, Friday, April 24 ==

My mother has finally &quot;let go&quot; of me. She is now reading by herself and 

she said &quot;I begin my study.&quot; in Greek and 
I replied &quot;Me too!&quot; because I probably am too tense when talking with my mother. I probably need to get better at communicating with my mother.

I didn't do anything yet. Instead I fell headlong into &quot;my idea&quot;. Me and my mother need a stable technique and I think we reading on our own while saying which days we are going to study and what hours is a good thing but ie. I might say I'll study Monday, Wednesday and on Sunday and my mother might go I might study on Tuesday, Wednesday and Saturday and &quot;during these hours&quot;.

That make us independent from each other and then my plan was that on Mondays we talk about how the week went.

Using &quot;AI Mode&quot; by Google to create a plan for myself. Now I no longer have the worry of how the studying will go with my mother. I can study on my own terms and my mother can too so I can finally be interested maybe in the parts that interest me and let go of the other parts that I find irrelevant regardless of what the person who created the course on this for-profit site Coursera think.

It's important to think for yourself! Anyway now I'll go to the top of the page to add an {{tl|AI-generated}} Template

It's possible I'm &quot;allergic&quot; to the word &quot;studying&quot;. Maybe school was traumatizing for my learning experience. What if everyone just said &quot;read&quot; instead. This may be a purely &quot;psychological&quot; phenomenon going on with me.

Thanks to &quot;AI Mode&quot; I got &quot;word&quot; of [[Wikiversity:Learning goals]]. I can share my learning goals. Thanks to &quot;AI Mode&quot; i learnt that they are talking there. I just needed a program that I can give the ball to, it can give it back to me, back and forth etc. etc. because on my own I go back into my &quot;mental health&quot; and &quot;depression&quot; but &quot;AI Mode&quot; actually helps me to be here with you all.

== w 17, Saturday, April 25 ==

She needs to reset her password...that's cool.

I started &quot;studying&quot; on Coursera when I had not published my idea here, when {{tl|AI-generated}} template was probably not created yet, unless I don't know what I'm talking about.

Now what I'd like to put my focus on is the idea. I don't seem to have infinite time.

== w 18, Tuesday, April 28 ==

We are continuing on independence. Tomorrow she invites me a certain time to get in touch with her as she has accelerated her learning/studying methods and I've let her &quot;run&quot;...because she learns better that way.

I have stalled. I think that in the long run it's best for me to &quot;find my way&quot; rather than going forward through &quot;hand-holding&quot; that slows my mother down. She has experience with studies and I'm sort of a &quot;liability&quot; in that regard.

Time to find my own way. Time to leave whatever PTSD-like I have behind me and see:
* How do I learn?
* What's the best way for me to learn?
* Maybe I should make more notes about exactly what I'm doing.
** When I'm spending time with things am I doing it to &quot;flee from the studies&quot;?
*** Can I make &quot;logs&quot; of what's happening or is this a &quot;psychiatric issue&quot; that is hard to solve?</text>
      <sha1>r4zddvf1s8og9bm4yvbz25xwwgzrhgi</sha1>
    </revision>
  </page>
  <page>
    <title>Draft:The Neurodiversity-inspired Idea</title>
    <ns>118</ns>
    <id>329157</id>
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      <id>2806966</id>
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      <contributor>
        <username>ThinkingScience</username>
        <id>3061446</id>
      </contributor>
      <comment>/* Naming Suggestions */ some of the names may seem less serious than other name ideas</comment>
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      <text bytes="7483" sha1="kurpmw4e85f0noyebydo11psgu8i4jd" xml:space="preserve">{{Research project|status=draft}}
{{AI-generated}}

{{Notice|'''Please excuse mistakes and problems''' this is a work in progress and pages may be published which are unfinished and that contain unfinished sentences and repetitions}}

== Explanation regarding {{tl|AI-generated}} template presence on this page ==

* Some questions that can be asked have been generated where there is a note about it. (&quot;AI Mode&quot; by Google)
* If something has been generated by an &quot;AI&quot;/LLM then please make a note of that so the reader knows. Also please document the specific &quot;AI name&quot;, ie. &quot;AI Mode&quot;, &quot;GPT-5 mini&quot; etc. as long as that name is enough to find the AI/LLM on Wikidata or on Wikipedia.

For questions that have been &quot;AI-Generated&quot; this section has been created to document the queries/input:
* [[Draft:The Neurodiversity-inspired Idea/AI Prompt History for Questions|Please document your input and output here when interacting with an AI/LLM]]

== Hypothes-is/-es ==

Preliminary date for basics of hypothesis/hypotheses being developed: June 16, 2026.

== Original Motivations ==

This section can list motivations by each user who contributes content or questions to this page.
* '''User:ThinkingScience''' My motivation is related to perceived limited progress by psychiatry and getting inspired by writers exploring Neurodiversity topics. I don't feel I have a right to have an opinion about psychiatry considering this idea's methodology is being developed during the publishing(and before) of this edit. It is my hope that if this idea develops how I expect it to it will be an &quot;extra parameter&quot; that some other sciences can use, including psychiatry. It is my hope that this idea will thus help psychiatry develop in a great way though my personal hope is it will help sociology more.
* Example user x
* Example user y etc.

== &quot;Do no harm&quot; ==

This section can list ideas/comments by users who are trying to do no harm while using their methodology(which should be documented on this page, if possible!):
* '''User:ThinkingScience''' Considering I am watching videos of famous people in interviews. I am making notes...my goal should be that not only my public notes are following the &quot;Do no harm&quot; but that my private notes do as well. That can be my goal for now. We'll see how this develops...
* Example user x
* Example user y etc.

Do no harm links from [[Wikiversity:Research ethics]] into Wikipedia with no examples for Wikiversity users specifically, yet.

== &quot;Research projects must fully document the methods&quot; ==

{{quote|Safety - Research must be conducted in a safe and lawful manner. Do no harm.}}

An idea for the minimum age of the subjects which are &quot;studied&quot; in a &quot;Do no harm&quot; way, if even possible, might be 29 years of age. There is disagreement regarding the exact age of a person when their brain matures and it may benefit the student/contributor if the &quot;starting age&quot; or &quot;minimum age&quot; is high enough so that a person with a &quot;fully matured brain&quot; also has some experience living with that fully matured brain.

💡💡💡'''Suggestion: 29 minimum years of age for anyone participating in this project.'''💡💡💡 for the sake of &quot;Do no harm&quot;?

which is described on [[Wikiversity:Research ethics]] that links to Wikipedia. This section needs work.

One of the methodologies is to watch a video, ie. a video interview of famous people or footage where the researcher has gotten legal access and specific consent from any person appearing in the video footage.

Methodologies need to be developed where data is gathered in a way that adheres to &quot;Do no harm&quot;.

=== Focusing on creating a &quot;Do no harm&quot;-compliant method ===

This needs to be developed.

This sub-page is created so we can make video notes. We can watch a video and then we can make video notes. We must do the video notes by following [[Wikiversity:Research ethics]] and &quot;Do no harm&quot;. How to do that can be tricky. &quot;Do no harm&quot; links to Wikipedia because we don't have our own resource where we help you how to do that.

=== method of interacting with draft and other pages on Wikiversity ===

&quot;AI Mode&quot; by Google can be used to get inspired by what kind of things to focus on, including if one thinks they started &quot;blathering&quot; and the text started to grow 'for no apparent reason' because the user landed in a &quot;non-productive behavior&quot; and the repeating themselves kept going on and on.

Prompts that generate questions and other things could be added into a subsection of this draft research

=== Video Notes before the creation of a more 'stable' method that adheres to &quot;Do no harm&quot; ===

* [[/Method_development_through_video_notes|Video Notes]]

== Questions that might encourage the development of this idea and its methodology ==

Questions that encouraged certain dates of time:
* June 16: Two months after the creation of the project some 'basic form' of a/- hypothes-is/-es should exist. ie. measuring rapport. How rapport can/will be measured. What tools can be used. How one makes &quot;first contact&quot; with people. Challenges and work to be done regarding the &quot;main hypothesis&quot;. This is a prediction. Time will tell if a &quot;main hypothesis&quot; will be ready by June 16.

'''The New Way: 100% human generated questions and answers''':

If a student/contributor figures out their own questions without the help of an AI/&quot;AI&quot;/LLM please contribute in this section. Otherwise go to &quot;The Old Way&quot; section and contribute there.

Questions in no particular order:
* Is this &quot;idea&quot; testable?
- It is in the plans for it to be testable '''but not testable yet'''.

* Is it possible to create one or more hypotheses based on this &quot;idea&quot;?
- Multiple hypotheses have been made but never published. Plan is to publish hypotheses related to the &quot;idea&quot; in the future.
* Is there a hypothesis or a number of hypotheses related to the idea?
** Can this idea be proven false?
- It can not yet be proven false considering a hypothesis is not currently described. Goal is to have a hypothesis at least before June 16, 2026 which is 2 months roughly after the first creation of the page. 
A basic hypothesis structure may/should exist by then but a complete hypothesis being created that is &quot;perfect in some ways&quot; that can't be predicted yet.

'''The Old Way'''

This is for any AI/LLM that is available to you. Please document the output here and if you want to contribute the input prompts and more specific details go to the subpage [[Draft:The_Neurodiversity-inspired_Idea/AI_Prompt_History_for_Questions]]

Questions and 'follow up'-/improved questions generated by Google &quot;AI Mode&quot;:
* What is missing right now?
** &quot;Improved&quot; version: &quot;What key sections are missing from this research draft to meet Wikiversity standards?&quot;
* How will we know if the idea is working?

== Naming Suggestions ==

Feel free to edit/modify or remove content in this section.

Example name ideas:
* Pre-research: Observations made inside psychiatry spectrums
* Psychiatric Spectrum Specifications

''These names should probably not be used but were kept for inspiring questions in readers'':
* Next-Gen Brain Types
* Next-Gen Sociology
* Next-Gen Neurotypes

== References/in-Wikiversity-Wikilinks ==

This section is for linking to specific references:

* [2] - [[Draft:The_Neurodiversity-inspired_Idea/AI_Prompt_History_for_Questions#Reference_2]]


== Future references to this draft ==

In the event that other publications start referring to this draft in the future, the template &quot;findsources&quot; is added:

{{findsources}}</text>
      <sha1>kurpmw4e85f0noyebydo11psgu8i4jd</sha1>
    </revision>
    <revision>
      <id>2806973</id>
      <parentid>2806966</parentid>
      <timestamp>2026-04-29T09:59:55Z</timestamp>
      <contributor>
        <username>ThinkingScience</username>
        <id>3061446</id>
      </contributor>
      <comment>/* Hypothes-is/-es */ Publishing this so I don't change anything accidentally</comment>
      <origin>2806973</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8728" sha1="be9fl7gpjz272ibm5l1inigpv3g6cc3" xml:space="preserve">{{Research project|status=draft}}
{{AI-generated}}

{{Notice|'''Please excuse mistakes and problems''' this is a work in progress and pages may be published which are unfinished and that contain unfinished sentences and repetitions}}

== Explanation regarding {{tl|AI-generated}} template presence on this page ==

* Some questions that can be asked have been generated where there is a note about it. (&quot;AI Mode&quot; by Google)
* If something has been generated by an &quot;AI&quot;/LLM then please make a note of that so the reader knows. Also please document the specific &quot;AI name&quot;, ie. &quot;AI Mode&quot;, &quot;GPT-5 mini&quot; etc. as long as that name is enough to find the AI/LLM on Wikidata or on Wikipedia.

For questions that have been &quot;AI-Generated&quot; this section has been created to document the queries/input:
* [[Draft:The Neurodiversity-inspired Idea/AI Prompt History for Questions|Please document your input and output here when interacting with an AI/LLM]]

== Hypothes-is/-es ==

Preliminary date for basics of hypothesis/hypotheses being developed: June 16, 2026.

=== In preparation for Hypothes-is/-es: Core Assumption Example ===

Google's &quot;AI Mode&quot; was used on 2026-04-29 and the formation of an &quot;example Core Assumption&quot; was made. The following is input sent to &quot;AI Mode&quot;:

{{quote|This I posted previously:

{{quote|Example of a &quot;Core Assumption&quot;: That not &quot;Everybody is different&quot; in brain structure but that some humen show so great similarities between each other that one might say &quot;it is the same brain&quot;. If that fails, my idea fails completely.}}

This I'm thinking about posting now and I want your output based on this input:

Example of a &quot;Core Assumption&quot;: That not &quot;Everybody is different&quot; in brain structure but that some humen show so great similarities between each other that one might say &quot;it is the same brain&quot;. I assume that that my core assumption and &quot;main idea&quot; can be fully broken by measurement of Rapport. I assume that I can predict better rapport between individuals who interact for the first time compared to a random sample. If that fails, my idea fails completely.}}

The system in which predictions are made regarding the related hypotheses are assumed to be found in the data. Studying the data should lead to understanding how to make the predictions in the hypotheses.

== Original Motivations ==

This section can list motivations by each user who contributes content or questions to this page.
* '''User:ThinkingScience''' My motivation is related to perceived limited progress by psychiatry and getting inspired by writers exploring Neurodiversity topics. I don't feel I have a right to have an opinion about psychiatry considering this idea's methodology is being developed during the publishing(and before) of this edit. It is my hope that if this idea develops how I expect it to it will be an &quot;extra parameter&quot; that some other sciences can use, including psychiatry. It is my hope that this idea will thus help psychiatry develop in a great way though my personal hope is it will help sociology more.
* Example user x
* Example user y etc.

== &quot;Do no harm&quot; ==

This section can list ideas/comments by users who are trying to do no harm while using their methodology(which should be documented on this page, if possible!):
* '''User:ThinkingScience''' Considering I am watching videos of famous people in interviews. I am making notes...my goal should be that not only my public notes are following the &quot;Do no harm&quot; but that my private notes do as well. That can be my goal for now. We'll see how this develops...
* Example user x
* Example user y etc.

Do no harm links from [[Wikiversity:Research ethics]] into Wikipedia with no examples for Wikiversity users specifically, yet.

== &quot;Research projects must fully document the methods&quot; ==

{{quote|Safety - Research must be conducted in a safe and lawful manner. Do no harm.}}

An idea for the minimum age of the subjects which are &quot;studied&quot; in a &quot;Do no harm&quot; way, if even possible, might be 29 years of age. There is disagreement regarding the exact age of a person when their brain matures and it may benefit the student/contributor if the &quot;starting age&quot; or &quot;minimum age&quot; is high enough so that a person with a &quot;fully matured brain&quot; also has some experience living with that fully matured brain.

💡💡💡'''Suggestion: 29 minimum years of age for anyone participating in this project.'''💡💡💡 for the sake of &quot;Do no harm&quot;?

which is described on [[Wikiversity:Research ethics]] that links to Wikipedia. This section needs work.

One of the methodologies is to watch a video, ie. a video interview of famous people or footage where the researcher has gotten legal access and specific consent from any person appearing in the video footage.

Methodologies need to be developed where data is gathered in a way that adheres to &quot;Do no harm&quot;.

=== Focusing on creating a &quot;Do no harm&quot;-compliant method ===

This needs to be developed.

This sub-page is created so we can make video notes. We can watch a video and then we can make video notes. We must do the video notes by following [[Wikiversity:Research ethics]] and &quot;Do no harm&quot;. How to do that can be tricky. &quot;Do no harm&quot; links to Wikipedia because we don't have our own resource where we help you how to do that.

=== method of interacting with draft and other pages on Wikiversity ===

&quot;AI Mode&quot; by Google can be used to get inspired by what kind of things to focus on, including if one thinks they started &quot;blathering&quot; and the text started to grow 'for no apparent reason' because the user landed in a &quot;non-productive behavior&quot; and the repeating themselves kept going on and on.

Prompts that generate questions and other things could be added into a subsection of this draft research

=== Video Notes before the creation of a more 'stable' method that adheres to &quot;Do no harm&quot; ===

* [[/Method_development_through_video_notes|Video Notes]]

== Questions that might encourage the development of this idea and its methodology ==

Questions that encouraged certain dates of time:
* June 16: Two months after the creation of the project some 'basic form' of a/- hypothes-is/-es should exist. ie. measuring rapport. How rapport can/will be measured. What tools can be used. How one makes &quot;first contact&quot; with people. Challenges and work to be done regarding the &quot;main hypothesis&quot;. This is a prediction. Time will tell if a &quot;main hypothesis&quot; will be ready by June 16.

'''The New Way: 100% human generated questions and answers''':

If a student/contributor figures out their own questions without the help of an AI/&quot;AI&quot;/LLM please contribute in this section. Otherwise go to &quot;The Old Way&quot; section and contribute there.

Questions in no particular order:
* Is this &quot;idea&quot; testable?
- It is in the plans for it to be testable '''but not testable yet'''.

* Is it possible to create one or more hypotheses based on this &quot;idea&quot;?
- Multiple hypotheses have been made but never published. Plan is to publish hypotheses related to the &quot;idea&quot; in the future.
* Is there a hypothesis or a number of hypotheses related to the idea?
** Can this idea be proven false?
- It can not yet be proven false considering a hypothesis is not currently described. Goal is to have a hypothesis at least before June 16, 2026 which is 2 months roughly after the first creation of the page. 
A basic hypothesis structure may/should exist by then but a complete hypothesis being created that is &quot;perfect in some ways&quot; that can't be predicted yet.

'''The Old Way'''

This is for any AI/LLM that is available to you. Please document the output here and if you want to contribute the input prompts and more specific details go to the subpage [[Draft:The_Neurodiversity-inspired_Idea/AI_Prompt_History_for_Questions]]

Questions and 'follow up'-/improved questions generated by Google &quot;AI Mode&quot;:
* What is missing right now?
** &quot;Improved&quot; version: &quot;What key sections are missing from this research draft to meet Wikiversity standards?&quot;
* How will we know if the idea is working?

== Naming Suggestions ==

Feel free to edit/modify or remove content in this section.

Example name ideas:
* Pre-research: Observations made inside psychiatry spectrums
* Psychiatric Spectrum Specifications

''These names should probably not be used but were kept for inspiring questions in readers'':
* Next-Gen Brain Types
* Next-Gen Sociology
* Next-Gen Neurotypes

== References/in-Wikiversity-Wikilinks ==

This section is for linking to specific references:

* [2] - [[Draft:The_Neurodiversity-inspired_Idea/AI_Prompt_History_for_Questions#Reference_2]]


== Future references to this draft ==

In the event that other publications start referring to this draft in the future, the template &quot;findsources&quot; is added:

{{findsources}}</text>
      <sha1>be9fl7gpjz272ibm5l1inigpv3g6cc3</sha1>
    </revision>
    <revision>
      <id>2806974</id>
      <parentid>2806973</parentid>
      <timestamp>2026-04-29T10:01:46Z</timestamp>
      <contributor>
        <username>ThinkingScience</username>
        <id>3061446</id>
      </contributor>
      <comment>/* Hypothes-is/-es */ Mentioned data earlier. Making sure to express that there is no data, can probably not be any data without a prediction/hypothesis</comment>
      <origin>2806974</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8769" sha1="rhbmf046sd4u2as6b5spovyu8s0f4ub" xml:space="preserve">{{Research project|status=draft}}
{{AI-generated}}

{{Notice|'''Please excuse mistakes and problems''' this is a work in progress and pages may be published which are unfinished and that contain unfinished sentences and repetitions}}

== Explanation regarding {{tl|AI-generated}} template presence on this page ==

* Some questions that can be asked have been generated where there is a note about it. (&quot;AI Mode&quot; by Google)
* If something has been generated by an &quot;AI&quot;/LLM then please make a note of that so the reader knows. Also please document the specific &quot;AI name&quot;, ie. &quot;AI Mode&quot;, &quot;GPT-5 mini&quot; etc. as long as that name is enough to find the AI/LLM on Wikidata or on Wikipedia.

For questions that have been &quot;AI-Generated&quot; this section has been created to document the queries/input:
* [[Draft:The Neurodiversity-inspired Idea/AI Prompt History for Questions|Please document your input and output here when interacting with an AI/LLM]]

== Hypothes-is/-es ==

Preliminary date for basics of hypothesis/hypotheses being developed: June 16, 2026.

=== In preparation for Hypothes-is/-es: Core Assumption Example ===

Google's &quot;AI Mode&quot; was used on 2026-04-29 and the formation of an &quot;example Core Assumption&quot; was made. The following is input sent to &quot;AI Mode&quot;:

{{quote|This I posted previously:

{{quote|Example of a &quot;Core Assumption&quot;: That not &quot;Everybody is different&quot; in brain structure but that some humen show so great similarities between each other that one might say &quot;it is the same brain&quot;. If that fails, my idea fails completely.}}

This I'm thinking about posting now and I want your output based on this input:

Example of a &quot;Core Assumption&quot;: That not &quot;Everybody is different&quot; in brain structure but that some humen show so great similarities between each other that one might say &quot;it is the same brain&quot;. I assume that that my core assumption and &quot;main idea&quot; can be fully broken by measurement of Rapport. I assume that I can predict better rapport between individuals who interact for the first time compared to a random sample. If that fails, my idea fails completely.}}

The system in which predictions are made regarding the related hypotheses are assumed to be found in the data. Studying the data should lead to understanding how to make the predictions in the hypotheses.

== Data ==

There is currently no data.

== Original Motivations ==

This section can list motivations by each user who contributes content or questions to this page.
* '''User:ThinkingScience''' My motivation is related to perceived limited progress by psychiatry and getting inspired by writers exploring Neurodiversity topics. I don't feel I have a right to have an opinion about psychiatry considering this idea's methodology is being developed during the publishing(and before) of this edit. It is my hope that if this idea develops how I expect it to it will be an &quot;extra parameter&quot; that some other sciences can use, including psychiatry. It is my hope that this idea will thus help psychiatry develop in a great way though my personal hope is it will help sociology more.
* Example user x
* Example user y etc.

== &quot;Do no harm&quot; ==

This section can list ideas/comments by users who are trying to do no harm while using their methodology(which should be documented on this page, if possible!):
* '''User:ThinkingScience''' Considering I am watching videos of famous people in interviews. I am making notes...my goal should be that not only my public notes are following the &quot;Do no harm&quot; but that my private notes do as well. That can be my goal for now. We'll see how this develops...
* Example user x
* Example user y etc.

Do no harm links from [[Wikiversity:Research ethics]] into Wikipedia with no examples for Wikiversity users specifically, yet.

== &quot;Research projects must fully document the methods&quot; ==

{{quote|Safety - Research must be conducted in a safe and lawful manner. Do no harm.}}

An idea for the minimum age of the subjects which are &quot;studied&quot; in a &quot;Do no harm&quot; way, if even possible, might be 29 years of age. There is disagreement regarding the exact age of a person when their brain matures and it may benefit the student/contributor if the &quot;starting age&quot; or &quot;minimum age&quot; is high enough so that a person with a &quot;fully matured brain&quot; also has some experience living with that fully matured brain.

💡💡💡'''Suggestion: 29 minimum years of age for anyone participating in this project.'''💡💡💡 for the sake of &quot;Do no harm&quot;?

which is described on [[Wikiversity:Research ethics]] that links to Wikipedia. This section needs work.

One of the methodologies is to watch a video, ie. a video interview of famous people or footage where the researcher has gotten legal access and specific consent from any person appearing in the video footage.

Methodologies need to be developed where data is gathered in a way that adheres to &quot;Do no harm&quot;.

=== Focusing on creating a &quot;Do no harm&quot;-compliant method ===

This needs to be developed.

This sub-page is created so we can make video notes. We can watch a video and then we can make video notes. We must do the video notes by following [[Wikiversity:Research ethics]] and &quot;Do no harm&quot;. How to do that can be tricky. &quot;Do no harm&quot; links to Wikipedia because we don't have our own resource where we help you how to do that.

=== method of interacting with draft and other pages on Wikiversity ===

&quot;AI Mode&quot; by Google can be used to get inspired by what kind of things to focus on, including if one thinks they started &quot;blathering&quot; and the text started to grow 'for no apparent reason' because the user landed in a &quot;non-productive behavior&quot; and the repeating themselves kept going on and on.

Prompts that generate questions and other things could be added into a subsection of this draft research

=== Video Notes before the creation of a more 'stable' method that adheres to &quot;Do no harm&quot; ===

* [[/Method_development_through_video_notes|Video Notes]]

== Questions that might encourage the development of this idea and its methodology ==

Questions that encouraged certain dates of time:
* June 16: Two months after the creation of the project some 'basic form' of a/- hypothes-is/-es should exist. ie. measuring rapport. How rapport can/will be measured. What tools can be used. How one makes &quot;first contact&quot; with people. Challenges and work to be done regarding the &quot;main hypothesis&quot;. This is a prediction. Time will tell if a &quot;main hypothesis&quot; will be ready by June 16.

'''The New Way: 100% human generated questions and answers''':

If a student/contributor figures out their own questions without the help of an AI/&quot;AI&quot;/LLM please contribute in this section. Otherwise go to &quot;The Old Way&quot; section and contribute there.

Questions in no particular order:
* Is this &quot;idea&quot; testable?
- It is in the plans for it to be testable '''but not testable yet'''.

* Is it possible to create one or more hypotheses based on this &quot;idea&quot;?
- Multiple hypotheses have been made but never published. Plan is to publish hypotheses related to the &quot;idea&quot; in the future.
* Is there a hypothesis or a number of hypotheses related to the idea?
** Can this idea be proven false?
- It can not yet be proven false considering a hypothesis is not currently described. Goal is to have a hypothesis at least before June 16, 2026 which is 2 months roughly after the first creation of the page. 
A basic hypothesis structure may/should exist by then but a complete hypothesis being created that is &quot;perfect in some ways&quot; that can't be predicted yet.

'''The Old Way'''

This is for any AI/LLM that is available to you. Please document the output here and if you want to contribute the input prompts and more specific details go to the subpage [[Draft:The_Neurodiversity-inspired_Idea/AI_Prompt_History_for_Questions]]

Questions and 'follow up'-/improved questions generated by Google &quot;AI Mode&quot;:
* What is missing right now?
** &quot;Improved&quot; version: &quot;What key sections are missing from this research draft to meet Wikiversity standards?&quot;
* How will we know if the idea is working?

== Naming Suggestions ==

Feel free to edit/modify or remove content in this section.

Example name ideas:
* Pre-research: Observations made inside psychiatry spectrums
* Psychiatric Spectrum Specifications

''These names should probably not be used but were kept for inspiring questions in readers'':
* Next-Gen Brain Types
* Next-Gen Sociology
* Next-Gen Neurotypes

== References/in-Wikiversity-Wikilinks ==

This section is for linking to specific references:

* [2] - [[Draft:The_Neurodiversity-inspired_Idea/AI_Prompt_History_for_Questions#Reference_2]]


== Future references to this draft ==

In the event that other publications start referring to this draft in the future, the template &quot;findsources&quot; is added:

{{findsources}}</text>
      <sha1>rhbmf046sd4u2as6b5spovyu8s0f4ub</sha1>
    </revision>
  </page>
  <page>
    <title>User:ThinkingScience/ND Inspired Idea Notebook</title>
    <ns>2</ns>
    <id>329177</id>
    <revision>
      <id>2806967</id>
      <parentid>2806789</parentid>
      <timestamp>2026-04-29T08:49:43Z</timestamp>
      <contributor>
        <username>ThinkingScience</username>
        <id>3061446</id>
      </contributor>
      <comment>/* April 27, 2026 */ == April 29, 2026 ==</comment>
      <origin>2806967</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3819" sha1="99m8tz4xzuya95pqcrqtbes8epl2dki" xml:space="preserve">Template Links:
* {{tl|Draft}}
* {{tl|underconstruction}}

'''On this page I plan to add daily notes regarding [[Draft:The Neurodiversity-inspired Idea]].'''

== &quot;Diary&quot; ==

== April 18, 2026 ==

A suggestion I got was that it may help the project if I provide some questions along with the idea. Also to make a main space where I gather info about my progress but that will probably be the draft itself if I move forward. Now if I write a &quot;diary&quot; that will be only regarding the project.

Turned &quot;me language&quot; into expressing that everyone is welcome, that I don't &quot;own&quot; [[Draft:The Neurodiversity-inspired Idea]]. Now everything that says &quot;I did this&quot; &quot;I did that&quot; should be gone. I think this was an improvement of some sort.

Plan for next edits on the draft page: Add an &lt;nowiki&gt;&quot; == Old Methodology needing updating == &quot;&lt;/nowiki&gt; where I will add old methodologies where I had not planned ahead too good and the &quot;Do no harm&quot; I did not know about or could not focus on. That was before I created my own Draft that feels like it only happened some days ago.

Interaction I thought was an efficient method but how would methodology be modified today with what I know now and will know in the future?

== April 20, 2026 ==

Why does it seem like I'm the only one using the word &quot;methodology&quot;? Did the [[Wikiversity:Research ethics]] mention it?
* I cannot find it! I checked all infoboxes! It must have been generated and I probably never questioned it...until now.

== April 21, 2026 ==

I think I put a new subsection on the Draft space something that was related to developing my method/methodology into the &quot;Do no harm&quot;. Considering I have almost not developed anything but I still gotta work on this...to do...

== April 22, 2026 ==

I am yearning / looking forward to working on video notes in a &quot;Do no harm&quot; way. I don't feel like it has happened yet. I did make modifications but it may have increased complexity. A complexity that will make it harder for me to work or just different.

== April 27, 2026 ==

I met with my father and he is a friend of the sciences. One word: hypothesis. He asked what my main hypothesis is. Of course hypothesis is a way to test if the idea is sound or if it's for the trashcan. I'm glad he gave me this feedback or interest in trying to learn more.

Today I woke up being inspired by that:

* Is there a hypothesis or a number of hypotheses related to the idea?
** Can this idea be proven false?

Why it's important to prove a hypothesis false: so that we don't waste time on trash ideas. If they are not provable we give time to the creator of the suggestion/idea to prove make a hypothesis. Only when the creator fails to provide any sort of hypothesis and maybe suggests their own idea be removed because as hard as they tried they couldn't make a functioning hypothesis...then I guess that's one of the more 'natural' ways for a project to more naturally leave Wikiversity.

Deadlines etc. can help me keep moving...

What a hypothesis is: &quot;I predict this will happen&quot; and then checking results what happens and whether it fulfilled the prediction or not and sometimes we stumble upon new things we did not expect.

[[Operationalization]] is then also needed to make &quot;ambiguous ideas&quot; measurable. My father remembers that I &quot;wanted to save the world&quot; but it was nothing other than an observation I wanted to share with other people that I had made that began years earlier. June 16 should be 2 months roughly after the creation of the Draft:Idea...

== April 29, 2026 ==

Just made my father aware of this &quot;idea&quot; that so far doesn't have a published hypothesis of any kind, not even in a &quot;basic stage&quot;. Only the deadline for June 16 exists right now.

I'm happy he did not reject me working on it. He encourages me to work harder. I got a &quot;Great job so far!&quot; compliment.</text>
      <sha1>99m8tz4xzuya95pqcrqtbes8epl2dki</sha1>
    </revision>
    <revision>
      <id>2806968</id>
      <parentid>2806967</parentid>
      <timestamp>2026-04-29T09:03:34Z</timestamp>
      <contributor>
        <username>ThinkingScience</username>
        <id>3061446</id>
      </contributor>
      <comment>/* April 29, 2026 */ so my goal should be to try to break my own idea and to try to prove it wrong as much as I can?</comment>
      <origin>2806968</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4161" sha1="ps66q91ydckjt939xveorjkiixerchh" xml:space="preserve">Template Links:
* {{tl|Draft}}
* {{tl|underconstruction}}

'''On this page I plan to add daily notes regarding [[Draft:The Neurodiversity-inspired Idea]].'''

== &quot;Diary&quot; ==

== April 18, 2026 ==

A suggestion I got was that it may help the project if I provide some questions along with the idea. Also to make a main space where I gather info about my progress but that will probably be the draft itself if I move forward. Now if I write a &quot;diary&quot; that will be only regarding the project.

Turned &quot;me language&quot; into expressing that everyone is welcome, that I don't &quot;own&quot; [[Draft:The Neurodiversity-inspired Idea]]. Now everything that says &quot;I did this&quot; &quot;I did that&quot; should be gone. I think this was an improvement of some sort.

Plan for next edits on the draft page: Add an &lt;nowiki&gt;&quot; == Old Methodology needing updating == &quot;&lt;/nowiki&gt; where I will add old methodologies where I had not planned ahead too good and the &quot;Do no harm&quot; I did not know about or could not focus on. That was before I created my own Draft that feels like it only happened some days ago.

Interaction I thought was an efficient method but how would methodology be modified today with what I know now and will know in the future?

== April 20, 2026 ==

Why does it seem like I'm the only one using the word &quot;methodology&quot;? Did the [[Wikiversity:Research ethics]] mention it?
* I cannot find it! I checked all infoboxes! It must have been generated and I probably never questioned it...until now.

== April 21, 2026 ==

I think I put a new subsection on the Draft space something that was related to developing my method/methodology into the &quot;Do no harm&quot;. Considering I have almost not developed anything but I still gotta work on this...to do...

== April 22, 2026 ==

I am yearning / looking forward to working on video notes in a &quot;Do no harm&quot; way. I don't feel like it has happened yet. I did make modifications but it may have increased complexity. A complexity that will make it harder for me to work or just different.

== April 27, 2026 ==

I met with my father and he is a friend of the sciences. One word: hypothesis. He asked what my main hypothesis is. Of course hypothesis is a way to test if the idea is sound or if it's for the trashcan. I'm glad he gave me this feedback or interest in trying to learn more.

Today I woke up being inspired by that:

* Is there a hypothesis or a number of hypotheses related to the idea?
** Can this idea be proven false?

Why it's important to prove a hypothesis false: so that we don't waste time on trash ideas. If they are not provable we give time to the creator of the suggestion/idea to prove make a hypothesis. Only when the creator fails to provide any sort of hypothesis and maybe suggests their own idea be removed because as hard as they tried they couldn't make a functioning hypothesis...then I guess that's one of the more 'natural' ways for a project to more naturally leave Wikiversity.

Deadlines etc. can help me keep moving...

What a hypothesis is: &quot;I predict this will happen&quot; and then checking results what happens and whether it fulfilled the prediction or not and sometimes we stumble upon new things we did not expect.

[[Operationalization]] is then also needed to make &quot;ambiguous ideas&quot; measurable. My father remembers that I &quot;wanted to save the world&quot; but it was nothing other than an observation I wanted to share with other people that I had made that began years earlier. June 16 should be 2 months roughly after the creation of the Draft:Idea...

== April 29, 2026 ==

Just made my father aware of this &quot;idea&quot; that so far doesn't have a published hypothesis of any kind, not even in a &quot;basic stage&quot;. Only the deadline for June 16 exists right now.

I'm happy he did not reject me working on it. He encourages me to work harder. I got a &quot;Great job so far!&quot; compliment.

=== Google's &quot;AI Mode&quot; ===

This was part was 100% AI-generated:

{{quote|What you are describing is a core concept in the philosophy of science called Falsification. It was popularized by Karl Popper, who argued that science doesn't progress by &quot;proving things right,&quot; but by rigorously trying to prove things wrong and failing to do so.}}</text>
      <sha1>ps66q91ydckjt939xveorjkiixerchh</sha1>
    </revision>
  </page>
  <page>
    <title>For-profit Internet companies harm children and threaten democracy</title>
    <ns>0</ns>
    <id>329315</id>
    <redirect title="Online platforms&#039; effects on public health, safety and democracy" />
    <revision>
      <id>2806873</id>
      <parentid>2806828</parentid>
      <timestamp>2026-04-28T14:51:40Z</timestamp>
      <contributor>
        <username>DavidMCEddy</username>
        <id>218607</id>
      </contributor>
      <comment>wdsmth</comment>
      <origin>2806873</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10513" sha1="45ds0ml3gu77tgg0hmjlwzdfgcnbrg2" xml:space="preserve">:''This discusses a 2026-05-06 interview with [[w:Yael Eisenstat|Yael Eisenstat]] about how some for-profit Internet companies have contributed to harming children and undermining democracy. A video and 29:00 mm:ss podcast excerpted from the interview will be added when available. The podcast will be released 2026-05-16 to the fortnightly &quot;Media &amp; Democracy&quot; show&lt;ref name=M&amp;D&gt;&lt;!--Media &amp; Democracy--&gt;{{cite Q|Q127839818}}&lt;/ref&gt; syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]&lt;ref&gt;&lt;!--Pacifica Radio Network--&gt;{{cite Q|Q2045587}}&lt;/ref&gt; Network of [[w:List of Pacifica Radio stations and affiliates|over 200 community radio stations]].''&lt;ref&gt;&lt;!--list of Pacifica Radio stations and affiliates--&gt;{{cite Q|Q6593294}}&lt;/ref&gt;

:''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]&lt;ref name=NPOV&gt;The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs&lt;/ref&gt; and treating others with respect.''&lt;ref name=AGF&gt;[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] is different: Contributors there are asked to [[Wikinews:Never assume|&quot;Don't assume things; be skeptical about everything.&quot;]] That's wise. However, we should still treat others with respect while being skeptical.&lt;/ref&gt;
&lt;!--[[File:2026-05-06 interview with Yaël Eisenstat.WebM|thumb|2026-05-06 interview with Yaël Eisenstat about how for-profit Internet companies harm children and threaten democracy.]]--&gt; 
&lt;!--[[File:2026-05-06 interview with Yaël Eisenstat.ogg|thumb|29:00 mm:ss excerpts from a 2026-05-06 interview with Yaël Eisenstat about how for-profit Internet companies harm children and threaten democracy.]]--&gt;

[[w:Yaël Eisenstat|Yaël Eisenstat]] discusses how some for-profit Internet companies have contributed to harming children and undermining democracy. Eisenstat is currently&lt;ref&gt;&lt;!--Yael Eisenstat--&gt;{{cite Q|Q82046593}}&lt;/ref&gt; the Director of Policy and Impact at Cybersecurity for Democracy(C4D),&lt;ref&gt;&lt;!--Cybersecurity for Democracy--&gt;{{cite Q|Q139568543}}&lt;/ref&gt; working on policy solutions for how to hold social media and other online platforms accountable for their effects on public safety and democracy. Previously, she was Vice President at the [[w:Anti-Defamation League|Anti-Defamation League]] (ADL) Center for Technology &amp; Society (CTS).  She was a Facebook election integrity head in 2018 and later became a whistleblower, speaking publicly about the dangers to democracy stemming from the company's decisions and product. She held other positions protecting democracy including as an intelligence officer, diplomat, and White House advisor.  C4D contributed to the recent March 24, 2026, jury verdicts in a civil case against Internet companies in [[w:New Mexico|New Mexico]].&lt;ref&gt;McQue (2026), Allyn (2026), &lt;!--C4D and the Courts: Meta Guilty Verdicts--&gt;{{cite Q|Q139572464}}&lt;/ref&gt;  

Eisenstat is interviewed by Spencer Graves.&lt;ref name=Graves&gt;&lt;!--Spencer Graves--&gt;{{cite Q|Q56452480}}&lt;/ref&gt; 

== Eisenstat's work == 

Eisenstat's work includes a [[w:TED (conference)|TED Talk]] and an [[w:South by Southwest|SXSW panel]] in 2020 and a 2024 Tech platforms and political violence. 

=== 2020 TED talk === 

In Eisenstat's (2020) TED talk, she said that around 2015 she began to notice that she was losing the ability to engage with others who were thought differently. Conversations with others in the US were becoming more difficult than conversations she had had as a CIA officer and diplomat drinking tea and talking with outspoken anti-Western clerics and suspected terrorists in Africa. Many of those engagements began with mutual suspicion but none degenerated into shouting or insults. In some cases she built collaboration on areas of mutual interest. Her most powerful tools were to listen, learn and build empathy. Most of her contacts wanted to feel heard, validated and respected. But social media companies like [[w:Facebook|Facebook]] incentivize inflammatory content contributing to a culture of political polarization and mistrust. This generates revenue for Facebook and similar companies that make money from clicks, &quot;because the shortest path to a click is anger or hate&quot;, in the words of [[Facebook whistleblower Frances Haugen says|Facebook whistleblower Frances Haugen]], interviewed 2024-08-19 for this Media &amp; Democracy series. When Facebook asked Eisenstat in 2018 to lead their work to support global elections integrity for political ads, she agreed. She left six months later, speaking openly about Facebook's inability to meet its responsibility to secure elections, subsequently documented, e.g., in the thousands of internal Facebook documents that [[w:Frances Haugen|Haugen]] released to the [[w:Securities and Exchange Commission|Securities and Exchange Commission]] and ''[[w:The Wall Street Journal|The Wall Street Journal]]'' in 2021.

=== 2020 SXSW panel === 

Eisenstat was part of a &quot;panel about the Future of Tech Responsibility&quot; for the 2020 [[w:South by Southwest|South by Southwest]] festival. The festival was cancelled due to COVID-19, but the panel was held virtually. This panel included a discussion of [[w:Section 230|Section 230]] of the [[w:Communications Act of 1934|Communications Act of 1934]], as amended by the [[w:Communications Decency Act|Communications Decency Act]] of 1996.&lt;ref&gt;Reid (2020).&lt;/ref&gt; It was &quot;written before platforms such as [[w:Facebook|Facebook]], [[w:YouTube|YouTube]] and [[w:X (social network)|Twitter]] existed&quot; -- written while [[w:Google|Google]] was a research project by [[w:Stanford University|Stanford]] [[w:Doctor of Philosophy|PhD]] students [[w:Larry Page|Larry Page]] and [[w:Sergey Brin|Sergey Brin]]. Section 230 includes, &quot;No provider or user of an interactive computer service shall be treated as the publisher or speaker of any information provided by another information content provider.&quot;&lt;ref&gt;&lt;!--47 U.S. Code § 230 - Protection for private blocking and screening of offensive material--&gt;{{cite Q|Q139570261}}&lt;/ref&gt; Eisenstat says that it's time to revisit Section 230, to demand accountability where Internet companies promote or suppress information based on the content while protecting web freedom otherwise. This is similar to the [[Dean Baker on Internet companies threatening democracy internationally and how to fix that|recommendations of]] [[w:Dean Baker|Dean Baker]] that when Internet companies make money by promoting information differentially based on content, they should be liable as are legacy media under the US Supreme Court decision in ''[[w:New York Times Co. v. Sullivan|NYT v. Sullivan]]'' (1964). In other cases, they should be treated as [[w:Common carrier|common carrier]]s like telephone companies. 

=== 2024: Tech platforms and political violence === 

More recently, Eisenstat et al. (2024a, b) are insisting that, &quot;Tech Platforms Must Do More to Avoid Contributing to Potential Political Violence&quot;. ''The New York Times'' had reported that, &quot;a steady undercurrent of violence and physical risk has become a new normal,&quot; particularly targeting public officials and democratic institutions. A survey from the Brennan Center found that 38% of election officials have experienced violent threats. They attributed these threats primarily to tech platforms and gave seven recommendations in four themes &quot;congruent with any number of papers that academics and civil society leaders have published over the years.&quot; They said that platforms 
* must develop robust standards for threat assessment and engage in scenario planning, crisis training, and engagement with external stakeholders, with as much transparency as possible.
* should enforce clear and actionable content moderation policies that address election integrity.
* should enforce their rules uniformly, not exempting politicians and other political influencers. 
* must clearly explain important content moderation decisions, ensuring transparency especially when it comes to high profile accounts.

They hope that increasing demands for accountability will prompt platforms to act more responsibly and prioritize the risk of political violence both in the United States and abroad. 

== The need for media reform to improve democracy == 

This article is part of [[:category:Media reform to improve democracy]]. A summary of episodes to 2025-11-15 is available in [[Media &amp; Democracy lessons for the future]]. 

==Discussion ==

:''[Interested readers are invite to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]&lt;ref name=NPOV/&gt; and treating others with respect.&lt;ref name=AGF/&gt;]''

== Notes ==
{{reflist}}

== Bibliography ==

* &lt;!--Bobby Allyn (2026-03-25) &quot;Jury finds Meta and Google negligent in social media harms trial--&gt;{{cite Q|Q139572103}}
* &lt;!--Yaël Eisenstat (2020-08) &quot;Dear Facebook, this is how you're breaking democracy&quot;--&gt;{{cite Q|Q138844363}}
* &lt;!--Yaël Eisenstat (2021) &quot;Section 230 Revisited: Web Freedom vs Accountability--&gt;{{cite Q|Q139568755|date=2021}}
* &lt;!-- Yaël Eisenstat, Justin Hendrix, and Daniel Kreiss (2024-05-22, 2024a) &quot;
Preventing Tech-Fueled Political Violence: What online platforms can do to ensure that they do not contribute to election-related violence&quot;, The Bulletin of Technology &amp; Public Life--&gt;{{cite Q|Q139571027|date=2024a}} 
* &lt;!-- Yaël Eisenstat, Justin Hendrix, and Daniel Kreiss (2024-05-22, 2024b) &quot;
Tech Platforms Must Do More to Avoid Contributing to Potential Political Violence
&quot;, Tech Policy Press and Just Security--&gt;{{cite Q|Q139571163}}
* &lt;!--Katie McQue (2026-04-24) &quot; Meta ordered to pay $375m after being found liable in child exploitation case--&gt;{{cite Q|Q139572337}}
* &lt;!--Blake E. Reid (2020-09-04) &quot;Section 230 of… what?--&gt;{{cite Q|Q139570229}} 

[[Category:Media]]
[[Category:News]]
[[Category:Democracy]]
[[Category:Suicide]]
[[Category:Politics]]
[[Category:Google]]
[[Category:Internet]]
[[Category:Social media]]
[[Category:Media reform to improve democracy]]
&lt;!--list of categories
https://en.wikiversity.org/wiki/Wikiversity:Category_Review
[[Wikiversity:Category Review]]--&gt;</text>
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Abstract text goes here 
}}

==1. Introduction==

[[file:example image.png|thumb|left
| Image caption text goes here
(attribution: name of image creator, [https://creativecommons.org/licenses/by/3.0/deed.en CC-BY 3.0])
]]

'χξϛ'. I asked an AI chatbot how people of the Greek-speaking world wrote the number 666 in New Testament times, and that was what it said. I knew that, of course, but I really wanted to ask the next question about what I had been taught many years ago, that χξϛ was Satan's visual parody of the title of God’s Messiah, looking like χριστος on the outside but ξ (with its snake-like appearance) replacing everything in the middle between the two outer letters. I have wondered why I never come across it in any commentaries or dictionaries I consult, except perhaps in a few places on the internet. I asked the AI to evaluate the idea, and this is what it said:&lt;blockquote&gt;You’re not alone in noticing the visual resemblance between χξϛ (666) and χριστός (Christos, 'Christ') in Greek.… From a literary-symbolic standpoint, it’s an interesting idea.… However, this visual-letterplay interpretation is speculative and post hoc — there’s no strong evidence that early readers or the author intended the shape or graphic similarity of the letters to carry symbolic meaning. Greek readers were trained to read by sound and meaning, not by visually analysing the shape of words as we might today in a world of logos and brands.&lt;ref&gt;{{Cite web|url=https://chatgpt.com/share/681a24c3-0dbc-8012-9caa-aa81652de95a|title=Greek Numerals 666|last=ChatGPT|first=chatbot|date=8 May 2025|website=ChatGPT}}&lt;/ref&gt;&lt;/blockquote&gt;This got me thinking. Is it true that this visual interpretation is 'speculative and post hoc'? Is there really no evidence to consider?

The number 666 in Revelation 13:18 has been interpreted traditionally through the lens of gematria, usually proposing to link it to Nero Caesar or θηρίον (thērion, beast), while 'it has also been thought a parody on the divine number, seven, given Revelation’s use of seven and given other demonic parodies of the divine in Revelation'.&lt;ref&gt;{{Cite book|title=IVP Cultural Background Commentary (electronic edition for Olive Tree Bible software)|last=Keener|first=Craig S.|publisher=InterVarsity Press|year=2014|location=Downers Grove, IL}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://www.thegospelcoalition.org/article/why-is-the-number-of-the-beast-666/|title=Why Is the Number of the Beast 666?|last=Beale|first=G. K.|date=11 Feb. 2011|website=The Gospel Coalition}}&lt;/ref&gt; Another proposal has been made to interpret the number visually by taking χξϛ as seemingly consisting of 'the initial and final letters of the word Xριστος (Christos), Christ, … with the symbol of the serpent between them'.&lt;ref&gt;{{Cite web|url=https://levendwater.org/books/numbers/number_in_scripture_bullinger.pdf|title=Number in Scripture: Its Supernatural Design and Spiritual Significance, 4th ed. PDF file, (London: Eyre &amp; Spottiswoode Ltd., 1921), p. 49|last=Bullinger|first=E. W.|date=1921}}&lt;/ref&gt; Yet, it does not appear to have received as a credible option. This study proposes that there is adequate evidence from the context, both literary and historical, for interpreting χξϛ as a visual symbol and that taking the visual appearance of χξϛ as a layer of its symbolic meaning is neither speculative nor post hoc but will add to our understanding of what John saw.

== 2. Examining the Text in its Literary Context ==

The number χξϛ is found in Revelation 13, where John continues to narrate a vision he saw. It is a 'mark' which John saw the people who worshipped the beast receive on their right hands or foreheads. As they received it to bear on their body, it must have meant something to them, but what did it mean to them? John understood it, and as he described it, he expected his readers in Asia Minor to understand it also. For us today, our goal must be to establish the perception of the mark which first existed in the minds of those who received it, and the method for that is by analysing how the mark functioned in the scene of the vision as John reports it.

=== 2.1 Visual Nature of Revelation and of the Mark ===
Revelation finds itself in the Jewish apocalyptic tradition, characterised by imagery and symbolism. It opens by identifying itself as 'the revelation from Jesus Christ, which God gave him to show his servants what must soon take place' (1:1). John saw 'a door standing open in heaven' (4:1) and again 'heaven standing open' (19:11). The repeated invitation, 'Come…, I will show you…' (4:1; 17:1; 21:9), led John to report many things he saw. Thus, what John wrote to convey is fundamentally visual in nature. The Apocalypse, therefore, is not just a documented text of heard words but a documentary report of seen visions. It is a literary description of prophetic visions that are rich in imagery from the Hebrew Scriptures. 

In particular, χξϛ was the mark of the beast, i.e. a visual symbol of allegiance to be seen on the body of those who worshipped the beast. In the context of the whole story of Revelation, it is apparent that this was Satan's mimicry of God's sealing (i.e. marking) of his servants (7:3). Also, against the whole story of the Bible, it can be seen as a mimicry of the Jewish practice of visible demonstration of their allegiance to God (cf. Deut. 6:8). The visual nature of the mark, its function and its literary context suggests the importance of how χξϛ looked, i.e. its visual appearance to human eyes. 

=== 2.2 Visual Form of ΧΞϚ and its Function as Pictogram ===
The mark of the beast was the name of the beast, which was in turn the number of the name, and this number was 666. In the text of the latest edition of the Greek New Testament by the United Bible Societies, this number is written out fully in words as 'ἑξακόσιοι ἑξήκοντα ἕξ'.&lt;ref&gt;{{Cite book|title=The Greek New Testament, Fifth Revised Edition|last=ed. by Aland|first=Barbara (and others)|publisher=United Bible Societies|year=2014|location=Stuttgart}}&lt;/ref&gt; However, the Greek New Testament by Tyndale House makes a different choice, because the earliest manuscript witness (Papyrus 47 or P47, from mid-third century) shows that the number was written in an abbreviated form in ancient times.&lt;ref&gt;{{Cite book|title=The Greek New Testament|last=ed. by Jongkind|first=Dirk (and others)|publisher=Tyndale House|year=2017|location=Cainmbridge}}&lt;/ref&gt;

==== 2.2.1 Greek Numeral System ====
Today in English, numbers are commonly written using Arabic numerals, like 666, as a kind of shorthand notation, rather than writing fully in words, like 'six hundred and sixty-six'. The same was true in ancient Greek, except that they used letters from the Greek alphabet as numerals rather than the Arabic numerals, which incidentally should probably be described more accurately as Hindu-Arabic numerals, as they first developed in India before becoming adopted into the Arabic system around the seventh century or some time before that.&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Arabic_numerals&amp;oldid=1296826253|title=Arabic numerals|last=Wikipedia contributors|date=22 June 2025|website=Wikipedia, The Free Encyclopeida}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=History_of_the_Hindu%E2%80%93Arabic_numeral_system&amp;oldid=1264812857|title=History of the Hindu–Arabic numeral system|last=Wikipedia contributors,|date=23 December 2024|website=Wikipedia, The Free Encyclopeida}}&lt;/ref&gt;

The Greek system is the first attested alphabetic numeral system in the world, dating back to the sixth century BC, and called Ionic or Milesian because of its origin in west Asia Minor around Miletus in Ionia.&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Alphabetic_numeral_system&amp;oldid=1222860822|title=Alphabetic numeral system|last=Wikipedia contributors|date=8 May 2024|website=Wikipedia, The Free Encyclopedia}}&lt;/ref&gt; This numeral system continued to be used in Asia Minor well into the Roman period, which is directly relevant for the present study of Revelation, and these numerals were marked by a line above them (overline or overbar) to distinguish them from normal letters.&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Greek_numerals&amp;oldid=1295786959|title=Greek numerals|last=Wikipedia contributors|date=15 June 2025|website=Wikipedia, The Free Encyclopedia}}&lt;/ref&gt;

ΧΞϚ (with overline) is how the number appears in early manuscripts. Each of the three Greek letters employed to represent the number 666 had their numerical values as shown in Table 1 below:
{| class=&quot;wikitable&quot;
|+Table 1: Numerical Values of ΧΞϚ
!Letter
!Letter Name
!Numerical Value
|-
|Χ
|chi
|600
|-
|Ξ
|xi
|60
|-
|Ϛ
|stigma
|6
|}
Yet. it is their visual forms that need our particular attention, because the number was a visual symbol to be seen on the openly visible parts of the body of the beast-followers.

==== 2.2.2 Handwritten Form in Majuscule ====
At this point, it is important to note that lowercase letters had not yet been developed in the first century. What John saw and wrote down would have been in uppercase letters. And, of course, everything was handwritten, as it was long before the days of typesetting. As such, any consideration of the Greek letters for their visual forms must bear in mind how they appeared when handwritten in majuscule as found in early manuscripts. 

==== 2.2.3 Σ (sigma) and Ϛ (stigma) ====
Commonly, the Greek letter sigma is considered to have three forms: uppercase Σ, medial lowercase σ, and final lowercase ς. However, there were two extra lesser-known forms. Lunate sigma (uppercase Ϲ and lowercase ϲ), so called because of its visual resemblance to a crescent moon, came into usage from about fourth century BC and became a standard form of sigma during the late antiquity and Middle Ages.&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Sigma&amp;oldid=1298495170|title=Sigma|last=Wikipedia contributors|date=2 July 2025|website=Wikipedia, The Free Encyclopedia}}&lt;/ref&gt; As such, it is commonly found in many early manuscripts of the New Testament.

The image (Figure 1) below shows folio 7 of Papyrus 47, which contains the text from Revelation 13:16–14:10. The text on the ninth line says:

ΕϹΤΙΝ ΔΕ ΧΞϚ ΚΑΙ ΕΙΔΟΝ, ΚΑΙ ΕΙΔΟῪ ΑΡ(ΝΙΟΝ)

which looks a little more like this in a font designed for a greater visual resemblance to the handwritten text in the early manuscripts:&lt;ref&gt;{{Cite web|url=https://github.com/Center-for-New-Testament-Restoration/font|title=Koine Greek Font|last=Bunning|first=Alan|date=9 Oct. 2022|archive-url=Center for New Testament Restoration}}&lt;/ref&gt;

The number χξϛ is found in Revelation 13, where John continues to narrate a vision he saw. It is a 'mark' which John saw the people who worshipped the beast receive on their right hands or foreheads. As they received it to bear on their body, it must have meant something to them, but what did it mean to them? John understood it, and as he described it, he expected his readers in Asia Minor to understand it also. For us today, our goal must be to establish the perception of the mark which first existed in the minds of those who received it, and the method for that is by analysing how the mark functioned in the scene of the vision as John reports it.

=== 2.1 Visual Nature of Revelation and of the Mark ===
Revelation finds itself in the Jewish apocalyptic tradition, characterised by imagery and symbolism. It opens by identifying itself as 'the revelation from Jesus Christ, which God gave him to show his servants what must soon take place' (1:1). John saw 'a door standing open in heaven' (4:1) and again 'heaven standing open' (19:11). The repeated invitation, 'Come…, I will show you…' (4:1; 17:1; 21:9), led John to report many things he saw. Thus, what John wrote to convey is fundamentally visual in nature. The Apocalypse, therefore, is not just a documented text of heard words but a documentary report of seen visions. It is a literary description of prophetic visions that are rich in imagery from the Hebrew Scriptures. 

In particular, χξϛ was the mark of the beast, i.e. a visual symbol of allegiance to be seen on the body of those who worshipped the beast. In the context of the whole story of Revelation, it is apparent that this was Satan's mimicry of God's sealing (i.e. marking) of his servants (7:3). Also, against the whole story of the Bible, it can be seen as a mimicry of the Jewish practice of visible demonstration of their allegiance to God (cf. Deut. 6:8). The visual nature of the mark, its function and its literary context suggests the importance of how χξϛ looked, i.e. its visual appearance to human eyes. 

=== 2.2 Visual Form of ΧΞϚ and its Function as Pictogram ===
The mark of the beast was the name of the beast, which was in turn the number of the name, and this number was 666. In the text of the latest edition of the Greek New Testament by the United Bible Societies, this number is written out fully in words as 'ἑξακόσιοι ἑξήκοντα ἕξ'.&lt;ref&gt;{{Cite book|title=The Greek New Testament, Fifth Revised Edition|last=ed. by Aland|first=Barbara (and others)|publisher=United Bible Societies|year=2014|location=Stuttgart}}&lt;/ref&gt; However, the Greek New Testament by Tyndale House makes a different choice, because the earliest manuscript witness (Papyrus 47 or P47, from mid-third century) shows that the number was written in an abbreviated form in ancient times.&lt;ref&gt;{{Cite book|title=The Greek New Testament|last=ed. by Jongkind|first=Dirk (and others)|publisher=Tyndale House|year=2017|location=Cainmbridge}}&lt;/ref&gt;

==== 2.2.1 Greek Numeral System ====
Today in English, numbers are commonly written using Arabic numerals, like 666, as a kind of shorthand notation, rather than writing fully in words, like 'six hundred and sixty-six'. The same was true in ancient Greek, except that they used letters from the Greek alphabet as numerals rather than the Arabic numerals, which incidentally should probably be described more accurately as Hindu-Arabic numerals, as they first developed in India before becoming adopted into the Arabic system around the seventh century or some time before that.&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Arabic_numerals&amp;oldid=1296826253|title=Arabic numerals|last=Wikipedia contributors|date=22 June 2025|website=Wikipedia, The Free Encyclopeida}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=History_of_the_Hindu%E2%80%93Arabic_numeral_system&amp;oldid=1264812857|title=History of the Hindu–Arabic numeral system|last=Wikipedia contributors,|date=23 December 2024|website=Wikipedia, The Free Encyclopeida}}&lt;/ref&gt;

The Greek system is the first attested alphabetic numeral system in the world, dating back to the sixth century BC, and called Ionic or Milesian because of its origin in west Asia Minor around Miletus in Ionia.&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Alphabetic_numeral_system&amp;oldid=1222860822|title=Alphabetic numeral system|last=Wikipedia contributors|date=8 May 2024|website=Wikipedia, The Free Encyclopedia}}&lt;/ref&gt; This numeral system continued to be used in Asia Minor well into the Roman period, which is directly relevant for the present study of Revelation, and these numerals were marked by a line above them (overline or overbar) to distinguish them from normal letters.&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Greek_numerals&amp;oldid=1295786959|title=Greek numerals|last=Wikipedia contributors|date=15 June 2025|website=Wikipedia, The Free Encyclopedia}}&lt;/ref&gt;

ΧΞϚ (with overline) is how the number appears in early manuscripts. Each of the three Greek letters employed to represent the number 666 had their numerical values as shown in Table 1 below:
{| class=&quot;wikitable&quot;
|+Table 1: Numerical Values of ΧΞϚ
!Letter
!Letter Name
!Numerical Value
|-
|Χ
|chi
|600
|-
|Ξ
|xi
|60
|-
|Ϛ
|stigma
|6
|}
Yet. it is their visual forms that need our particular attention, because the number was a visual symbol to be seen on the openly visible parts of the body of the beast-followers.

==== 2.2.2 Handwritten Form in Majuscule ====
At this point, it is important to note that lowercase letters had not yet been developed in the first century. What John saw and wrote down would have been in uppercase letters. And, of course, everything was handwritten, as it was long before the days of typesetting. As such, any consideration of the Greek letters for their visual forms must bear in mind how they appeared when handwritten in majuscule as found in early manuscripts. 

==== 2.2.3 Σ (sigma) and Ϛ (stigma) ====
Commonly, the Greek letter sigma is considered to have three forms: uppercase Σ, medial lowercase σ, and final lowercase ς. However, there were two extra lesser-known forms. Lunate sigma (uppercase Ϲ and lowercase ϲ), so called because of its visual resemblance to a crescent moon, came into usage from about fourth century BC and became a standard form of sigma during the late antiquity and Middle Ages.&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Sigma&amp;oldid=1298495170|title=Sigma|last=Wikipedia contributors|date=2 July 2025|website=Wikipedia, The Free Encyclopedia}}&lt;/ref&gt; As such, it is commonly found in many early manuscripts of the New Testament.

The image (Figure 1) below shows folio 7 of Papyrus 47, which contains the text from Revelation 13:16–14:10.&lt;ref&gt;{{Cite web|url=https://viewer.cbl.ie/viewer/image/BP_III_f_7/1/|title=Revelation 13.16–14.4; 14.4–10|website=Chester Beatty Online Collections|access-date=16 May 2025}}&lt;/ref&gt; 

'''Figure 1: Papyrus 47 folio 7''' [[File:P47 folio 7 - Rev 13vv1-15.jpg|alt=Figure 1. P47 folio 7|frameless|1247x1247px]]

The text on the ninth line says:

ΕϹΤΙΝ ΔΕ ΧΞϚ ΚΑΙ ΕΙΔΟΝ, ΚΑΙ ΕΙΔΟῪ ΑΡ(ΝΙΟΝ)

which looks a little more like this in a font designed for a greater visual resemblance to the handwritten text in the early manuscripts:&lt;ref&gt;{{Cite web|url=https://github.com/Center-for-New-Testament-Restoration/font|title=Koine Greek Font|last=Bunning|first=Alan|date=9 Oct. 2022|archive-url=Center for New Testament Restoration}}&lt;/ref&gt;

[[File:Koine_Majuscule_text.png|frameless|380x380px]]

What should be observed here is the visual resemblance between Ϲ (crescent sigma, the second letter) and Ϛ (stigma, the tenth letter). This resemblance is perhaps not too surprising, considering the origin of Ϛ as a ligature of sigma (Σ) and tau (Τ).

==== 2.2.4 Nomina Sacra ====
Early Christians considered certain names and titles, like Θεός (Theos, God), Κύριος (Kyrios, Lord), Ἰησοῦς (Iēsous, Jesus), Χριστός (Christos, Christ), Υἱός (Huios, Son, referring to Jesus) and Πνεῦμα (Pneuma, Spirit, referring to the Holy Spirit), as nomina sacra (sacred names), to be treated with respect.&lt;ref name=&quot;:0&quot;&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Nomina_sacra&amp;oldid=1270217331|title=Nomina sacra|last=Wikipedia contributors|date=18 January 2025|website=Wikipedia, The Free Encyclopedia}}&lt;/ref&gt; Those names and titles were also words that occurred frequently in the manuscripts, and the scribes developed a practice of abbreviating them, usually by contraction, taking the first one or two letters and the last letter of the word, skipping all middle letters, and marking them with an overline to indicate abbreviation in the same way as when marking numbers written in Greek numerals.

Precisely when this practice arose is not known. However, the abbreviation practice in Greek literature predates Christian writings, going back to the fourth century BC, as the earliest known Western shorthand system was employed by the Greek historian Xenophon (a student of Socrates) in his work Ἀπομνημονεύματα (Memorabilia or Memoir of Socrates), which is considered to have been completed shortly after 371 BC.&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Scribal_abbreviation&amp;oldid=1283806604|title=Scribal abbreviation|last=Wikipedia contributors|date=3 April 2025|website=Wikipedia, The Free Encyclopedia}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Memorabilia_(Xenophon)&amp;oldid=1254008920|title=Memorabilia (Xenophon)|last=Wikipedia contributors|date=29 October 2024|website=Wikipedia, The Free Encyclopedia}}&lt;/ref&gt; In the light of this pre-Christian practice in the Greek literary tradition, it would have been natural for Christian writers to make use of it in their own writings. The manuscript evidence is that 'nomina sacra are consistently observed in even the earliest extant Christian writings, ... implying that when these were written, in approximately the second century, the practice had already been established for some time.'&lt;ref name=&quot;:0&quot; /&gt; It is, then, reasonable to estimate that origin of nomina sacra was early in the first century.

That, in turn, means that when Revelation was written toward the end of the first century, John and his readers would have been familiar with the practice of nomina sacra. More specifically, it would have been a normal and common experience for them to write or to see ΧϹ or ΧΡϹ for ΧΡΙϹΤΌϹ (Christos, Christ).

Initially, this practice was limited to only a handful of words, which are called nomina divina (divine names), as they all refer to persons of the Trinity as shown in the Table 2 below. However, the practice extended through the second and third centuries, and by the early Byzantine period in the fourth century, the extended practice was established to include the additional words in Table 3.&lt;ref&gt;{{Cite web|url=https://archive.org/details/bruce-m.-metzger-manuscripts-of-the-greek-bible.-an-introduction-to-palaeography/page/36/mode/1up|title=Manuscripts of the Greek Bible: An Introduction to Palaeography (Oxford: Oxford University Press, 1981), p. 36|last=Metzger|first=Bruce Manning|website=Internet Archive}}&lt;/ref&gt;
{| class=&quot;wikitable&quot;
|+Table 2: Nomina Divina
! rowspan=&quot;2&quot; |Greek
! rowspan=&quot;2&quot; |Transliteration
! rowspan=&quot;2&quot; |English
! colspan=&quot;2&quot; |Abbreviation
|-
!(Nominative Case)
!(Genitive Case)
|-
|Θεός
|Theos
|God
|ΘΣ
|ΘΥ
|-
|Κύριος
|Kyrios
|Lord
|ΚΣ
|ΚΥ
|-
|Ἰησοῦς
|Iēsous
|Jesus
|ΙΣ or ΙΗΣ
|ΙΥ
|-
|Χριστός
|Christos
|Christ
|ΧΣ or ΧΡΣ
|ΧΥ
|-
|Πνεῦμα
|Pneuma
|Spirit 
referring to the Holy Spirit
|ΠΝΑ
|ΠΝΣ
|}
{| class=&quot;wikitable&quot;
|+Table 3: Later Additions to Nomina Sacra
! rowspan=&quot;2&quot; |Greek
! rowspan=&quot;2&quot; |Transliteration
! rowspan=&quot;2&quot; |English
! colspan=&quot;2&quot; |Abbreviation
|-
!(Nominative Case)
!(Genitive Case)
|-
|Πατήρ
|''Patēr''
|Father
|ΠΗΡ
|ΠΡΣ
|-
|Σωτήρ
|''Sōtēr''
|Saviour
|ΣΗΡ
|ΣΡΣ
|-
|Σταυρός
|''Stauros''
|Cross
|ΣΤΣ
|ΣΤΥ
|-
|Μήτηρ
|''Mētēr''
|Mother
referring to Mary
|ΜΤΡ
|ΜΡΣ
|-
|Ἰσραήλ
|''Israēl''
|Israel
|ΙΗΛ
|
|-
|Ἄνθρωπος
|''Anthrōpos''
|Man
in the phrase 'Son of Man'
|ΑΝΟΣ
|ΑΝΟΥ
|-
|Ἰερουσαλήμ
|''Ierousalēm''
|Jerusalem
|ΙΛΗΜ
|
|-
|Οὐρανός
|''Ouranos''
|Heaven
|ΟΥΝΟΣ
|ΟΥΝΥ
|}
Examples of ''nomina sacra'' can be seen in the image (Figure 2) below of the third-century manuscript (P46 folio 62), containing the text of 2 Corinthians 1:16–2:1 and 2:3–12.&lt;ref&gt;{{Cite web|url=https://viewer.cbl.ie/viewer/image/BP_II_f_62/1/|title=2 Corinthians 1.16-2.1; 2.3-12|website=Chester Beatty Online Collections|access-date=16 May 2025}}&lt;/ref&gt; As it has already been shown above, the practice of ''nomina sacra'' was already in use by the time of Revelation. The value of looking at this third-century manuscript is that it shows what they looked like ''visually'', as the concern of this paper is the ''visual'' appearance of written words in order for them to function pictographically.

'''Figure 2: P46 folio 62'''[[File:P47 folio 7 - Rev 13vv1-15.jpg|alt=Figure 2: P46 folio 62|frameless|1247x1247px]]

In the manuscript, the text on the ninth line reads:

ΓΑΡ ΘΥ ΥΙϹ ΙΗϹ ΧΡϹ Ο ΕΝ ΥΜΕΙ͂Ν ΔΙ Η(ΜΩ͂Ν)

which looks more like:

[[File:Koine Majuscule text 2.png|frameless|380x380px]]

where ΘΕΟΥ͂ ΥΙΟϹ ΙΗϹΟΥ͂Ϲ ΧΡΙϹΤῸϹ ('God's Son Jesus Christ') is written in shorthand as ΘΥ ΥΙϹ ΙΗϹ ΧΡϹ (all with overline). The point to note here is that if the first-century readers were accustomed to seeing ΧϹ or ΧΡϹ (both with overline), written with an overline above it, as a shorthand for ΧΡΙϹΤΟϹ (''Christos'', Christ), then John and his readers would have been so familiar with [[File:Christ in shorthand.png|frameless|40x40px]] as a rightful title of their Lord that they would have been quick to see the blasphemy in [[File:Parody-christ in shorthand.png|frameless|42x42px]] being used by the beast as its mark.

==Third Heading, etc==


==Additional information==

===Acknowledgements===
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===Competing interests===
Any conflicts of interest that you would like to declare. Otherwise, a statement that the authors have no competing interest.

===Ethics statement===
An ethics statement, if appropriate, on any animal or human research performed should be included here or in the methods section.

==References==
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Abstract text goes here 
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==1. Introduction==

[[file:example image.png|thumb|left
| Image caption text goes here
(attribution: name of image creator, [https://creativecommons.org/licenses/by/3.0/deed.en CC-BY 3.0])
]]

'χξϛ'. I asked an AI chatbot how people of the Greek-speaking world wrote the number 666 in New Testament times, and that was what it said. I knew that, of course, but I really wanted to ask the next question about what I had been taught many years ago, that χξϛ was Satan's visual parody of the title of God’s Messiah, looking like χριστος on the outside but ξ (with its snake-like appearance) replacing everything in the middle between the two outer letters. I have wondered why I never come across it in any commentaries or dictionaries I consult, except perhaps in a few places on the internet. I asked the AI to evaluate the idea, and this is what it said:&lt;blockquote&gt;You’re not alone in noticing the visual resemblance between χξϛ (666) and χριστός (Christos, 'Christ') in Greek.… From a literary-symbolic standpoint, it’s an interesting idea.… However, this visual-letterplay interpretation is speculative and post hoc — there’s no strong evidence that early readers or the author intended the shape or graphic similarity of the letters to carry symbolic meaning. Greek readers were trained to read by sound and meaning, not by visually analysing the shape of words as we might today in a world of logos and brands.&lt;ref&gt;{{Cite web|url=https://chatgpt.com/share/681a24c3-0dbc-8012-9caa-aa81652de95a|title=Greek Numerals 666|last=ChatGPT|first=chatbot|date=8 May 2025|website=ChatGPT}}&lt;/ref&gt;&lt;/blockquote&gt;This got me thinking. Is it true that this visual interpretation is 'speculative and post hoc'? Is there really no evidence to consider?

The number 666 in Revelation 13:18 has been interpreted traditionally through the lens of gematria, usually proposing to link it to Nero Caesar or θηρίον (thērion, beast), while 'it has also been thought a parody on the divine number, seven, given Revelation’s use of seven and given other demonic parodies of the divine in Revelation'.&lt;ref&gt;{{Cite book|title=IVP Cultural Background Commentary (electronic edition for Olive Tree Bible software)|last=Keener|first=Craig S.|publisher=InterVarsity Press|year=2014|location=Downers Grove, IL}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://www.thegospelcoalition.org/article/why-is-the-number-of-the-beast-666/|title=Why Is the Number of the Beast 666?|last=Beale|first=G. K.|date=11 Feb. 2011|website=The Gospel Coalition}}&lt;/ref&gt; Another proposal has been made to interpret the number visually by taking χξϛ as seemingly consisting of 'the initial and final letters of the word Xριστος (Christos), Christ, … with the symbol of the serpent between them'.&lt;ref&gt;{{Cite web|url=https://levendwater.org/books/numbers/number_in_scripture_bullinger.pdf|title=Number in Scripture: Its Supernatural Design and Spiritual Significance, 4th ed. PDF file, (London: Eyre &amp; Spottiswoode Ltd., 1921), p. 49|last=Bullinger|first=E. W.|date=1921}}&lt;/ref&gt; Yet, it does not appear to have received as a credible option. This study proposes that there is adequate evidence from the context, both literary and historical, for interpreting χξϛ as a visual symbol and that taking the visual appearance of χξϛ as a layer of its symbolic meaning is neither speculative nor post hoc but will add to our understanding of what John saw.

== 2. Examining the Text in its Literary Context ==

The number χξϛ is found in Revelation 13, where John continues to narrate a vision he saw. It is a 'mark' which John saw the people who worshipped the beast receive on their right hands or foreheads. As they received it to bear on their body, it must have meant something to them, but what did it mean to them? John understood it, and as he described it, he expected his readers in Asia Minor to understand it also. For us today, our goal must be to establish the perception of the mark which first existed in the minds of those who received it, and the method for that is by analysing how the mark functioned in the scene of the vision as John reports it.

=== 2.1 Visual Nature of Revelation and of the Mark ===
Revelation finds itself in the Jewish apocalyptic tradition, characterised by imagery and symbolism. It opens by identifying itself as 'the revelation from Jesus Christ, which God gave him to show his servants what must soon take place' (1:1). John saw 'a door standing open in heaven' (4:1) and again 'heaven standing open' (19:11). The repeated invitation, 'Come…, I will show you…' (4:1; 17:1; 21:9), led John to report many things he saw. Thus, what John wrote to convey is fundamentally visual in nature. The Apocalypse, therefore, is not just a documented text of heard words but a documentary report of seen visions. It is a literary description of prophetic visions that are rich in imagery from the Hebrew Scriptures. 

In particular, χξϛ was the mark of the beast, i.e. a visual symbol of allegiance to be seen on the body of those who worshipped the beast. In the context of the whole story of Revelation, it is apparent that this was Satan's mimicry of God's sealing (i.e. marking) of his servants (7:3). Also, against the whole story of the Bible, it can be seen as a mimicry of the Jewish practice of visible demonstration of their allegiance to God (cf. Deut. 6:8). The visual nature of the mark, its function and its literary context suggests the importance of how χξϛ looked, i.e. its visual appearance to human eyes. 

=== 2.2 Visual Form of ΧΞϚ and its Function as Pictogram ===
The mark of the beast was the name of the beast, which was in turn the number of the name, and this number was 666. In the text of the latest edition of the Greek New Testament by the United Bible Societies, this number is written out fully in words as 'ἑξακόσιοι ἑξήκοντα ἕξ'.&lt;ref&gt;{{Cite book|title=The Greek New Testament, Fifth Revised Edition|last=ed. by Aland|first=Barbara (and others)|publisher=United Bible Societies|year=2014|location=Stuttgart}}&lt;/ref&gt; However, the Greek New Testament by Tyndale House makes a different choice, because the earliest manuscript witness (Papyrus 47 or P47, from mid-third century) shows that the number was written in an abbreviated form in ancient times.&lt;ref&gt;{{Cite book|title=The Greek New Testament|last=ed. by Jongkind|first=Dirk (and others)|publisher=Tyndale House|year=2017|location=Cainmbridge}}&lt;/ref&gt;

==== 2.2.1 Greek Numeral System ====
Today in English, numbers are commonly written using Arabic numerals, like 666, as a kind of shorthand notation, rather than writing fully in words, like 'six hundred and sixty-six'. The same was true in ancient Greek, except that they used letters from the Greek alphabet as numerals rather than the Arabic numerals, which incidentally should probably be described more accurately as Hindu-Arabic numerals, as they first developed in India before becoming adopted into the Arabic system around the seventh century or some time before that.&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Arabic_numerals&amp;oldid=1296826253|title=Arabic numerals|last=Wikipedia contributors|date=22 June 2025|website=Wikipedia, The Free Encyclopeida}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=History_of_the_Hindu%E2%80%93Arabic_numeral_system&amp;oldid=1264812857|title=History of the Hindu–Arabic numeral system|last=Wikipedia contributors,|date=23 December 2024|website=Wikipedia, The Free Encyclopeida}}&lt;/ref&gt;

The Greek system is the first attested alphabetic numeral system in the world, dating back to the sixth century BC, and called Ionic or Milesian because of its origin in west Asia Minor around Miletus in Ionia.&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Alphabetic_numeral_system&amp;oldid=1222860822|title=Alphabetic numeral system|last=Wikipedia contributors|date=8 May 2024|website=Wikipedia, The Free Encyclopedia}}&lt;/ref&gt; This numeral system continued to be used in Asia Minor well into the Roman period, which is directly relevant for the present study of Revelation, and these numerals were marked by a line above them (overline or overbar) to distinguish them from normal letters.&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Greek_numerals&amp;oldid=1295786959|title=Greek numerals|last=Wikipedia contributors|date=15 June 2025|website=Wikipedia, The Free Encyclopedia}}&lt;/ref&gt;

ΧΞϚ (with overline) is how the number appears in early manuscripts. Each of the three Greek letters employed to represent the number 666 had their numerical values as shown in Table 1 below:
{| class=&quot;wikitable&quot;
|+Table 1: Numerical Values of ΧΞϚ
!Letter
!Letter Name
!Numerical Value
|-
|Χ
|chi
|600
|-
|Ξ
|xi
|60
|-
|Ϛ
|stigma
|6
|}
Yet. it is their visual forms that need our particular attention, because the number was a visual symbol to be seen on the openly visible parts of the body of the beast-followers.

==== 2.2.2 Handwritten Form in Majuscule ====
At this point, it is important to note that lowercase letters had not yet been developed in the first century. What John saw and wrote down would have been in uppercase letters. And, of course, everything was handwritten, as it was long before the days of typesetting. As such, any consideration of the Greek letters for their visual forms must bear in mind how they appeared when handwritten in majuscule as found in early manuscripts. 

==== 2.2.3 Σ (sigma) and Ϛ (stigma) ====
Commonly, the Greek letter sigma is considered to have three forms: uppercase Σ, medial lowercase σ, and final lowercase ς. However, there were two extra lesser-known forms. Lunate sigma (uppercase Ϲ and lowercase ϲ), so called because of its visual resemblance to a crescent moon, came into usage from about fourth century BC and became a standard form of sigma during the late antiquity and Middle Ages.&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Sigma&amp;oldid=1298495170|title=Sigma|last=Wikipedia contributors|date=2 July 2025|website=Wikipedia, The Free Encyclopedia}}&lt;/ref&gt; As such, it is commonly found in many early manuscripts of the New Testament.

The image (Figure 1) below shows folio 7 of Papyrus 47, which contains the text from Revelation 13:16–14:10. The text on the ninth line says:

ΕϹΤΙΝ ΔΕ ΧΞϚ ΚΑΙ ΕΙΔΟΝ, ΚΑΙ ΕΙΔΟῪ ΑΡ(ΝΙΟΝ)

which looks a little more like this in a font designed for a greater visual resemblance to the handwritten text in the early manuscripts:&lt;ref&gt;{{Cite web|url=https://github.com/Center-for-New-Testament-Restoration/font|title=Koine Greek Font|last=Bunning|first=Alan|date=9 Oct. 2022|archive-url=Center for New Testament Restoration}}&lt;/ref&gt;

The number χξϛ is found in Revelation 13, where John continues to narrate a vision he saw. It is a 'mark' which John saw the people who worshipped the beast receive on their right hands or foreheads. As they received it to bear on their body, it must have meant something to them, but what did it mean to them? John understood it, and as he described it, he expected his readers in Asia Minor to understand it also. For us today, our goal must be to establish the perception of the mark which first existed in the minds of those who received it, and the method for that is by analysing how the mark functioned in the scene of the vision as John reports it.

=== 2.1 Visual Nature of Revelation and of the Mark ===
Revelation finds itself in the Jewish apocalyptic tradition, characterised by imagery and symbolism. It opens by identifying itself as 'the revelation from Jesus Christ, which God gave him to show his servants what must soon take place' (1:1). John saw 'a door standing open in heaven' (4:1) and again 'heaven standing open' (19:11). The repeated invitation, 'Come…, I will show you…' (4:1; 17:1; 21:9), led John to report many things he saw. Thus, what John wrote to convey is fundamentally visual in nature. The Apocalypse, therefore, is not just a documented text of heard words but a documentary report of seen visions. It is a literary description of prophetic visions that are rich in imagery from the Hebrew Scriptures. 

In particular, χξϛ was the mark of the beast, i.e. a visual symbol of allegiance to be seen on the body of those who worshipped the beast. In the context of the whole story of Revelation, it is apparent that this was Satan's mimicry of God's sealing (i.e. marking) of his servants (7:3). Also, against the whole story of the Bible, it can be seen as a mimicry of the Jewish practice of visible demonstration of their allegiance to God (cf. Deut. 6:8). The visual nature of the mark, its function and its literary context suggests the importance of how χξϛ looked, i.e. its visual appearance to human eyes. 

=== 2.2 Visual Form of ΧΞϚ and its Function as Pictogram ===
The mark of the beast was the name of the beast, which was in turn the number of the name, and this number was 666. In the text of the latest edition of the Greek New Testament by the United Bible Societies, this number is written out fully in words as 'ἑξακόσιοι ἑξήκοντα ἕξ'.&lt;ref&gt;{{Cite book|title=The Greek New Testament, Fifth Revised Edition|last=ed. by Aland|first=Barbara (and others)|publisher=United Bible Societies|year=2014|location=Stuttgart}}&lt;/ref&gt; However, the Greek New Testament by Tyndale House makes a different choice, because the earliest manuscript witness (Papyrus 47 or P47, from mid-third century) shows that the number was written in an abbreviated form in ancient times.&lt;ref&gt;{{Cite book|title=The Greek New Testament|last=ed. by Jongkind|first=Dirk (and others)|publisher=Tyndale House|year=2017|location=Cainmbridge}}&lt;/ref&gt;

==== 2.2.1 Greek Numeral System ====
Today in English, numbers are commonly written using Arabic numerals, like 666, as a kind of shorthand notation, rather than writing fully in words, like 'six hundred and sixty-six'. The same was true in ancient Greek, except that they used letters from the Greek alphabet as numerals rather than the Arabic numerals, which incidentally should probably be described more accurately as Hindu-Arabic numerals, as they first developed in India before becoming adopted into the Arabic system around the seventh century or some time before that.&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Arabic_numerals&amp;oldid=1296826253|title=Arabic numerals|last=Wikipedia contributors|date=22 June 2025|website=Wikipedia, The Free Encyclopeida}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=History_of_the_Hindu%E2%80%93Arabic_numeral_system&amp;oldid=1264812857|title=History of the Hindu–Arabic numeral system|last=Wikipedia contributors,|date=23 December 2024|website=Wikipedia, The Free Encyclopeida}}&lt;/ref&gt;

The Greek system is the first attested alphabetic numeral system in the world, dating back to the sixth century BC, and called Ionic or Milesian because of its origin in west Asia Minor around Miletus in Ionia.&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Alphabetic_numeral_system&amp;oldid=1222860822|title=Alphabetic numeral system|last=Wikipedia contributors|date=8 May 2024|website=Wikipedia, The Free Encyclopedia}}&lt;/ref&gt; This numeral system continued to be used in Asia Minor well into the Roman period, which is directly relevant for the present study of Revelation, and these numerals were marked by a line above them (overline or overbar) to distinguish them from normal letters.&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Greek_numerals&amp;oldid=1295786959|title=Greek numerals|last=Wikipedia contributors|date=15 June 2025|website=Wikipedia, The Free Encyclopedia}}&lt;/ref&gt;

ΧΞϚ (with overline) is how the number appears in early manuscripts. Each of the three Greek letters employed to represent the number 666 had their numerical values as shown in Table 1 below:
{| class=&quot;wikitable&quot;
|+Table 1: Numerical Values of ΧΞϚ
!Letter
!Letter Name
!Numerical Value
|-
|Χ
|chi
|600
|-
|Ξ
|xi
|60
|-
|Ϛ
|stigma
|6
|}
Yet. it is their visual forms that need our particular attention, because the number was a visual symbol to be seen on the openly visible parts of the body of the beast-followers.

==== 2.2.2 Handwritten Form in Majuscule ====
At this point, it is important to note that lowercase letters had not yet been developed in the first century. What John saw and wrote down would have been in uppercase letters. And, of course, everything was handwritten, as it was long before the days of typesetting. As such, any consideration of the Greek letters for their visual forms must bear in mind how they appeared when handwritten in majuscule as found in early manuscripts. 

==== 2.2.3 Σ (sigma) and Ϛ (stigma) ====
Commonly, the Greek letter sigma is considered to have three forms: uppercase Σ, medial lowercase σ, and final lowercase ς. However, there were two extra lesser-known forms. Lunate sigma (uppercase Ϲ and lowercase ϲ), so called because of its visual resemblance to a crescent moon, came into usage from about fourth century BC and became a standard form of sigma during the late antiquity and Middle Ages.&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Sigma&amp;oldid=1298495170|title=Sigma|last=Wikipedia contributors|date=2 July 2025|website=Wikipedia, The Free Encyclopedia}}&lt;/ref&gt; As such, it is commonly found in many early manuscripts of the New Testament.

The image (Figure 1) below shows folio 7 of Papyrus 47, which contains the text from Revelation 13:16–14:10.&lt;ref&gt;{{Cite web|url=https://viewer.cbl.ie/viewer/image/BP_III_f_7/1/|title=Revelation 13.16–14.4; 14.4–10|website=Chester Beatty Online Collections|access-date=16 May 2025}}&lt;/ref&gt; 

'''Figure 1: Papyrus 47 folio 7''' [[File:P47 folio 7 - Rev 13vv1-15.jpg|alt=Figure 1. P47 folio 7|frameless|1247x1247px]]

The text on the ninth line says:

ΕϹΤΙΝ ΔΕ ΧΞϚ ΚΑΙ ΕΙΔΟΝ, ΚΑΙ ΕΙΔΟῪ ΑΡ(ΝΙΟΝ)

which looks a little more like this in a font designed for a greater visual resemblance to the handwritten text in the early manuscripts:&lt;ref&gt;{{Cite web|url=https://github.com/Center-for-New-Testament-Restoration/font|title=Koine Greek Font|last=Bunning|first=Alan|date=9 Oct. 2022|archive-url=Center for New Testament Restoration}}&lt;/ref&gt;

[[File:Koine_Majuscule_text.png|frameless|380x380px]]

What should be observed here is the visual resemblance between Ϲ (crescent sigma, the second letter) and Ϛ (stigma, the tenth letter). This resemblance is perhaps not too surprising, considering the origin of Ϛ as a ligature of sigma (Σ) and tau (Τ).

==== 2.2.4 Nomina Sacra ====
Early Christians considered certain names and titles, like Θεός (Theos, God), Κύριος (Kyrios, Lord), Ἰησοῦς (Iēsous, Jesus), Χριστός (Christos, Christ), Υἱός (Huios, Son, referring to Jesus) and Πνεῦμα (Pneuma, Spirit, referring to the Holy Spirit), as nomina sacra (sacred names), to be treated with respect.&lt;ref name=&quot;:0&quot;&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Nomina_sacra&amp;oldid=1270217331|title=Nomina sacra|last=Wikipedia contributors|date=18 January 2025|website=Wikipedia, The Free Encyclopedia}}&lt;/ref&gt; Those names and titles were also words that occurred frequently in the manuscripts, and the scribes developed a practice of abbreviating them, usually by contraction, taking the first one or two letters and the last letter of the word, skipping all middle letters, and marking them with an overline to indicate abbreviation in the same way as when marking numbers written in Greek numerals.

Precisely when this practice arose is not known. However, the abbreviation practice in Greek literature predates Christian writings, going back to the fourth century BC, as the earliest known Western shorthand system was employed by the Greek historian Xenophon (a student of Socrates) in his work Ἀπομνημονεύματα (Memorabilia or Memoir of Socrates), which is considered to have been completed shortly after 371 BC.&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Scribal_abbreviation&amp;oldid=1283806604|title=Scribal abbreviation|last=Wikipedia contributors|date=3 April 2025|website=Wikipedia, The Free Encyclopedia}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Memorabilia_(Xenophon)&amp;oldid=1254008920|title=Memorabilia (Xenophon)|last=Wikipedia contributors|date=29 October 2024|website=Wikipedia, The Free Encyclopedia}}&lt;/ref&gt; In the light of this pre-Christian practice in the Greek literary tradition, it would have been natural for Christian writers to make use of it in their own writings. The manuscript evidence is that 'nomina sacra are consistently observed in even the earliest extant Christian writings, ... implying that when these were written, in approximately the second century, the practice had already been established for some time.'&lt;ref name=&quot;:0&quot; /&gt; It is, then, reasonable to estimate that origin of nomina sacra was early in the first century.

That, in turn, means that when Revelation was written toward the end of the first century, John and his readers would have been familiar with the practice of nomina sacra. More specifically, it would have been a normal and common experience for them to write or to see ΧϹ or ΧΡϹ for ΧΡΙϹΤΌϹ (Christos, Christ).

Initially, this practice was limited to only a handful of words, which are called nomina divina (divine names), as they all refer to persons of the Trinity as shown in the Table 2 below. However, the practice extended through the second and third centuries, and by the early Byzantine period in the fourth century, the extended practice was established to include the additional words in Table 3.&lt;ref&gt;{{Cite web|url=https://archive.org/details/bruce-m.-metzger-manuscripts-of-the-greek-bible.-an-introduction-to-palaeography/page/36/mode/1up|title=Manuscripts of the Greek Bible: An Introduction to Palaeography (Oxford: Oxford University Press, 1981), p. 36|last=Metzger|first=Bruce Manning|website=Internet Archive}}&lt;/ref&gt;
{| class=&quot;wikitable&quot;
|+Table 2: Nomina Divina
! rowspan=&quot;2&quot; |Greek
! rowspan=&quot;2&quot; |Transliteration
! rowspan=&quot;2&quot; |English
! colspan=&quot;2&quot; |Abbreviation
|-
!(Nominative Case)
!(Genitive Case)
|-
|Θεός
|Theos
|God
|ΘΣ
|ΘΥ
|-
|Κύριος
|Kyrios
|Lord
|ΚΣ
|ΚΥ
|-
|Ἰησοῦς
|Iēsous
|Jesus
|ΙΣ or ΙΗΣ
|ΙΥ
|-
|Χριστός
|Christos
|Christ
|ΧΣ or ΧΡΣ
|ΧΥ
|-
|Πνεῦμα
|Pneuma
|Spirit 
referring to the Holy Spirit
|ΠΝΑ
|ΠΝΣ
|}
{| class=&quot;wikitable&quot;
|+Table 3: Later Additions to Nomina Sacra
! rowspan=&quot;2&quot; |Greek
! rowspan=&quot;2&quot; |Transliteration
! rowspan=&quot;2&quot; |English
! colspan=&quot;2&quot; |Abbreviation
|-
!(Nominative Case)
!(Genitive Case)
|-
|Πατήρ
|''Patēr''
|Father
|ΠΗΡ
|ΠΡΣ
|-
|Σωτήρ
|''Sōtēr''
|Saviour
|ΣΗΡ
|ΣΡΣ
|-
|Σταυρός
|''Stauros''
|Cross
|ΣΤΣ
|ΣΤΥ
|-
|Μήτηρ
|''Mētēr''
|Mother
referring to Mary
|ΜΤΡ
|ΜΡΣ
|-
|Ἰσραήλ
|''Israēl''
|Israel
|ΙΗΛ
|
|-
|Ἄνθρωπος
|''Anthrōpos''
|Man
in the phrase 'Son of Man'
|ΑΝΟΣ
|ΑΝΟΥ
|-
|Ἰερουσαλήμ
|''Ierousalēm''
|Jerusalem
|ΙΛΗΜ
|
|-
|Οὐρανός
|''Ouranos''
|Heaven
|ΟΥΝΟΣ
|ΟΥΝΥ
|}
Examples of ''nomina sacra'' can be seen in the image (Figure 2) below of the third-century manuscript (P46 folio 62), containing the text of 2 Corinthians 1:16–2:1 and 2:3–12.&lt;ref&gt;{{Cite web|url=https://viewer.cbl.ie/viewer/image/BP_II_f_62/1/|title=2 Corinthians 1.16-2.1; 2.3-12|website=Chester Beatty Online Collections|access-date=16 May 2025}}&lt;/ref&gt; As it has already been shown above, the practice of ''nomina sacra'' was already in use by the time of Revelation. The value of looking at this third-century manuscript is that it shows what they looked like ''visually'', as the concern of this paper is the ''visual'' appearance of written words in order for them to function pictographically.

'''Figure 2: P46 folio 62'''[[File:P47 folio 7 - Rev 13vv1-15.jpg|alt=Figure 2: P46 folio 62|frameless|1247x1247px]]

In the manuscript, the text on the ninth line reads:

ΓΑΡ ΘΥ ΥΙϹ ΙΗϹ ΧΡϹ Ο ΕΝ ΥΜΕΙ͂Ν ΔΙ Η(ΜΩ͂Ν)

which looks more like:

[[File:Koine Majuscule text 2.png|frameless|380x380px]]

where ΘΕΟΥ͂ ΥΙΟϹ ΙΗϹΟΥ͂Ϲ ΧΡΙϹΤῸϹ ('God's Son Jesus Christ') is written in shorthand as ΘΥ ΥΙϹ ΙΗϹ ΧΡϹ (all with overline). The point to note here is that if the first-century readers were accustomed to seeing ΧϹ or ΧΡϹ (both with overline), written with an overline above it, as a shorthand for ΧΡΙϹΤΟϹ (''Christos'', Christ), then John and his readers would have been so familiar with [[File:Christ in shorthand.png|frameless|40x40px]] as a rightful title of their Lord that they would have been quick to see the blasphemy in [[File:Parody-christ in shorthand.png|frameless|42x42px]] being used by the beast as its mark.

==== 2.2.5 ΧΞϚ as Visual Parody ====
The mark of the beast in Revelation 13:18 was written as ΧΞϚ. In the vision, John saw it on the right hands or foreheads of those who followed the beast. What was its function and significance in the vision? What did John understand it meant for the beast's followers? 

Given the visual similarity between Ϛ and Ϲ in handwritten form, it is difficult to imagine that the general outward resemblance between ΧΞϚ and ΧΡϹ (both with overline) could have escaped the attention of the first-century readers. What stood out would have been the only notable visual difference, Ξ standing in the middle, in place of Ρ. Figure 3 below shows the side-by-side comparison of the images of these two words from early manuscripts.

'''Figure 3: Side-by-Side Comparison of ΧΞϚ (in P47 f.7) and ΧΡϹ (in P46 f.62)'''[[File:Visual Parody.png|alt=Figure 3: Side-by-Side Comparison of ΧΞϚ (in P47 f.7) and ΧΡϹ (in P46 f.62)|frameless|800x800px]]Ξ, when handwritten, often looked like an asymmetric and wavy zig-zag ξ. The question is if John and his original readers in Asia Minor might have perceived its shape as snake-like and seen it as a symbol of a serpent, or more specifically, 'the ancient snake' (Rev. 12:9; 20:2). The answer to this question is not found in available ancient manuscripts. Why? It is possible that such a visual association was considered too obvious to discuss and unnecessary for documentation. 

Today, English serves in a similar role as Greek did in the ancient world, as a convenient tool for international communication. Like Greek, English uses a phonetic alphabet. As such, teaching phonics to children is a part of literacy education in many parts of the English-speaking world, and Letterland is one method that is widely used for the purpose.&lt;ref&gt;Letterland is a phonics-based method for teaching literacy, originally developed in UK but now used globally in English-speaking world. For more information, see &lt;nowiki&gt;https://www.letterland.com/company&lt;/nowiki&gt;.&lt;/ref&gt; In their system, the letter S is taught as 'Sammy Snake', as shown in the picture (Figure 4) below.&lt;ref&gt;The image of the Letterland character, Sammy Snake, here used by permission, is copyrighted by Letterland.&lt;/ref&gt; 

[[File:Sammy Snake in Classroom.jpg|frameless|449x449px]]

There is no documented discussion or explanation about why a snake should represent the letter S, presumably because the visual association between the letter shape and the creature's image is accepted naturally. It is not difficult to imagine the same for the image association between ξ and a snake among the Koine speaking Christians in the first-century Graeco-Roman world. The proposal is that such a visual association is indeed likely to have existed.

If, then, people in the ancient Greek speaking world considered ξ serpent-like, as people in today's English-speaking world naturally call the letter S 'Sammy Snake', ΧΞϚ (χξϛ) would have functioned effectively as a pictogram for immediate visual perception of its meaning. The ''seeming'' resemblance of its two outer letters to those of ΧΡϹ (χrs), given that Ϲ looked similar to Ϛ in handwritten form, and the snake-symbol Ξ (ξ) replacing the middle would have signified a satanic parody of the title of God's Messiah, ΧΡΙϹΤΟϹ. 

==Third Heading, etc==


==Additional information==

===Acknowledgements===
Any people, organisations, or funding sources that you would like to thank.

===Competing interests===
Any conflicts of interest that you would like to declare. Otherwise, a statement that the authors have no competing interest.

===Ethics statement===
An ethics statement, if appropriate, on any animal or human research performed should be included here or in the methods section.

==References==
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==1. Introduction==

[[file:example image.png|thumb|left
| Image caption text goes here
(attribution: name of image creator, [https://creativecommons.org/licenses/by/3.0/deed.en CC-BY 3.0])
]]

'χξϛ'. I asked an AI chatbot how people of the Greek-speaking world wrote the number 666 in New Testament times, and that was what it said. I knew that, of course, but I really wanted to ask the next question about what I had been taught many years ago, that χξϛ was Satan's visual parody of the title of God’s Messiah, looking like χριστος on the outside but ξ (with its snake-like appearance) replacing everything in the middle between the two outer letters. I have wondered why I never come across it in any commentaries or dictionaries I consult, except perhaps in a few places on the internet. I asked the AI to evaluate the idea, and this is what it said:&lt;blockquote&gt;You’re not alone in noticing the visual resemblance between χξϛ (666) and χριστός (Christos, 'Christ') in Greek.… From a literary-symbolic standpoint, it’s an interesting idea.… However, this visual-letterplay interpretation is speculative and post hoc — there’s no strong evidence that early readers or the author intended the shape or graphic similarity of the letters to carry symbolic meaning. Greek readers were trained to read by sound and meaning, not by visually analysing the shape of words as we might today in a world of logos and brands.&lt;ref&gt;{{Cite web|url=https://chatgpt.com/share/681a24c3-0dbc-8012-9caa-aa81652de95a|title=Greek Numerals 666|last=ChatGPT|first=chatbot|date=8 May 2025|website=ChatGPT}}&lt;/ref&gt;&lt;/blockquote&gt;This got me thinking. Is it true that this visual interpretation is 'speculative and post hoc'? Is there really no evidence to consider?

The number 666 in Revelation 13:18 has been interpreted traditionally through the lens of gematria, usually proposing to link it to Nero Caesar or θηρίον (thērion, beast), while 'it has also been thought a parody on the divine number, seven, given Revelation’s use of seven and given other demonic parodies of the divine in Revelation'.&lt;ref&gt;{{Cite book|title=IVP Cultural Background Commentary (electronic edition for Olive Tree Bible software)|last=Keener|first=Craig S.|publisher=InterVarsity Press|year=2014|location=Downers Grove, IL}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://www.thegospelcoalition.org/article/why-is-the-number-of-the-beast-666/|title=Why Is the Number of the Beast 666?|last=Beale|first=G. K.|date=11 Feb. 2011|website=The Gospel Coalition}}&lt;/ref&gt; Another proposal has been made to interpret the number visually by taking χξϛ as seemingly consisting of 'the initial and final letters of the word Xριστος (Christos), Christ, … with the symbol of the serpent between them'.&lt;ref&gt;{{Cite web|url=https://levendwater.org/books/numbers/number_in_scripture_bullinger.pdf|title=Number in Scripture: Its Supernatural Design and Spiritual Significance, 4th ed. PDF file, (London: Eyre &amp; Spottiswoode Ltd., 1921), p. 49|last=Bullinger|first=E. W.|date=1921}}&lt;/ref&gt; Yet, it does not appear to have received as a credible option. This study proposes that there is adequate evidence from the context, both literary and historical, for interpreting χξϛ as a visual symbol and that taking the visual appearance of χξϛ as a layer of its symbolic meaning is neither speculative nor post hoc but will add to our understanding of what John saw.

== 2. Examining the Text in its Literary Context ==

The number χξϛ is found in Revelation 13, where John continues to narrate a vision he saw. It is a 'mark' which John saw the people who worshipped the beast receive on their right hands or foreheads. As they received it to bear on their body, it must have meant something to them, but what did it mean to them? John understood it, and as he described it, he expected his readers in Asia Minor to understand it also. For us today, our goal must be to establish the perception of the mark which first existed in the minds of those who received it, and the method for that is by analysing how the mark functioned in the scene of the vision as John reports it.

=== 2.1 Visual Nature of Revelation and of the Mark ===
Revelation finds itself in the Jewish apocalyptic tradition, characterised by imagery and symbolism. It opens by identifying itself as 'the revelation from Jesus Christ, which God gave him to show his servants what must soon take place' (1:1). John saw 'a door standing open in heaven' (4:1) and again 'heaven standing open' (19:11). The repeated invitation, 'Come…, I will show you…' (4:1; 17:1; 21:9), led John to report many things he saw. Thus, what John wrote to convey is fundamentally visual in nature. The Apocalypse, therefore, is not just a documented text of heard words but a documentary report of seen visions. It is a literary description of prophetic visions that are rich in imagery from the Hebrew Scriptures. 

In particular, χξϛ was the mark of the beast, i.e. a visual symbol of allegiance to be seen on the body of those who worshipped the beast. In the context of the whole story of Revelation, it is apparent that this was Satan's mimicry of God's sealing (i.e. marking) of his servants (7:3). Also, against the whole story of the Bible, it can be seen as a mimicry of the Jewish practice of visible demonstration of their allegiance to God (cf. Deut. 6:8). The visual nature of the mark, its function and its literary context suggests the importance of how χξϛ looked, i.e. its visual appearance to human eyes. 

=== 2.2 Visual Form of ΧΞϚ and its Function as Pictogram ===
The mark of the beast was the name of the beast, which was in turn the number of the name, and this number was 666. In the text of the latest edition of the Greek New Testament by the United Bible Societies, this number is written out fully in words as 'ἑξακόσιοι ἑξήκοντα ἕξ'.&lt;ref&gt;{{Cite book|title=The Greek New Testament, Fifth Revised Edition|last=ed. by Aland|first=Barbara (and others)|publisher=United Bible Societies|year=2014|location=Stuttgart}}&lt;/ref&gt; However, the Greek New Testament by Tyndale House makes a different choice, because the earliest manuscript witness (Papyrus 47 or P47, from mid-third century) shows that the number was written in an abbreviated form in ancient times.&lt;ref&gt;{{Cite book|title=The Greek New Testament|last=ed. by Jongkind|first=Dirk (and others)|publisher=Tyndale House|year=2017|location=Cainmbridge}}&lt;/ref&gt;

==== 2.2.1 Greek Numeral System ====
Today in English, numbers are commonly written using Arabic numerals, like 666, as a kind of shorthand notation, rather than writing fully in words, like 'six hundred and sixty-six'. The same was true in ancient Greek, except that they used letters from the Greek alphabet as numerals rather than the Arabic numerals, which incidentally should probably be described more accurately as Hindu-Arabic numerals, as they first developed in India before becoming adopted into the Arabic system around the seventh century or some time before that.&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Arabic_numerals&amp;oldid=1296826253|title=Arabic numerals|last=Wikipedia contributors|date=22 June 2025|website=Wikipedia, The Free Encyclopeida}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=History_of_the_Hindu%E2%80%93Arabic_numeral_system&amp;oldid=1264812857|title=History of the Hindu–Arabic numeral system|last=Wikipedia contributors,|date=23 December 2024|website=Wikipedia, The Free Encyclopeida}}&lt;/ref&gt;

The Greek system is the first attested alphabetic numeral system in the world, dating back to the sixth century BC, and called Ionic or Milesian because of its origin in west Asia Minor around Miletus in Ionia.&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Alphabetic_numeral_system&amp;oldid=1222860822|title=Alphabetic numeral system|last=Wikipedia contributors|date=8 May 2024|website=Wikipedia, The Free Encyclopedia}}&lt;/ref&gt; This numeral system continued to be used in Asia Minor well into the Roman period, which is directly relevant for the present study of Revelation, and these numerals were marked by a line above them (overline or overbar) to distinguish them from normal letters.&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Greek_numerals&amp;oldid=1295786959|title=Greek numerals|last=Wikipedia contributors|date=15 June 2025|website=Wikipedia, The Free Encyclopedia}}&lt;/ref&gt;

ΧΞϚ (with overline) is how the number appears in early manuscripts. Each of the three Greek letters employed to represent the number 666 had their numerical values as shown in Table 1 below:
{| class=&quot;wikitable&quot;
|+Table 1: Numerical Values of ΧΞϚ
!Letter
!Letter Name
!Numerical Value
|-
|Χ
|chi
|600
|-
|Ξ
|xi
|60
|-
|Ϛ
|stigma
|6
|}
Yet. it is their visual forms that need our particular attention, because the number was a visual symbol to be seen on the openly visible parts of the body of the beast-followers.

==== 2.2.2 Handwritten Form in Majuscule ====
At this point, it is important to note that lowercase letters had not yet been developed in the first century. What John saw and wrote down would have been in uppercase letters. And, of course, everything was handwritten, as it was long before the days of typesetting. As such, any consideration of the Greek letters for their visual forms must bear in mind how they appeared when handwritten in majuscule as found in early manuscripts. 

==== 2.2.3 Σ (sigma) and Ϛ (stigma) ====
Commonly, the Greek letter sigma is considered to have three forms: uppercase Σ, medial lowercase σ, and final lowercase ς. However, there were two extra lesser-known forms. Lunate sigma (uppercase Ϲ and lowercase ϲ), so called because of its visual resemblance to a crescent moon, came into usage from about fourth century BC and became a standard form of sigma during the late antiquity and Middle Ages.&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Sigma&amp;oldid=1298495170|title=Sigma|last=Wikipedia contributors|date=2 July 2025|website=Wikipedia, The Free Encyclopedia}}&lt;/ref&gt; As such, it is commonly found in many early manuscripts of the New Testament.

The image (Figure 1) below shows folio 7 of Papyrus 47, which contains the text from Revelation 13:16–14:10. The text on the ninth line says:

ΕϹΤΙΝ ΔΕ ΧΞϚ ΚΑΙ ΕΙΔΟΝ, ΚΑΙ ΕΙΔΟῪ ΑΡ(ΝΙΟΝ)

which looks a little more like this in a font designed for a greater visual resemblance to the handwritten text in the early manuscripts:&lt;ref&gt;{{Cite web|url=https://github.com/Center-for-New-Testament-Restoration/font|title=Koine Greek Font|last=Bunning|first=Alan|date=9 Oct. 2022|archive-url=Center for New Testament Restoration}}&lt;/ref&gt;

The number χξϛ is found in Revelation 13, where John continues to narrate a vision he saw. It is a 'mark' which John saw the people who worshipped the beast receive on their right hands or foreheads. As they received it to bear on their body, it must have meant something to them, but what did it mean to them? John understood it, and as he described it, he expected his readers in Asia Minor to understand it also. For us today, our goal must be to establish the perception of the mark which first existed in the minds of those who received it, and the method for that is by analysing how the mark functioned in the scene of the vision as John reports it.

=== 2.1 Visual Nature of Revelation and of the Mark ===
Revelation finds itself in the Jewish apocalyptic tradition, characterised by imagery and symbolism. It opens by identifying itself as 'the revelation from Jesus Christ, which God gave him to show his servants what must soon take place' (1:1). John saw 'a door standing open in heaven' (4:1) and again 'heaven standing open' (19:11). The repeated invitation, 'Come…, I will show you…' (4:1; 17:1; 21:9), led John to report many things he saw. Thus, what John wrote to convey is fundamentally visual in nature. The Apocalypse, therefore, is not just a documented text of heard words but a documentary report of seen visions. It is a literary description of prophetic visions that are rich in imagery from the Hebrew Scriptures. 

In particular, χξϛ was the mark of the beast, i.e. a visual symbol of allegiance to be seen on the body of those who worshipped the beast. In the context of the whole story of Revelation, it is apparent that this was Satan's mimicry of God's sealing (i.e. marking) of his servants (7:3). Also, against the whole story of the Bible, it can be seen as a mimicry of the Jewish practice of visible demonstration of their allegiance to God (cf. Deut. 6:8). The visual nature of the mark, its function and its literary context suggests the importance of how χξϛ looked, i.e. its visual appearance to human eyes. 

=== 2.2 Visual Form of ΧΞϚ and its Function as Pictogram ===
The mark of the beast was the name of the beast, which was in turn the number of the name, and this number was 666. In the text of the latest edition of the Greek New Testament by the United Bible Societies, this number is written out fully in words as 'ἑξακόσιοι ἑξήκοντα ἕξ'.&lt;ref&gt;{{Cite book|title=The Greek New Testament, Fifth Revised Edition|last=ed. by Aland|first=Barbara (and others)|publisher=United Bible Societies|year=2014|location=Stuttgart}}&lt;/ref&gt; However, the Greek New Testament by Tyndale House makes a different choice, because the earliest manuscript witness (Papyrus 47 or P47, from mid-third century) shows that the number was written in an abbreviated form in ancient times.&lt;ref&gt;{{Cite book|title=The Greek New Testament|last=ed. by Jongkind|first=Dirk (and others)|publisher=Tyndale House|year=2017|location=Cainmbridge}}&lt;/ref&gt;

==== 2.2.1 Greek Numeral System ====
Today in English, numbers are commonly written using Arabic numerals, like 666, as a kind of shorthand notation, rather than writing fully in words, like 'six hundred and sixty-six'. The same was true in ancient Greek, except that they used letters from the Greek alphabet as numerals rather than the Arabic numerals, which incidentally should probably be described more accurately as Hindu-Arabic numerals, as they first developed in India before becoming adopted into the Arabic system around the seventh century or some time before that.&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Arabic_numerals&amp;oldid=1296826253|title=Arabic numerals|last=Wikipedia contributors|date=22 June 2025|website=Wikipedia, The Free Encyclopeida}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=History_of_the_Hindu%E2%80%93Arabic_numeral_system&amp;oldid=1264812857|title=History of the Hindu–Arabic numeral system|last=Wikipedia contributors,|date=23 December 2024|website=Wikipedia, The Free Encyclopeida}}&lt;/ref&gt;

The Greek system is the first attested alphabetic numeral system in the world, dating back to the sixth century BC, and called Ionic or Milesian because of its origin in west Asia Minor around Miletus in Ionia.&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Alphabetic_numeral_system&amp;oldid=1222860822|title=Alphabetic numeral system|last=Wikipedia contributors|date=8 May 2024|website=Wikipedia, The Free Encyclopedia}}&lt;/ref&gt; This numeral system continued to be used in Asia Minor well into the Roman period, which is directly relevant for the present study of Revelation, and these numerals were marked by a line above them (overline or overbar) to distinguish them from normal letters.&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Greek_numerals&amp;oldid=1295786959|title=Greek numerals|last=Wikipedia contributors|date=15 June 2025|website=Wikipedia, The Free Encyclopedia}}&lt;/ref&gt;

ΧΞϚ (with overline) is how the number appears in early manuscripts. Each of the three Greek letters employed to represent the number 666 had their numerical values as shown in Table 1 below:
{| class=&quot;wikitable&quot;
|+Table 1: Numerical Values of ΧΞϚ
!Letter
!Letter Name
!Numerical Value
|-
|Χ
|chi
|600
|-
|Ξ
|xi
|60
|-
|Ϛ
|stigma
|6
|}
Yet. it is their visual forms that need our particular attention, because the number was a visual symbol to be seen on the openly visible parts of the body of the beast-followers.

==== 2.2.2 Handwritten Form in Majuscule ====
At this point, it is important to note that lowercase letters had not yet been developed in the first century. What John saw and wrote down would have been in uppercase letters. And, of course, everything was handwritten, as it was long before the days of typesetting. As such, any consideration of the Greek letters for their visual forms must bear in mind how they appeared when handwritten in majuscule as found in early manuscripts. 

==== 2.2.3 Σ (sigma) and Ϛ (stigma) ====
Commonly, the Greek letter sigma is considered to have three forms: uppercase Σ, medial lowercase σ, and final lowercase ς. However, there were two extra lesser-known forms. Lunate sigma (uppercase Ϲ and lowercase ϲ), so called because of its visual resemblance to a crescent moon, came into usage from about fourth century BC and became a standard form of sigma during the late antiquity and Middle Ages.&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Sigma&amp;oldid=1298495170|title=Sigma|last=Wikipedia contributors|date=2 July 2025|website=Wikipedia, The Free Encyclopedia}}&lt;/ref&gt; As such, it is commonly found in many early manuscripts of the New Testament.

The image (Figure 1) below shows folio 7 of Papyrus 47, which contains the text from Revelation 13:16–14:10.&lt;ref&gt;{{Cite web|url=https://viewer.cbl.ie/viewer/image/BP_III_f_7/1/|title=Revelation 13.16–14.4; 14.4–10|website=Chester Beatty Online Collections|access-date=16 May 2025}}&lt;/ref&gt; 

'''Figure 1: Papyrus 47 folio 7''' [[File:P47 folio 7 – Rev 13v16–14v10.jpg|alt=Figure 1. P47 folio 7|frameless|1247x1247px]]

The text on the ninth line says:

ΕϹΤΙΝ ΔΕ ΧΞϚ ΚΑΙ ΕΙΔΟΝ, ΚΑΙ ΕΙΔΟῪ ΑΡ(ΝΙΟΝ)

which looks a little more like this in a font designed for a greater visual resemblance to the handwritten text in the early manuscripts:&lt;ref&gt;{{Cite web|url=https://github.com/Center-for-New-Testament-Restoration/font|title=Koine Greek Font|last=Bunning|first=Alan|date=9 Oct. 2022|archive-url=Center for New Testament Restoration}}&lt;/ref&gt;

[[File:Koine_Majuscule_text.png|frameless|380x380px]]

What should be observed here is the visual resemblance between Ϲ (crescent sigma, the second letter) and Ϛ (stigma, the tenth letter). This resemblance is perhaps not too surprising, considering the origin of Ϛ as a ligature of sigma (Σ) and tau (Τ).

==== 2.2.4 Nomina Sacra ====
Early Christians considered certain names and titles, like Θεός (Theos, God), Κύριος (Kyrios, Lord), Ἰησοῦς (Iēsous, Jesus), Χριστός (Christos, Christ), Υἱός (Huios, Son, referring to Jesus) and Πνεῦμα (Pneuma, Spirit, referring to the Holy Spirit), as nomina sacra (sacred names), to be treated with respect.&lt;ref name=&quot;:0&quot;&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Nomina_sacra&amp;oldid=1270217331|title=Nomina sacra|last=Wikipedia contributors|date=18 January 2025|website=Wikipedia, The Free Encyclopedia}}&lt;/ref&gt; Those names and titles were also words that occurred frequently in the manuscripts, and the scribes developed a practice of abbreviating them, usually by contraction, taking the first one or two letters and the last letter of the word, skipping all middle letters, and marking them with an overline to indicate abbreviation in the same way as when marking numbers written in Greek numerals.

Precisely when this practice arose is not known. However, the abbreviation practice in Greek literature predates Christian writings, going back to the fourth century BC, as the earliest known Western shorthand system was employed by the Greek historian Xenophon (a student of Socrates) in his work Ἀπομνημονεύματα (Memorabilia or Memoir of Socrates), which is considered to have been completed shortly after 371 BC.&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Scribal_abbreviation&amp;oldid=1283806604|title=Scribal abbreviation|last=Wikipedia contributors|date=3 April 2025|website=Wikipedia, The Free Encyclopedia}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://en.wikipedia.org/w/index.php?title=Memorabilia_(Xenophon)&amp;oldid=1254008920|title=Memorabilia (Xenophon)|last=Wikipedia contributors|date=29 October 2024|website=Wikipedia, The Free Encyclopedia}}&lt;/ref&gt; In the light of this pre-Christian practice in the Greek literary tradition, it would have been natural for Christian writers to make use of it in their own writings. The manuscript evidence is that 'nomina sacra are consistently observed in even the earliest extant Christian writings, ... implying that when these were written, in approximately the second century, the practice had already been established for some time.'&lt;ref name=&quot;:0&quot; /&gt; It is, then, reasonable to estimate that origin of nomina sacra was early in the first century.

That, in turn, means that when Revelation was written toward the end of the first century, John and his readers would have been familiar with the practice of nomina sacra. More specifically, it would have been a normal and common experience for them to write or to see ΧϹ or ΧΡϹ for ΧΡΙϹΤΌϹ (Christos, Christ).

Initially, this practice was limited to only a handful of words, which are called nomina divina (divine names), as they all refer to persons of the Trinity as shown in the Table 2 below. However, the practice extended through the second and third centuries, and by the early Byzantine period in the fourth century, the extended practice was established to include the additional words in Table 3.&lt;ref&gt;{{Cite web|url=https://archive.org/details/bruce-m.-metzger-manuscripts-of-the-greek-bible.-an-introduction-to-palaeography/page/36/mode/1up|title=Manuscripts of the Greek Bible: An Introduction to Palaeography (Oxford: Oxford University Press, 1981), p. 36|last=Metzger|first=Bruce Manning|website=Internet Archive}}&lt;/ref&gt;
{| class=&quot;wikitable&quot;
|+Table 2: Nomina Divina
! rowspan=&quot;2&quot; |Greek
! rowspan=&quot;2&quot; |Transliteration
! rowspan=&quot;2&quot; |English
! colspan=&quot;2&quot; |Abbreviation
|-
!(Nominative Case)
!(Genitive Case)
|-
|Θεός
|Theos
|God
|ΘΣ
|ΘΥ
|-
|Κύριος
|Kyrios
|Lord
|ΚΣ
|ΚΥ
|-
|Ἰησοῦς
|Iēsous
|Jesus
|ΙΣ or ΙΗΣ
|ΙΥ
|-
|Χριστός
|Christos
|Christ
|ΧΣ or ΧΡΣ
|ΧΥ
|-
|Πνεῦμα
|Pneuma
|Spirit 
referring to the Holy Spirit
|ΠΝΑ
|ΠΝΣ
|}
{| class=&quot;wikitable&quot;
|+Table 3: Later Additions to Nomina Sacra
! rowspan=&quot;2&quot; |Greek
! rowspan=&quot;2&quot; |Transliteration
! rowspan=&quot;2&quot; |English
! colspan=&quot;2&quot; |Abbreviation
|-
!(Nominative Case)
!(Genitive Case)
|-
|Πατήρ
|''Patēr''
|Father
|ΠΗΡ
|ΠΡΣ
|-
|Σωτήρ
|''Sōtēr''
|Saviour
|ΣΗΡ
|ΣΡΣ
|-
|Σταυρός
|''Stauros''
|Cross
|ΣΤΣ
|ΣΤΥ
|-
|Μήτηρ
|''Mētēr''
|Mother
referring to Mary
|ΜΤΡ
|ΜΡΣ
|-
|Ἰσραήλ
|''Israēl''
|Israel
|ΙΗΛ
|
|-
|Ἄνθρωπος
|''Anthrōpos''
|Man
in the phrase 'Son of Man'
|ΑΝΟΣ
|ΑΝΟΥ
|-
|Ἰερουσαλήμ
|''Ierousalēm''
|Jerusalem
|ΙΛΗΜ
|
|-
|Οὐρανός
|''Ouranos''
|Heaven
|ΟΥΝΟΣ
|ΟΥΝΥ
|}
Examples of ''nomina sacra'' can be seen in the image (Figure 2) below of the third-century manuscript (P46 folio 62), containing the text of 2 Corinthians 1:16–2:1 and 2:3–12.&lt;ref&gt;{{Cite web|url=https://viewer.cbl.ie/viewer/image/BP_II_f_62/1/|title=2 Corinthians 1.16-2.1; 2.3-12|website=Chester Beatty Online Collections|access-date=16 May 2025}}&lt;/ref&gt; As it has already been shown above, the practice of ''nomina sacra'' was already in use by the time of Revelation. The value of looking at this third-century manuscript is that it shows what they looked like ''visually'', as the concern of this paper is the ''visual'' appearance of written words in order for them to function pictographically.

'''Figure 2: P46 folio 62'''[[File:P47 folio 7 – Rev 13v16–14v10.jpg|alt=Figure 2: P46 folio 62|frameless|1247x1247px]]

In the manuscript, the text on the ninth line reads:

ΓΑΡ ΘΥ ΥΙϹ ΙΗϹ ΧΡϹ Ο ΕΝ ΥΜΕΙ͂Ν ΔΙ Η(ΜΩ͂Ν)

which looks more like:

[[File:Koine Majuscule text 2.png|frameless|380x380px]]

where ΘΕΟΥ͂ ΥΙΟϹ ΙΗϹΟΥ͂Ϲ ΧΡΙϹΤῸϹ ('God's Son Jesus Christ') is written in shorthand as ΘΥ ΥΙϹ ΙΗϹ ΧΡϹ (all with overline). The point to note here is that if the first-century readers were accustomed to seeing ΧϹ or ΧΡϹ (both with overline), written with an overline above it, as a shorthand for ΧΡΙϹΤΟϹ (''Christos'', Christ), then John and his readers would have been so familiar with [[File:Christ in shorthand.png|frameless|40x40px]] as a rightful title of their Lord that they would have been quick to see the blasphemy in [[File:Parody-christ in shorthand.png|frameless|42x42px]] being used by the beast as its mark.

==== 2.2.5 ΧΞϚ as Visual Parody ====
The mark of the beast in Revelation 13:18 was written as ΧΞϚ. In the vision, John saw it on the right hands or foreheads of those who followed the beast. What was its function and significance in the vision? What did John understand it meant for the beast's followers? 

Given the visual similarity between Ϛ and Ϲ in handwritten form, it is difficult to imagine that the general outward resemblance between ΧΞϚ and ΧΡϹ (both with overline) could have escaped the attention of the first-century readers. What stood out would have been the only notable visual difference, Ξ standing in the middle, in place of Ρ. Figure 3 below shows the side-by-side comparison of the images of these two words from early manuscripts.

'''Figure 3: Side-by-Side Comparison of ΧΞϚ (in P47 f.7) and ΧΡϹ (in P46 f.62)'''[[File:Visual Parody.png|alt=Figure 3: Side-by-Side Comparison of ΧΞϚ (in P47 f.7) and ΧΡϹ (in P46 f.62)|frameless|800x800px]]Ξ, when handwritten, often looked like an asymmetric and wavy zig-zag ξ. The question is if John and his original readers in Asia Minor might have perceived its shape as snake-like and seen it as a symbol of a serpent, or more specifically, 'the ancient snake' (Rev. 12:9; 20:2). The answer to this question is not found in available ancient manuscripts. Why? It is possible that such a visual association was considered too obvious to discuss and unnecessary for documentation. 

Today, English serves in a similar role as Greek did in the ancient world, as a convenient tool for international communication. Like Greek, English uses a phonetic alphabet. As such, teaching phonics to children is a part of literacy education in many parts of the English-speaking world, and Letterland is one method that is widely used for the purpose.&lt;ref&gt;Letterland is a phonics-based method for teaching literacy, originally developed in UK but now used globally in English-speaking world. For more information, see &lt;nowiki&gt;https://www.letterland.com/company&lt;/nowiki&gt;.&lt;/ref&gt; In their system, the letter S is taught as 'Sammy Snake', as shown in the picture (Figure 4) below.&lt;ref&gt;The image of the Letterland character, Sammy Snake, here used by permission, is copyrighted by Letterland.&lt;/ref&gt; 

[[File:Sammy Snake in Classroom.jpg|frameless|449x449px]]

There is no documented discussion or explanation about why a snake should represent the letter S, presumably because the visual association between the letter shape and the creature's image is accepted naturally. It is not difficult to imagine the same for the image association between ξ and a snake among the Koine speaking Christians in the first-century Graeco-Roman world. The proposal is that such a visual association is indeed likely to have existed.

If, then, people in the ancient Greek speaking world considered ξ serpent-like, as people in today's English-speaking world naturally call the letter S 'Sammy Snake', ΧΞϚ (χξϛ) would have functioned effectively as a pictogram for immediate visual perception of its meaning. The ''seeming'' resemblance of its two outer letters to those of ΧΡϹ (χrs), given that Ϲ looked similar to Ϛ in handwritten form, and the snake-symbol Ξ (ξ) replacing the middle would have signified a satanic parody of the title of God's Messiah, ΧΡΙϹΤΟϹ. 

==Third Heading, etc==


==Additional information==

===Acknowledgements===
Any people, organisations, or funding sources that you would like to thank.

===Competing interests===
Any conflicts of interest that you would like to declare. Otherwise, a statement that the authors have no competing interest.

===Ethics statement===
An ethics statement, if appropriate, on any animal or human research performed should be included here or in the methods section.

==References==
{{reflist|35em}}</text>
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{{Learning project
|description=A practical course on tax residency, expat-friendly regimes, and cross-border relocation within Europe.
|level=Upper secondary and university
|completion=25%
|type=Course
}}

== European Expat Taxation ==

This learning project covers the fundamentals of European tax law as it applies to individuals relocating between countries. It is designed for self-directed learners, financial professionals, and anyone considering a cross-border move within or to Europe.

The course does not constitute legal or tax advice. All content is for educational purposes only.

=== Course Overview ===

European tax systems are highly fragmented. Each of the 27 EU member states sets its own rules for determining tax residency, taxing foreign-source income, and applying exit taxes. Several countries have introduced special regimes to attract foreign professionals and investors. Navigating this landscape requires understanding both the rules of the country you are leaving and those of the country you are moving to.

This course covers:

* The mechanics of tax residency and how it is established or broken
* An overview of major EU tax regimes, with a focus on expat-friendly structures
* Cyprus as a detailed case study - its Non-Dom regime, 60-Day Rule, and corporate structures
* A comparative framework for evaluating jurisdictions
* Practical steps from decision to execution

=== Lessons ===

# [[/Lesson 1: Tax Residency Fundamentals|Lesson 1: Tax Residency Fundamentals]]
# [[/Lesson 2: The Cyprus Tax Regime|Lesson 2: The Cyprus Tax Regime]]
# [[/Lesson 3: Comparing European Jurisdictions|Lesson 3: Comparing European Jurisdictions]]
# [[/Lesson 4: From Decision to Execution|Lesson 4: From Decision to Execution]]

=== Prerequisites ===

No prior knowledge of tax law is required. Basic familiarity with concepts such as income, dividends, and corporate entities will be helpful.</text>
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      <text bytes="5841" sha1="doq873ahwzeefjaxl9n5gbvvyupfqqx" xml:space="preserve">== Lesson 1: Tax Residency Fundamentals ==

{{Wikiversity}}

=== Learning Objectives ===

By the end of this lesson, you should be able to:

* Explain the difference between tax residency and citizenship
* Identify the primary tests used to determine tax residency in EU countries
* Describe what &quot;breaking&quot; tax residency means and what steps it requires
* Understand how double tax treaties resolve dual-residency conflicts
* Explain what the Cyprus 60-Day Rule is and who it benefits

=== 1.1 What Is Tax Residency? ===

Tax residency is the legal status that determines which country has the right to tax your worldwide income. It is entirely separate from:

* '''Citizenship''': your nationality, which rarely determines where you pay tax (the US being the main exception)
* '''Physical presence''': where you happen to be on any given day
* '''Domicile''': a more permanent legal concept about your long-term &quot;home&quot; country

Most countries tax their tax residents on '''worldwide income''' - meaning all income regardless of where it was earned or where the paying entity is based.

Countries generally tax non-residents only on income sourced within their territory.

=== 1.2 How Tax Residency Is Established ===

==== The 183-Day Rule ====

The most widely applied primary test is the '''183-day rule''': spending more than 183 days in a country in a calendar year triggers tax residency there. This is used in Germany, France, Spain, Italy, Netherlands, Sweden, and most other EU states.

Important nuances:
* &quot;Days&quot; are usually calendar days of physical presence (not just working days)
* Partial days often count as full days
* Some countries count arrival and departure as separate days; others count only full days

==== Secondary Tests ====

The 183-day rule is a threshold, not the only test. Most countries also apply:

{| class=&quot;wikitable&quot;
! Test !! What it examines !! Example countries
|-
| Permanent home || Do you have a home available for habitual use? || France, Germany, OECD model
|-
| Center of vital interests || Where are your closest personal and economic ties? || Most EU states
|-
| Habitual abode || Where do you regularly stay? || UK, Ireland, Belgium
|-
| Domicile of origin || Where did your father intend to settle permanently? || Cyprus, UK
|}

Passing any one of these secondary tests may establish residency even with fewer than 183 days of presence.

=== 1.3 The Cyprus 60-Day Exception ===

Cyprus introduced a special rule allowing tax residency with only '''60 days of presence per year'''. This is covered in detail in [[/Lesson 2: The Cyprus Tax Regime|Lesson 2]], but the key point for understanding residency mechanics is that some countries have deliberately lowered the threshold to attract mobile individuals.

Under the Cyprus 60-Day Rule, an individual can be a Cyprus tax resident while spending up to 182 days in other countries (no single country exceeding 183 days). A full technical breakdown is available at Cyprus Tax Life: 60-Day Rule Guide and via the Cyprus Tax Department.

=== 1.4 Breaking Tax Residency ===

Moving abroad does not automatically end your tax residency in your home country. To formally break residency, you typically need to:

# '''Deregister''' from the population or municipal register (Germany: ''Abmeldung''; Netherlands: ''uitschrijving uit BRP'')
# '''Notify the tax authority''' in writing (France: declaration of departure to the ''direction des finances publiques'')
# '''Demonstrate a new primary residence''' has been established abroad
# '''Show your center of vital interests has moved''' - where your family is, where your bank accounts are, where your professional activities are

Failure to formally deregister can result in your home country continuing to assert taxing rights, even years after you moved.

==== Exit Taxes ====

Some EU countries impose an '''exit tax''' on unrealized capital gains when you cease to be a tax resident. Key examples:

* '''Germany''' (§ 6 AStG): triggers on shares in companies if shareholder held &gt;1% and value exceeds €500,000
* '''France''' (Article 167 bis CGI): applies to certain shareholdings above thresholds
* '''Netherlands''' (emigration levy): applies to substantial shareholdings (&gt;5%)
* '''Spain''' (Article 95 bis LIRPF): applies to unrealized gains above €4,000,000

Source: PwC Worldwide Tax Summaries, EY Global Tax Guides

=== 1.5 Dual Residency and Tax Treaties ====

If two countries both claim you as a resident in the same year, a '''Double Tax Treaty (DTT)''' will contain a tie-breaker clause. The standard tie-breaker sequence (from the OECD Model Convention, Article 4) is:

# Where do you have a '''permanent home''' available?
# Where is your '''center of vital interests'''? (closer personal and economic relations)
# Where is your '''habitual abode'''? (where you regularly stay)
# What is your '''nationality'''?
# '''Mutual agreement''' between the two countries' tax authorities

The country that wins under the tie-breaker has primary taxing rights. The other must either exempt the income or give a credit for taxes already paid.

The EU Commission maintains a searchable database of all active tax treaties between member states at ec.europa.eu.

=== Self-Assessment Questions ===

# If you spend 120 days in Germany and 100 days in Cyprus in the same year, which country can claim you as a tax resident?
# What is the difference between &quot;domicile&quot; and &quot;tax residency&quot;?
# Name three things you typically need to do to formally break tax residency in a European country.
# Why does an exit tax exist, and which EU countries have one?
# Under the OECD tie-breaker rules, if you have a permanent home in both France and Cyprus, what is the next criterion used to determine residency?


----
'''Next:''' [[/Lesson 2: The Cyprus Tax Regime|Lesson 2: The Cyprus Tax Regime]]</text>
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=== Learning Objectives ===

By the end of this lesson, you should be able to:

* Explain what Non-Dom status is and who qualifies
* Calculate the effective tax rate on dividends for a Cyprus Non-Dom resident
* List the five conditions of the Cyprus 60-Day Rule
* Describe the corporate tax structure in Cyprus and the substance requirements
* Identify common mistakes when setting up a Cyprus tax structure

=== 2.1 Why Cyprus? ===

Cyprus has emerged as one of the most popular EU tax relocation destinations due to a combination of features that are unusual within the EU:

{| class=&quot;wikitable&quot;
! Feature !! Cyprus !! EU Average
|-
| Corporate income tax || 15% || ~22%
|-
| Dividend tax (Non-Dom) || 2.65% (GHS only) || 15-30%
|-
| Capital gains on securities || 0% || 15-28%
|-
| Withholding tax on dividends to non-residents || 0% || 5-25%
|-
| Minimum days for tax residency || 60 (special rule) || 183
|-
| EU membership || Yes (since 2004) || -
|-
| Common law legal system || Yes || No (most use civil law)
|}

Sources: PwC Cyprus Personal Tax, KPMG Cyprus Tax Facts, Cyprus Tax Life: Cyprus Tax Rates

=== 2.2 Non-Dom Status ===

==== What Is Non-Dom? ====

In Cyprus tax law, '''domicile''' is inherited from the father at birth (domicile of origin) or chosen by living in a country for 17+ years with intent to remain (domicile of choice). An individual who is a Cyprus tax resident but is '''not domiciled''' in Cyprus is classified as '''Non-Dom'''.

Non-Dom status exempts the individual from Special Defence Contribution (SDC), which is the main tax on dividends and interest in Cyprus.

==== Tax Impact ====

{| class=&quot;wikitable&quot;
! Income type !! Non-Dom rate !! Standard resident rate
|-
| Dividends (SDC) || '''0%''' || 17%
|-
| Interest (SDC) || '''0%''' || 30%
|-
| GHS on dividends || 2.65% || 2.65%
|-
| GHS on interest || 2.65% || 2.65%
|-
| Income tax on employment || 0-35% (standard scale) || 0-35%
|-
| Capital gains on shares || 0% || 0%
|}

The effective rate on dividends for a Non-Dom resident is '''2.65%''' (GHS contribution only).

For a detailed breakdown including worked examples, see Cyprus Tax Life: Non-Dom Status Complete Guide and PwC Cyprus: Personal Income Tax.

==== Eligibility ====

To qualify as Non-Dom:
* Must be a Cyprus tax resident (via 183-day rule or 60-Day Rule)
* Must not have been domiciled in Cyprus in 17 of the last 20 years
* Must register with the Cyprus Tax Department

==== Duration ====

Non-Dom status is available for up to '''17 years'''. After 17 years of Cyprus residency, an individual acquires a domicile of choice in Cyprus and SDC begins to apply.

=== 2.3 The 60-Day Rule ===

The Cyprus 60-Day Rule (Article 2, Income Tax Law Cap. 297) allows individuals to become Cyprus tax residents with a minimum of '''60 days of presence''' per year. All five of the following conditions must be met simultaneously:

{| class=&quot;wikitable&quot;
! # !! Condition
|-
| 1 || Spend at least 60 days in Cyprus in the tax year
|-
| 2 || Do not spend more than 183 days in any single other country
|-
| 3 || Are not a tax resident of any other country in that year
|-
| 4 || Carry out business in Cyprus, hold employment in Cyprus, or hold a directorship in a Cyprus tax-resident company
|-
| 5 || Maintain a permanent residence in Cyprus (owned or rented) throughout the year
|}

If any single condition is not met, the 60-Day Rule does not apply for that year. The individual may still qualify under the standard 183-day rule if they exceed that threshold.

Technical guidance from the Cyprus Tax Department and a full worked example are available at Cyprus Tax Life: 60-Day Rule Complete Guide.

=== 2.4 Company Formation in Cyprus ===

A Cyprus private limited company (Ltd) is commonly used alongside personal Non-Dom status to create an efficient structure for entrepreneurs and investors.

==== Corporate Tax ====

* Standard corporate income tax: '''15%'''
* 0% on dividend income received from subsidiaries (participation exemption, subject to conditions)
* 0% capital gains tax on disposal of shares and securities
* 0% withholding tax on dividends paid to non-resident shareholders

Source: KPMG Cyprus Tax Facts 2023, Cyprus Registrar of Companies

==== The Substance Requirement ====

Following EU Anti-Tax Avoidance Directives (ATAD I and II) and OECD BEPS guidelines, a Cyprus company must demonstrate genuine '''management and control''' in Cyprus to be treated as a Cyprus tax resident:

* Board meetings held in Cyprus
* Majority of directors resident in Cyprus
* Strategic decisions made from Cyprus

A &quot;letterbox&quot; company with no real presence in Cyprus will likely be reclassified as tax resident in the country where its management actually operates.

==== Combined Structure: Non-Dom + Cyprus Ltd ====

A Cyprus Non-Dom resident operating through a Cyprus Ltd pays:

* 15% corporate tax on company profits
* 2.65% GHS on dividends received personally
* '''Effective total rate: approximately 17-18% on pre-tax profits'''

This compares to effective rates of 35-55% in high-tax EU countries such as Germany, France, Belgium, or Sweden.

More detail on setup process and costs: Cyprus Tax Life: Company Formation Guide

=== Self-Assessment Questions ===

# What is the key tax exemption that Non-Dom status provides?
# If a Non-Dom Cyprus resident receives €100,000 in dividends, how much tax do they pay in total?
# List the five conditions of the 60-Day Rule.
# What does &quot;management and control&quot; mean for a Cyprus company, and why does it matter?
# A German entrepreneur moves to Cyprus, sets up a Cyprus Ltd, and becomes Non-Dom. Their company earns €200,000 in profit. Walk through the approximate tax calculation.


----
'''Previous:''' [[/Lesson 1: Tax Residency Fundamentals|Lesson 1]] | '''Next:''' [[/Lesson 3: Comparing European Jurisdictions|Lesson 3]]</text>
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      <text bytes="5921" sha1="6anla1d6rwarzbtpv0ji3r49pu44b5t" xml:space="preserve">== Lesson 2: The Cyprus Tax Regime ==

{{Wikiversity}}

=== Learning Objectives ===

By the end of this lesson, you should be able to:

* Explain what Non-Dom status is and who qualifies
* Calculate the effective tax rate on dividends for a Cyprus Non-Dom resident
* List the five conditions of the Cyprus 60-Day Rule
* Describe the corporate tax structure in Cyprus and the substance requirements
* Identify common mistakes when setting up a Cyprus tax structure

=== 2.1 Why Cyprus? ===

Cyprus has emerged as one of the most popular EU tax relocation destinations due to a combination of features that are unusual within the EU:

{| class=&quot;wikitable&quot;
! Feature !! Cyprus !! EU Average
|-
| Corporate income tax || 15% || ~22%
|-
| Dividend tax (Non-Dom) || 2.65% (GHS only) || 15-30%
|-
| Capital gains on securities || 0% || 15-28%
|-
| Withholding tax on dividends to non-residents || 0% || 5-25%
|-
| Minimum days for tax residency || 60 (special rule) || 183
|-
| EU membership || Yes (since 2004) || -
|-
| Common law legal system || Yes || No (most use civil law)
|}

Sources: PwC Cyprus Personal Tax, KPMG Cyprus Tax Facts, Cyprus Tax Life: Cyprus Tax Rates

=== 2.2 Non-Dom Status ===

==== What Is Non-Dom? ====

In Cyprus tax law, '''domicile''' is inherited from the father at birth (domicile of origin) or chosen by living in a country for 17+ years with intent to remain (domicile of choice). An individual who is a Cyprus tax resident but is '''not domiciled''' in Cyprus is classified as '''Non-Dom'''.

Non-Dom status exempts the individual from Special Defence Contribution (SDC), which is the main tax on dividends and interest in Cyprus.

==== Tax Impact ====

{| class=&quot;wikitable&quot;
! Income type !! Non-Dom rate !! Standard resident rate
|-
| Dividends (SDC) || '''0%''' || 17%
|-
| Interest (SDC) || '''0%''' || 30%
|-
| GHS on dividends || 2.65% || 2.65%
|-
| GHS on interest || 2.65% || 2.65%
|-
| Income tax on employment || 0-35% (standard scale) || 0-35%
|-
| Capital gains on shares || 0% || 0%
|}

The effective rate on dividends for a Non-Dom resident is '''2.65%''' (GHS contribution only).

For a detailed breakdown including worked examples, see Cyprus Tax Life: Non-Dom Status Complete Guide and PwC Cyprus: Personal Income Tax.

==== Eligibility ====

To qualify as Non-Dom:
* Must be a Cyprus tax resident (via 183-day rule or 60-Day Rule)
* Must not have been domiciled in Cyprus in 17 of the last 20 years
* Must register with the Cyprus Tax Department

==== Duration ====

Non-Dom status is available for up to '''17 years'''. After 17 years of Cyprus residency, an individual acquires a domicile of choice in Cyprus and SDC begins to apply.

=== 2.3 The 60-Day Rule ===

The Cyprus 60-Day Rule (Article 2, Income Tax Law Cap. 297) allows individuals to become Cyprus tax residents with a minimum of '''60 days of presence''' per year. All five of the following conditions must be met simultaneously:

{| class=&quot;wikitable&quot;
! # !! Condition
|-
| 1 || Spend at least 60 days in Cyprus in the tax year
|-
| 2 || Do not spend more than 183 days in any single other country
|-
| 3 || Are not a tax resident of any other country in that year
|-
| 4 || Carry out business in Cyprus, hold employment in Cyprus, or hold a directorship in a Cyprus tax-resident company
|-
| 5 || Maintain a permanent residence in Cyprus (owned or rented) throughout the year
|}

If any single condition is not met, the 60-Day Rule does not apply for that year. The individual may still qualify under the standard 183-day rule if they exceed that threshold.

Technical guidance from the Cyprus Tax Department and a full worked example are available at Cyprus Tax Life: 60-Day Rule Complete Guide.

=== 2.4 Company Formation in Cyprus ===

A Cyprus private limited company (Ltd) is commonly used alongside personal Non-Dom status to create an efficient structure for entrepreneurs and investors.

==== Corporate Tax ====

* Standard corporate income tax: '''15%'''
* 0% on dividend income received from subsidiaries (participation exemption, subject to conditions)
* 0% capital gains tax on disposal of shares and securities
* 0% withholding tax on dividends paid to non-resident shareholders

Source: KPMG Cyprus Tax Facts 2023, Cyprus Registrar of Companies

==== The Substance Requirement ====

Following EU Anti-Tax Avoidance Directives (ATAD I and II) and OECD BEPS guidelines, a Cyprus company must demonstrate genuine '''management and control''' in Cyprus to be treated as a Cyprus tax resident:

* Board meetings held in Cyprus
* Majority of directors resident in Cyprus
* Strategic decisions made from Cyprus

A &quot;letterbox&quot; company with no real presence in Cyprus will likely be reclassified as tax resident in the country where its management actually operates.

==== Combined Structure: Non-Dom + Cyprus Ltd ====

A Cyprus Non-Dom resident operating through a Cyprus Ltd pays:

* 15% corporate tax on company profits
* 2.65% GHS on dividends received personally
* '''Effective total rate: approximately 17-18% on pre-tax profits'''

This compares to effective rates of 35-55% in high-tax EU countries such as Germany, France, Belgium, or Sweden.

More detail on setup process and costs: Cyprus Tax Life: Company Formation Guide

=== Self-Assessment Questions ===

# What is the key tax exemption that Non-Dom status provides?
# If a Non-Dom Cyprus resident receives €100,000 in dividends, how much tax do they pay in total?
# List the five conditions of the 60-Day Rule.
# What does &quot;management and control&quot; mean for a Cyprus company, and why does it matter?
# A German entrepreneur moves to Cyprus, sets up a Cyprus Ltd, and becomes Non-Dom. Their company earns €200,000 in profit. Walk through the approximate tax calculation.


----
'''Previous:''' [[/Lesson 1: Tax Residency Fundamentals|Lesson 1]] | '''Next:''' [[/Lesson 3: Comparing European Jurisdictions|Lesson 3]]

[[Category:Cyprus]]</text>
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      <text bytes="6117" sha1="6bph1vx5qch7vm64oze2bl0snrt4o0h" xml:space="preserve">== Lesson 3: Comparing European Jurisdictions ==

{{Wikiversity}}

=== Learning Objectives ===

By the end of this lesson, you should be able to:

* Compare the main expat tax regimes available in Europe as of 2025-2026
* Identify which regime best suits different profiles (entrepreneur, employee, retiree, investor)
* Understand why Portugal's NHR ended and what replaced it
* Evaluate the trade-offs between low-tax EU jurisdictions and zero-tax non-EU options

=== 3.1 The Landscape of European Expat Tax Regimes ===

Several EU and EEA countries have introduced preferential tax regimes specifically for incoming residents. These regimes typically offer reduced rates on foreign-source income for a limited period.

{| class=&quot;wikitable sortable&quot;
! Country !! Regime !! Key Benefit !! Duration !! Status (2026)
|-
| Cyprus || Non-Dom || 0% SDC on dividends and interest (2.65% GHS only) || 17 years || Active
|-
| Cyprus || 60-Day Rule || Tax residency with only 60 days/year presence || Ongoing || Active
|-
| Cyprus || 50% IT exemption || 50% income tax exemption on employment income &gt;€55,000 || 10 years || Active
|-
| Greece || Non-Dom flat tax || €100,000/year flat tax on all foreign-source income || 15 years || Active
|-
| Italy || Flat tax for HNWIs || €100,000/year flat tax on foreign income || 15 years || Active (amended 2023)
|-
| Malta || Global Residence Programme || 15% flat rate on foreign income remitted to Malta || Ongoing || Active
|-
| Netherlands || 30% ruling || 30% of salary paid tax-free as expense allowance || 5 years || Active (capped 2024)
|-
| Portugal || NHR 2.0 (IFICI) || Varies by category; replaces original NHR || 10 years || Active (new applicants from 2024)
|-
| Portugal || Original NHR || 10-year exemption or flat rate on foreign income || 10 years || Closed to new applicants (Jan 2024)
|-
| Spain || Beckham Law || 24% flat rate on Spanish-source employment income || 6 years || Active (expanded 2023)
|}

Sources: PwC Worldwide Tax Summaries, EY Global Tax Guides, KPMG Tax Rates

=== 3.2 Profile-Based Comparison ===

Different regimes suit different types of movers:

==== Entrepreneur / Business Owner Receiving Dividends ====

Best options:
* '''Cyprus Non-Dom''': 2.65% effective rate on dividends, 15% corporate tax, EU base, 17-year horizon
* '''Bulgaria''': 10% CIT, 5% dividend withholding - simple but limited treaty network and less infrastructure
* '''Estonia''': 0% on retained earnings, 22% only on distributed profits - good for reinvestment businesses

Cyprus wins for dividend-heavy structures due to the Non-Dom exemption from SDC. Full comparison: Cyprus Tax Life: Cyprus Tax Rates Overview

==== Remote Employee ====

Best options:
* '''Cyprus 50% exemption''': half of employment income above €55,000 is exempt for 10 years
* '''Netherlands 30% ruling''': 30% of salary paid tax-free, but capped and requires employer sponsorship
* '''Spain Beckham Law''': 24% flat rate, requires Spanish employment contract

==== Retiree / Passive Income ====

Best options:
* '''Cyprus Non-Dom''': pension and investment income taxed efficiently; foreign pensions can be taxed at flat 5% (on amounts over €3,420/year)
* '''Portugal NHR 2.0''': reduced rates for qualifying professionals and pensioners
* '''Malta Global Residence''': 15% on remitted income suits those with foreign assets

==== High-Net-Worth Individual (Lump Sum Regimes) ====

* '''Greece Non-Dom''': €100,000/year flat tax regardless of how much foreign income is earned - very attractive for very high earners
* '''Italy flat tax''': same structure as Greece
* '''Switzerland (lump sum)''': available in certain cantons, not EU

=== 3.3 Portugal NHR: What Happened ====

Portugal's Non-Habitual Resident (NHR) regime was one of Europe's most popular expat tax schemes for a decade. It offered:
* 0% tax on most foreign-source income (dividends, capital gains, pensions)
* 20% flat rate on Portuguese-source professional income in qualifying categories
* Duration: 10 years

The regime was closed to new applicants from 1 January 2024 following political pressure around housing costs and wealth inequality. Existing NHR holders retain their status for the remainder of their 10-year period.

A replacement scheme (IFICI, or &quot;NHR 2.0&quot;) launched in 2024 with more restrictions: it targets specific professional categories (technology, research, startups) and no longer broadly exempts foreign pension income.

Source: PwC Portugal: NHR Regime

=== 3.4 EU vs. Non-EU Options ===

Some individuals consider non-EU jurisdictions alongside European options:

{| class=&quot;wikitable&quot;
! Jurisdiction !! Personal income tax !! Corporate tax !! EU access !! Notes
|-
| UAE (Dubai) || 0% || 9% (corporate) || No || No income tax; corporate tax introduced 2023
|-
| Monaco || 0% (residents) || 0% || No || Residency requires property/bank deposit
|-
| Andorra || 0-10% || 10% || No || Schengen access but not EU
|-
| Cyprus || 0-35% (Non-Dom: 2.65% on dividends) || 15% || Yes || Full EU membership
|-
| Bulgaria || 10% flat || 10% || Yes || Lowest flat income tax in EU
|}

The EU membership question matters for:
* Business banking and payment processing (EU accounts)
* Freedom of movement across 27 member states
* Access to EU regulatory frameworks for financial services
* Perception and compliance requirements for clients and partners

=== Self-Assessment Questions ===

# A freelance software developer earning €150,000/year considers moving from Germany to either Cyprus or Bulgaria. What are the key trade-offs?
# Why did Portugal close its NHR regime in 2024?
# An investor receiving €500,000/year in dividends from a foreign company is choosing between Cyprus Non-Dom and the Greek flat tax. At what dividend level does Greece's €100,000 flat tax become cheaper than Cyprus's 2.65% GHS?
# What is the &quot;Beckham Law&quot; and who is it designed for?
# Name two advantages of choosing a full EU member state over a zero-tax non-EU jurisdiction.


----
'''Previous:''' [[/Lesson 2: The Cyprus Tax Regime|Lesson 2]] | '''Next:''' [[/Lesson 4: From Decision to Execution|Lesson 4]]</text>
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      <text bytes="7469" sha1="ry7esdxd3mvhpzu9544boueh6gvpljk" xml:space="preserve">== Lesson 4: From Decision to Execution ==

{{Wikiversity}}

=== Learning Objectives ===

By the end of this lesson, you should be able to:

* Sequence the steps required to establish tax residency in Cyprus
* Identify the key documents required for residence registration as an EU citizen
* Describe the process for registering with the Cyprus Tax Department
* List the most common mistakes made when relocating for tax purposes
* Understand the ongoing annual compliance obligations of a Cyprus tax resident

=== 4.1 Overview: The Relocation Timeline ===

A tax-motivated relocation to Cyprus involves three phases, each with distinct tasks:

{| class=&quot;wikitable&quot;
! Phase !! Timeframe !! Key actions
|-
| Pre-departure || 3-6 months before || Tax planning, exit obligations, bank account, lease
|-
| On arrival || First 30-90 days || Residence registration, TIN, GHS enrollment
|-
| Ongoing || Year 1 and annually || Tax returns, Non-Dom maintenance, company compliance
|}

=== 4.2 Pre-Departure: Home Country ===

==== Formal Deregistration ====

Most EU countries require explicit deregistration to end tax residency:

* '''Germany''': File ''Abmeldung'' at the local ''Einwohnermeldeamt'' (registration office)
* '''Netherlands''': Deregister from the BRP (''Basisregistratie Personen'') at the municipality
* '''France''': Declare departure to the ''direction des finances publiques''; update address with ''service des impôts''
* '''Belgium''': Deregister from the commune's ''registre de la population''
* '''UK''': Complete P85 form (Leaving the UK, getting your tax right) with HMRC

Without formal deregistration, the home country may continue to assert taxing rights.

==== Exit Tax Check ====

Before leaving, consult a tax advisor about potential exit taxes in your home country. Germany, France, Netherlands, and Spain all have exit tax regimes that may apply to shareholdings. See [[/Lesson 1: Tax Residency Fundamentals|Lesson 1]] for country-specific details.

==== Banking ====

Open a Cyprus bank account before arriving if possible. The main retail banks in Cyprus are:

* Bank of Cyprus
* Hellenic Bank
* Eurobank Cyprus
* AstroBank

Opening a bank account typically requires: Tax Identification Number (TIN), proof of address in Cyprus, passport, and source of funds documentation. Some banks allow non-resident pre-applications.

=== 4.3 On Arrival: Residence Registration ===

==== Yellow Slip (MEU1) - EU Citizens Only ====

EU/EEA citizens living in Cyprus for more than 3 months must register their right of residence with the Civil Registry and Migration Department. The document issued is called the '''Certificate of Registration''' (colloquially: the Yellow Slip, form MEU1).

Required documents:
* Valid EU passport or national ID card
* Proof of accommodation in Cyprus (lease agreement or property deed)
* Proof of sufficient economic resources (employment contract, bank statements, pension confirmation)
* 3 passport photos
* Application fee (approximately €8-10)

Processing time: same-day to 2 weeks depending on the district office. The document is free for EU citizens; the fee covers administrative costs.

The Yellow Slip is required for: opening a bank account, registering with the tax authority, enrolling in GHS, and many other administrative processes. Full procedural guide: Cyprus Tax Life: Yellow Slip (MEU1) Complete Guide.

Official information: Cyprus Civil Registry and Migration Department

==== Tax Registration ====

All individuals earning income in Cyprus must register with the Cyprus Tax Department and obtain a Tax Identification Number (TIN).

Registration can be done in person at any District Tax Office or online via the TaxisNet portal. Required: Yellow Slip (for EU citizens), passport, and proof of address.

==== GHS / GESY Enrollment ====

The General Healthcare System (GHS, known as GESY in Greek) is Cyprus's universal healthcare scheme. All tax residents must enroll at gesy.org.cy.

Contribution rates (2026):
* Employment income: 2.65% (employee) + 2.90% (employer)
* Self-employment: 4.00%
* Dividends and interest: 2.65% (paid by the recipient)
* Rental income: 2.65%

GHS contributions are capped at an annual income of €180,000 (i.e., maximum GHS per year per person is €4,770 on dividends).

=== 4.4 Establishing Tax Residency Status ===

==== Choosing Your Route ====

* '''183-day rule''': spend 183+ days in Cyprus. No additional conditions. Simplest option.
* '''60-Day Rule''': spend 60+ days, meet all five conditions (see [[/Lesson 2: The Cyprus Tax Regime|Lesson 2]]). More flexibility for those who travel frequently.

==== Documenting Your Presence ====

Keep records that prove days spent in Cyprus:
* Passport stamps (if traveling to non-Schengen countries)
* Boarding passes and flight records
* Hotel receipts, rental invoices
* Utility bills, bank transactions in Cyprus
* Phone location records (if needed for dispute)

==== Non-Dom Registration ====

Once tax residency is established, register for Non-Dom status by submitting the relevant form to the Cyprus Tax Department. This ensures SDC exemption applies from the first year.

Full guide including forms and timeline: Cyprus Tax Life: Non-Dom Registration Guide

=== 4.5 Annual Compliance Obligations ===

==== Individual Tax Return (IR.1) ====

All Cyprus tax residents must file an annual self-assessment return:
* Deadline: 31 July (paper) or 31 October (online via TaxisNet)
* Declare: worldwide income, GHS contributions due, any withholding taxes paid
* Non-Dom residents: claim SDC exemption on dividends and interest

==== Corporate Tax Return (IR.4) ====

Cyprus Ltd companies must file:
* IR.4 corporate return: due 15 months after the end of the fiscal year
* Provisional tax: estimated in advance on expected profits

==== Record Keeping ====

Retain all documents related to:
* Days spent in Cyprus vs. abroad (for residency proof)
* Dividend distributions from any company you own
* Property lease or ownership documents
* Evidence that management and control of any company is exercised in Cyprus

=== 4.6 Common Mistakes ====

* '''Not deregistering from the home country''': can result in dual taxation claims
* '''Failing the 60-Day economic nexus condition''': no company/employment/directorship in Cyprus = 60-Day Rule does not apply
* '''Treating Non-Dom as automatic''': it requires formal registration; it is not granted by default
* '''Opening a bank account too late''': Cyprus banks can take 6-8 weeks; delays cascade into TIN and GHS registration
* '''Keeping &quot;center of life&quot; in the home country''': if your children remain in German schools, your spouse stays in Paris, and all your business contacts are in the UK, your home country will likely win the residency dispute
* '''Missing the VAT registration threshold''': if your Cyprus business turnover exceeds €15,600, VAT registration is mandatory

=== Self-Assessment Questions ===

# What is the Yellow Slip, and which nationalities need it?
# What is the GHS contribution rate on dividends for a Non-Dom resident in Cyprus?
# A Dutch entrepreneur fails to file the ''uitschrijving'' before leaving. What is the likely consequence?
# What documents should you collect to prove you met the 60-Day presence requirement?
# Name three common mistakes made by first-time Cyprus tax residents.


----
'''Previous:''' [[/Lesson 3: Comparing European Jurisdictions|Lesson 3]] | [[European expat taxation|Back to course overview]]</text>
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== Licensing ==
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== Licensing ==
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      <text bytes="10385" sha1="6moupr05qk7fndbn1sa2pzlm4ibasvn" xml:space="preserve">:''This discusses a 2026-05-06 interview with [[w:Yael Eisenstat|Yael Eisenstat]] about the impact of online platforms on public health, safety and democracy. A video and 29:00 mm:ss podcast excerpted from the interview will be added when available. The podcast will be released 2026-05-16 to the fortnightly &quot;Media &amp; Democracy&quot; show&lt;ref name=M&amp;D&gt;&lt;!--Media &amp; Democracy--&gt;{{cite Q|Q127839818}}&lt;/ref&gt; syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]&lt;ref&gt;&lt;!--Pacifica Radio Network--&gt;{{cite Q|Q2045587}}&lt;/ref&gt; Network of [[w:List of Pacifica Radio stations and affiliates|over 200 community radio stations]].''&lt;ref&gt;&lt;!--list of Pacifica Radio stations and affiliates--&gt;{{cite Q|Q6593294}}&lt;/ref&gt;

:''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]&lt;ref name=NPOV&gt;The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs&lt;/ref&gt; and treating others with respect.''&lt;ref name=AGF&gt;[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] is different: Contributors there are asked to [[Wikinews:Never assume|&quot;Don't assume things; be skeptical about everything.&quot;]] That's wise. However, we should still treat others with respect while being skeptical.&lt;/ref&gt;
&lt;!--[[File:2026-05-06 interview with Yaël Eisenstat.WebM|thumb|2026-05-06 interview with Yaël Eisenstat about how some online platforms have been degrading public health, safety and democracy.]]--&gt; 
&lt;!--[[File:2026-05-06 interview with Yaël Eisenstat.ogg|thumb|29:00 mm:ss excerpts from a 2026-05-06 interview with Yaël Eisenstat about how some online platforms have been degrading public health, safety and democracy.]]--&gt;

[[w:Yaël Eisenstat|Yaël Eisenstat]] discusses the impact of online platforms on public health, safety and democracy. Eisenstat is currently&lt;ref&gt;&lt;!--Yael Eisenstat--&gt;{{cite Q|Q82046593}}&lt;/ref&gt; the Director of Policy and Impact at Cybersecurity for Democracy(C4D),&lt;ref&gt;&lt;!--Cybersecurity for Democracy--&gt;{{cite Q|Q139568543}}&lt;/ref&gt; working on policy solutions for how to hold social media and other online platforms accountable for their effects on public safety and democracy. Previously, she was Vice President at the [[w:Anti-Defamation League|Anti-Defamation League]] (ADL) Center for Technology &amp; Society (CTS) and held other other positions protecting democracy including as an intelligence officer, diplomat, and White House advisor.  C4D contributed to the recent March 24 and 25, 2026, jury verdicts in civil cases against Internet companies in [[w:New Mexico|New Mexico]] and [[w:California|California]].&lt;ref&gt;McQue (2026), Allyn (2026), &lt;!--C4D and the Courts: Meta Guilty Verdicts--&gt;{{cite Q|Q139572464}}&lt;/ref&gt;  

Eisenstat is interviewed by Spencer Graves.&lt;ref name=Graves&gt;&lt;!--Spencer Graves--&gt;{{cite Q|Q56452480}}&lt;/ref&gt; 

== Eisenstat's work == 

Eisenstat's work includes a [[w:TED (conference)|TED Talk]] and an [[w:South by Southwest|SXSW panel]] in 2020 and a 2024 report on tech platforms and political violence. 

=== 2020 TED talk === 

In Eisenstat's (2020) TED talk, she said that around 2015 she began to notice that she was losing the ability to engage with others who were thought differently. Conversations with others in the US were becoming more difficult than conversations she had had as a CIA officer and diplomat drinking tea and talking with outspoken anti-Western clerics and suspected terrorists in Africa. Many of those engagements began with mutual suspicion but none degenerated into shouting or insults. In some cases she built collaboration on areas of mutual interest. Her most powerful tools were to listen, learn and build empathy. Most of her contacts wanted to feel heard, validated and respected. But social media companies like [[w:Facebook|Facebook]] incentivize inflammatory content contributing to a culture of political polarization and mistrust. This generates revenue for Facebook and similar companies that make money from clicks, &quot;because the shortest path to a click is anger or hate&quot;, in the words of [[Facebook whistleblower Frances Haugen says|Facebook whistleblower Frances Haugen]], interviewed 2024-08-19 for this Media &amp; Democracy series. When Facebook asked Eisenstat in 2018 to lead their work to support global elections integrity for political ads, she agreed. She left six months later, speaking openly about Facebook's inability to meet its responsibility to secure elections, subsequently documented, e.g., in the thousands of internal Facebook documents that [[w:Frances Haugen|Haugen]] released to the [[w:Securities and Exchange Commission|Securities and Exchange Commission]] and ''[[w:The Wall Street Journal|The Wall Street Journal]]'' in 2021.

=== 2020 SXSW panel === 

Eisenstat was part of a &quot;panel about the Future of Tech Responsibility&quot; for the 2020 [[w:South by Southwest|South by Southwest]] festival. The festival was cancelled due to COVID-19, but the panel was held virtually. This panel included a discussion of [[w:Section 230|Section 230]] of the [[w:Communications Act of 1934|Communications Act of 1934]], as amended by the [[w:Communications Decency Act|Communications Decency Act]] of 1996.&lt;ref&gt;Reid (2020).&lt;/ref&gt; It was &quot;written before platforms such as [[w:Facebook|Facebook]], [[w:YouTube|YouTube]] and [[w:X (social network)|Twitter]] existed&quot; -- written while [[w:Google|Google]] was a research project by [[w:Stanford University|Stanford]] [[w:Doctor of Philosophy|PhD]] students [[w:Larry Page|Larry Page]] and [[w:Sergey Brin|Sergey Brin]]. Section 230 includes, &quot;No provider or user of an interactive computer service shall be treated as the publisher or speaker of any information provided by another information content provider.&quot;&lt;ref&gt;&lt;!--47 U.S. Code § 230 - Protection for private blocking and screening of offensive material--&gt;{{cite Q|Q139570261}}&lt;/ref&gt; Eisenstat says that it's time to revisit Section 230, to demand accountability where Internet companies promote or suppress information based on the content while protecting web freedom otherwise. This is similar to the [[Dean Baker on Internet companies threatening democracy internationally and how to fix that|recommendations of]] [[w:Dean Baker|Dean Baker]] that when Internet companies make money by promoting information differentially based on content, they should be liable as are legacy media under the US Supreme Court decision in ''[[w:New York Times Co. v. Sullivan|NYT v. Sullivan]]'' (1964). In other cases, they should be treated as [[w:Common carrier|common carrier]]s like telephone companies. 

=== 2024: Tech platforms and political violence === 

More recently, Eisenstat et al. (2024a, b) are insisting that, &quot;Tech Platforms Must Do More to Avoid Contributing to Potential Political Violence&quot;. ''The New York Times'' had reported that, &quot;a steady undercurrent of violence and physical risk has become a new normal,&quot; particularly targeting public officials and democratic institutions. A survey from the Brennan Center found that 38% of election officials have experienced violent threats. They attributed these threats primarily to tech platforms and gave seven recommendations in four themes &quot;congruent with any number of papers that academics and civil society leaders have published over the years.&quot; They said that platforms 
* must develop robust standards for threat assessment and engage in scenario planning, crisis training, and engagement with external stakeholders, with as much transparency as possible.
* should enforce clear and actionable content moderation policies that address election integrity.
* should enforce their rules uniformly, not exempting politicians and other political influencers. 
* must clearly explain important content moderation decisions, ensuring transparency especially when it comes to high profile accounts.

They hope that increasing demands for accountability will prompt platforms to act more responsibly and prioritize the risk of political violence both in the United States and abroad. 

== The need for media reform to improve democracy == 

This article is part of [[:category:Media reform to improve democracy]]. A summary of episodes to 2025-11-15 is available in [[Media &amp; Democracy lessons for the future]]. 

==Discussion ==

:''[Interested readers are invite to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]&lt;ref name=NPOV/&gt; and treating others with respect.&lt;ref name=AGF/&gt;]''

== Notes ==
{{reflist}}

== Bibliography ==

* &lt;!--Bobby Allyn (2026-03-25) &quot;Jury finds Meta and Google negligent in social media harms trial--&gt;{{cite Q|Q139572103}}
* &lt;!--Yaël Eisenstat (2020-08) &quot;Dear Facebook, this is how you're breaking democracy&quot;--&gt;{{cite Q|Q138844363}}
* &lt;!--Yaël Eisenstat (2021) &quot;Section 230 Revisited: Web Freedom vs Accountability--&gt;{{cite Q|Q139568755|date=2021}}
* &lt;!-- Yaël Eisenstat, Justin Hendrix, and Daniel Kreiss (2024-05-22, 2024a) &quot;
Preventing Tech-Fueled Political Violence: What online platforms can do to ensure that they do not contribute to election-related violence&quot;, The Bulletin of Technology &amp; Public Life--&gt;{{cite Q|Q139571027|date=2024a}} 
* &lt;!-- Yaël Eisenstat, Justin Hendrix, and Daniel Kreiss (2024-05-22, 2024b) &quot;
Tech Platforms Must Do More to Avoid Contributing to Potential Political Violence
&quot;, Tech Policy Press and Just Security--&gt;{{cite Q|Q139571163}}
* &lt;!--Katie McQue (2026-04-24) &quot; Meta ordered to pay $375m after being found liable in child exploitation case--&gt;{{cite Q|Q139572337}}
* &lt;!--Blake E. Reid (2020-09-04) &quot;Section 230 of… what?--&gt;{{cite Q|Q139570229}} 

[[Category:Media]]
[[Category:News]]
[[Category:Democracy]]
[[Category:Suicide]]
[[Category:Politics]]
[[Category:Safety]]
[[Category:Public health]]
[[Category:Google]]
[[Category:Internet]]
[[Category:Social media]]
[[Category:Media reform to improve democracy]]
&lt;!--list of categories
https://en.wikiversity.org/wiki/Wikiversity:Category_Review
[[Wikiversity:Category Review]]--&gt;</text>
      <sha1>6moupr05qk7fndbn1sa2pzlm4ibasvn</sha1>
    </revision>
    <revision>
      <id>2806882</id>
      <parentid>2806879</parentid>
      <timestamp>2026-04-28T18:07:59Z</timestamp>
      <contributor>
        <username>DavidMCEddy</username>
        <id>218607</id>
      </contributor>
      <comment>correct errors in bio</comment>
      <origin>2806882</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10508" sha1="4l3e9qkhf5amcn9bq6ewbz1gpzsvyrs" xml:space="preserve">:''This discusses a 2026-05-06 interview with [[w:Yael Eisenstat|Yael Eisenstat]] about the impact of online platforms on public health, safety and democracy. A video and 29:00 mm:ss podcast excerpted from the interview will be added when available. The podcast will be released 2026-05-16 to the fortnightly &quot;Media &amp; Democracy&quot; show&lt;ref name=M&amp;D&gt;&lt;!--Media &amp; Democracy--&gt;{{cite Q|Q127839818}}&lt;/ref&gt; syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]&lt;ref&gt;&lt;!--Pacifica Radio Network--&gt;{{cite Q|Q2045587}}&lt;/ref&gt; Network of [[w:List of Pacifica Radio stations and affiliates|over 200 community radio stations]].''&lt;ref&gt;&lt;!--list of Pacifica Radio stations and affiliates--&gt;{{cite Q|Q6593294}}&lt;/ref&gt;

:''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]&lt;ref name=NPOV&gt;The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs&lt;/ref&gt; and treating others with respect.''&lt;ref name=AGF&gt;[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] is different: Contributors there are asked to [[Wikinews:Never assume|&quot;Don't assume things; be skeptical about everything.&quot;]] That's wise. However, we should still treat others with respect while being skeptical.&lt;/ref&gt;
&lt;!--[[File:2026-05-06 interview with Yaël Eisenstat.WebM|thumb|2026-05-06 interview with Yaël Eisenstat about how some online platforms have been degrading public health, safety and democracy.]]--&gt; 
&lt;!--[[File:2026-05-06 interview with Yaël Eisenstat.ogg|thumb|29:00 mm:ss excerpts from a 2026-05-06 interview with Yaël Eisenstat about how some online platforms have been degrading public health, safety and democracy.]]--&gt;

[[w:Yaël Eisenstat|Yaël Eisenstat]] discusses the impact of online platforms on public health, safety and democracy. She is currently&lt;ref&gt;&lt;!--Yael Eisenstat--&gt;{{cite Q|Q82046593}}&lt;/ref&gt; the Director of Policy and Impact at Cybersecurity for Democracy (C4D),&lt;ref&gt;&lt;!--Cybersecurity for Democracy--&gt;{{cite Q|Q139568543}}&lt;/ref&gt; working on policy solutions for how to hold social media and other online platforms accountable for their effects on public safety and democracy. Previously, she was Vice President at the [[w:Anti-Defamation League|Anti-Defamation League]] (ADL) Center for Technology &amp; Society (CTS). She was a Facebook election integrity head in 2018 and later became a whistleblower, speaking publicly about the dangers to democracy stemming from the company's decisions and products. She has held other other positions protecting democracy including as an intelligence officer, diplomat, and White House advisor.  C4D contributed to the recent March 24, 2026, jury verdict in a civil case against Internet companies in [[w:New Mexico|New Mexico]].&lt;ref&gt;McQue (2026), &lt;!--C4D and the Courts: Meta Guilty Verdicts--&gt;{{cite Q|Q139572464}}&lt;/ref&gt;  

Eisenstat is interviewed by Spencer Graves.&lt;ref name=Graves&gt;&lt;!--Spencer Graves--&gt;{{cite Q|Q56452480}}&lt;/ref&gt; 

== Eisenstat's work == 

Eisenstat's work includes a [[w:TED (conference)|TED Talk]] and an [[w:South by Southwest|SXSW panel]] in 2020 and a 2024 Tech platforms and political violence. 

=== 2020 TED talk === 

In Eisenstat's (2020) TED talk, she said that around 2015 she began to notice that she was losing the ability to engage with others who were thought differently. Conversations with others in the US were becoming more difficult than conversations she had had as a CIA officer and diplomat drinking tea and talking with outspoken anti-Western clerics and suspected terrorists in Africa. Many of those engagements began with mutual suspicion but none degenerated into shouting or insults. In some cases she built collaboration on areas of mutual interest. Her most powerful tools were to listen, learn and build empathy. Most of her contacts wanted to feel heard, validated and respected. But social media companies like [[w:Facebook|Facebook]] incentivize inflammatory content contributing to a culture of political polarization and mistrust. This generates revenue for Facebook and similar companies that make money from clicks, &quot;because the shortest path to a click is anger or hate&quot;, in the words of [[Facebook whistleblower Frances Haugen says|Facebook whistleblower Frances Haugen]], interviewed 2024-08-19 for this Media &amp; Democracy series. When Facebook asked Eisenstat in 2018 to lead their work to support global elections integrity for political ads, she agreed. She left six months later, speaking openly about Facebook's inability to meet its responsibility to secure elections, subsequently documented, e.g., in the thousands of internal Facebook documents that [[w:Frances Haugen|Haugen]] released to the [[w:Securities and Exchange Commission|Securities and Exchange Commission]] and ''[[w:The Wall Street Journal|The Wall Street Journal]]'' in 2021.

=== 2020 SXSW panel === 

Eisenstat was part of a &quot;panel about the Future of Tech Responsibility&quot; for the 2020 [[w:South by Southwest|South by Southwest]] festival. The festival was cancelled due to COVID-19, but the panel was held virtually. This panel included a discussion of [[w:Section 230|Section 230]] of the [[w:Communications Act of 1934|Communications Act of 1934]], as amended by the [[w:Communications Decency Act|Communications Decency Act]] of 1996.&lt;ref&gt;Reid (2020).&lt;/ref&gt; It was &quot;written before platforms such as [[w:Facebook|Facebook]], [[w:YouTube|YouTube]] and [[w:X (social network)|Twitter]] existed&quot; -- written while [[w:Google|Google]] was a research project by [[w:Stanford University|Stanford]] [[w:Doctor of Philosophy|PhD]] students [[w:Larry Page|Larry Page]] and [[w:Sergey Brin|Sergey Brin]]. Section 230 includes, &quot;No provider or user of an interactive computer service shall be treated as the publisher or speaker of any information provided by another information content provider.&quot;&lt;ref&gt;&lt;!--47 U.S. Code § 230 - Protection for private blocking and screening of offensive material--&gt;{{cite Q|Q139570261}}&lt;/ref&gt; Eisenstat says that it's time to revisit Section 230, to demand accountability where Internet companies promote or suppress information based on the content while protecting web freedom otherwise. This is similar to the [[Dean Baker on Internet companies threatening democracy internationally and how to fix that|recommendations of]] [[w:Dean Baker|Dean Baker]] that when Internet companies make money by promoting information differentially based on content, they should be liable as are legacy media under the US Supreme Court decision in ''[[w:New York Times Co. v. Sullivan|NYT v. Sullivan]]'' (1964). In other cases, they should be treated as [[w:Common carrier|common carrier]]s like telephone companies. 

=== 2024: Tech platforms and political violence === 

More recently, Eisenstat et al. (2024a, b) are insisting that, &quot;Tech Platforms Must Do More to Avoid Contributing to Potential Political Violence&quot;. ''The New York Times'' had reported that, &quot;a steady undercurrent of violence and physical risk has become a new normal,&quot; particularly targeting public officials and democratic institutions. A survey from the Brennan Center found that 38% of election officials have experienced violent threats. They attributed these threats primarily to tech platforms and gave seven recommendations in four themes &quot;congruent with any number of papers that academics and civil society leaders have published over the years.&quot; They said that platforms 
* must develop robust standards for threat assessment and engage in scenario planning, crisis training, and engagement with external stakeholders, with as much transparency as possible.
* should enforce clear and actionable content moderation policies that address election integrity.
* should enforce their rules uniformly, not exempting politicians and other political influencers. 
* must clearly explain important content moderation decisions, ensuring transparency especially when it comes to high profile accounts.

They hope that increasing demands for accountability will prompt platforms to act more responsibly and prioritize the risk of political violence both in the United States and abroad. 

== The need for media reform to improve democracy == 

This article is part of [[:category:Media reform to improve democracy]]. A summary of episodes to 2025-11-15 is available in [[Media &amp; Democracy lessons for the future]]. 

==Discussion ==

:''[Interested readers are invite to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]&lt;ref name=NPOV/&gt; and treating others with respect.&lt;ref name=AGF/&gt;]''

== Notes ==
{{reflist}}

== Bibliography ==

* &lt;!--Bobby Allyn (2026-03-25) &quot;Jury finds Meta and Google negligent in social media harms trial--&gt;{{cite Q|Q139572103}}
* &lt;!--Yaël Eisenstat (2020-08) &quot;Dear Facebook, this is how you're breaking democracy&quot;--&gt;{{cite Q|Q138844363}}
* &lt;!--Yaël Eisenstat (2021) &quot;Section 230 Revisited: Web Freedom vs Accountability--&gt;{{cite Q|Q139568755|date=2021}}
* &lt;!-- Yaël Eisenstat, Justin Hendrix, and Daniel Kreiss (2024-05-22, 2024a) &quot;
Preventing Tech-Fueled Political Violence: What online platforms can do to ensure that they do not contribute to election-related violence&quot;, The Bulletin of Technology &amp; Public Life--&gt;{{cite Q|Q139571027|date=2024a}} 
* &lt;!-- Yaël Eisenstat, Justin Hendrix, and Daniel Kreiss (2024-05-22, 2024b) &quot;
Tech Platforms Must Do More to Avoid Contributing to Potential Political Violence
&quot;, Tech Policy Press and Just Security--&gt;{{cite Q|Q139571163}}
* &lt;!--Katie McQue (2026-04-24) &quot; Meta ordered to pay $375m after being found liable in child exploitation case--&gt;{{cite Q|Q139572337}}
* &lt;!--Blake E. Reid (2020-09-04) &quot;Section 230 of… what?--&gt;{{cite Q|Q139570229}} 

[[Category:Media]]
[[Category:News]]
[[Category:Democracy]]
[[Category:Suicide]]
[[Category:Politics]]
[[Category:Safety]]
[[Category:Public health]]
[[Category:Google]]
[[Category:Internet]]
[[Category:Social media]]
[[Category:Media reform to improve democracy]]
&lt;!--list of categories
https://en.wikiversity.org/wiki/Wikiversity:Category_Review
[[Wikiversity:Category Review]]--&gt;</text>
      <sha1>4l3e9qkhf5amcn9bq6ewbz1gpzsvyrs</sha1>
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  </page>
  <page>
    <title>Research: Tracking</title>
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    <revision>
      <id>2806943</id>
      <timestamp>2026-04-29T03:54:45Z</timestamp>
      <contributor>
        <username>UrAvgEMATer</username>
        <id>3044443</id>
      </contributor>
      <comment>Made the page</comment>
      <origin>2806943</origin>
      <model>wikitext</model>
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      <text bytes="17228" sha1="9mmten1luc9yuz99cztqisw9dcqfeu9" xml:space="preserve">== Intro ==
This paper will focus on the latest research on the impacts of tracking and results from detracking in the world of secondary school mathematics. This topic is of special interest to me, as throughout our program, criticisms of tracking have been sprinkled here and there in our readings. None of those readings had tracking as their focus, so I was never able to formulate my own educated opinion on it. In addition, even though NCTM’s president Robert Q. Berry, III has argued in favor of detracking, I am still skeptical about such a stance since nearly all students schools still operate under tracking structures. I also imagine that detracking would make it even more difficult for teachers to successfully differentiate in their classrooms, with differentiation in separate ability classrooms already a challenge for some. Regardless, I wonder why school districts still implement tracking in their schools when the research says otherwise.

== Claim 1 - Tracking results in an inequitable structure ==
The first point I would like to address is that tracking in secondary mathematics education results in an inequitable system for students, with students in “on-level/regular” classes often receiving more procedural based instruction, less rigorous curriculum, and being held to lower expectations when they could achieve more.

For starters, Huinker et al. (2020) states in page 785 that “Ability grouping and tracking place students in qualitatively different experiences with many marginalized learners experiencing mathematics that is less engaging, is less rigorous, and has lower expectations for their mathematics learning.” This can come from teacher tracking, where less experienced teachers are assigned on-level courses while experienced teachers are assigned to advanced/AP classes. This results in students receiving instruction from educators who may not be as experienced in differentiating their instruction, using rich mathematical tasks, or orchestrating productive mathematical discussions. This is further emphasized by Nirode &amp; Boyd (2023) where they are argue that student tracking doesn’t just separate students, but their opportunities and resources as well.

In addition, across Boaler’s different longitudinal studies, it was found that middle-low achieving students performed better when classes were detracked compared to their counterparts in tracked classrooms (Boaler, 2011). As for the high achieving students, they performed at the same levels regardless of the classes being tracked or detracked. Boaler even acknowledges critics who speculated that such results were due to exceptional teaching, but counters with a study from Oxford that found student achievement higher in detracked schools compared to tracked schools due to the higher achievement of middle-low achieving students in the detracked programs. So the choice to keep middle-low achieving students in tracked programs is also a choice to prevent them from their full potential, as was seen from Boaler’s studies.

Finally, tracking in schools also results in unequal status across students. I know this not just from the research online, but because I experienced this firsthand. In 6th grade, everyone took the same math class, but going into 7th grade you were either placed in math or advanced math. Not being placed in advanced math made me think that math wasn’t “my thing”, and that maybe I should focus or care less about it in order to play to my strengths. This is supported by Bush et al. (2024) as they share how deterministic labels such as advanced, honors, on-level, remedial, etc all signal that some students have different statuses across classrooms.

As can be seen, tracking has several negative impacts on students while attempts to detrack has resulted in higher overall achievement.

== Claim 2 - Detracking has various positive effects ==
The second idea I would like to address is the impact that detracking in our schools can have. Detracking can improve achievement in students who would otherwise be ignored and provide them access to higher mathematics, but only when done so with instructional support.

The most immediate example of the benefits of detracking can be observed from Boaler (2011). Here she discusses the work of Burris et al. who ran a detracking experiment where they observed the math education and outcomes of a student cohort from 7th to 9th grade both for a group that went through the grades tracked and then another that went through detracked. They found that the detracked group passed courses at higher rates, took more advanced courses, students performed better on achievement tests, and that even achievement improvements were found across all levels of achieving students.

Although one can be quick to point out how this contradicts the claim that high achieving students achieve the same in both tracked and detracked courses, one cannot ignore the extensive lists of positives that came from detracking the second cohort.

Another example of positive long term outcomes for those that experienced detracked learning includes how detracked students placed in Algebra 1 compared to their tracked peers in remedial-Algebra, outperforming them in math achievement, credits earned, attendance, and staying enrolled in school (Dee &amp; Huffaker, 2024). In this study, a California school district implemented a detracking initiative that aimed to place upcoming 9th grade students who achieved below grade level in math in Algebra 1 instead of remedial-Algebra. The difference here though is that this study also ensured that the schools that adopted this would also provide instructional coaching, collaborative planning time, and professional development for teachers to properly and intentionally execute the new detracking practice. The control group was placed in the remedial-Algebra course, with the treatment group in Algebra 1 and their teachers receiving enhanced instruction.

The researchers also followed the students for multiple years after the 9th grade, and by comparing outcomes of the randomly assigned groups, they could isolate the effect of detracking from other variables. Clearly by the results of this study though, students in the treatment group benefitted from detracking practices. This study is excellent in the tracking world as some people fear that by placing lower achieving students in “higher” classes, that will do more harm to them than good. The paper does note that half of the treatment group did fail Algebra 1 initially or had to take a bridge class in grade 10. The other half moved on to Geometry in grade 10 while in contrast almost none of the control group moved on to Geometry in grade 10. In the end, students from the treatment group were more likely to pass Algebra II compared to the control group, showing that even with early setbacks, detracking does prove fruitful in the long term.

With that said, opt-out policies vs. opt-in policies is another great place to further detrack schools in a way that is not as radical as some think detracking is. In the Dallas ISD, changing from an opt-in to an opt-out policy for advanced 6th grade math resulted in 13% higher enrollment in Algebra 1 before high school (Castaño et al., 2026). In 2019, the Dallas ISD switched to an opt-out policy where 5th grade students who scored in the upper 50% on the Math STAAR would be automatically enrolled in advanced 6th grade math, a pathway that would culminate in the student taking Algebra 1 in 8th grade. Using methods to isolate the effect of the policy, researchers found that the simple policy change resulted in an additional 1,600 students taking Algebra 1 before high school, raising the 8th graders in Algebra 1 population to 37%.  Hispanic students benefitted the most from this, with an enrollment increase of 16.4 points. This result is further homed in by another study that found that “students of lower socioeconomic status may be more likely to pursue advantageous mathematics course- work after eighth-grade algebra enrollment,” (Dwyer &amp; Kelly, 2025) page 120.

Pairing this study with the aforementioned one from Dee and Huffaker (2024) that showed that enrollment of lower achieving students in advanced courses contributed to their long term success in mathematics, it can be better understood why a change at the 6th grade math level resulted in the long term success of these students at the 8th grade level with their increased enrollment of Algebra 1. Adoption of opt-out policies that make it so families are proactive about having their child take a lower rigor class instead of the other way around presents another way to challenge detracking when detracking is not possible, further positioning students in pathways where they will succeed long term.

The effects of detracking are clear: they elevate low-middle achieving students that would not normally have access to advanced/rich mathematical pathways that result in their long term success in mathematics, even when such success may come after minor setbacks. In addition, detracking has been shown to at the very least have high achieving students perform the same, and in other cases, achieve even higher. This is all of course when it is done with instructional support, proper planning time, and professional development that prepares these teachers to execute detracked classrooms to their fullest potential.

== Claim 3 - Successful detracking requires intentional instructional supports ==
My third idea expands on what the second idea ended on, which is that the success of detracking depends not just on the policy change itself, but on the instructional supports that surround it. Teachers must undergo professional development that involves differentiation, a push for deep conceptual understanding through rich tasks and discussions, and student collaboration.

One important aspect that teachers would need to be thoroughly coached on would be differentiation. This can come in many forms, such as tiering which we reviewed in block 2 of our math ed program here at UGA. To take from an article we read in Dr.Miller’s class, “our goal in DI is to provide each student with challenges that are sensible to them so they are working at the edges of their current reasoning,” (Hackenberg et al., 2020). In this article, the authors provided a mathematically rich task that had multiple entry points by changing the numbers on the task to be appropriate for students at different ability levels within one classroom so that they could all engage with the task. By doing so, all students were met with a challenge when attempting the task, regardless of if they were in ability stage 1, 2, or 3. Since detracking will result in students of varying ability levels being in one classroom, it is imperative that teachers of these detracked classrooms be not just familiar, but fluent in differentiation in order to see the greatest success in their classroom.

When talking about what kind of teachers we want in these detracked classrooms, we would want to be looking for the following qualities: teachers with high expectations, those that push deep conceptual understanding and connected math learning through rich problem-solving tasks and discussions. I retrieved these properties from Bush et al. (2024)’s argument on why the tracking of teachers must be dismantled. It argued in its fourth shift that the teachers with the properties mentioned above are always assigned to the advanced classes in high school. When detracking schools, if original “on-level” teachers aren’t engaged in instructional coaching that boosts them to the level of the “advanced” teachers, then some detracked classes will still have students who have teachers that are not engaging in these good, mathematical education practices. So we should look to provide teachers with professional development and professional learning communities where all teachers are expected to become teachers who have high expectations, push deep conceptual understanding, and use rich mathematical tasks and discussions.

We can also look to Jo Boaler’s (2011) list of three critical components that all teachers who found success when teaching detracked courses shared. The first component was that teachers used open exploratory tasks so that students worked on mathematical tasks that were appropriate to their level. The second was that students were taught to respect and work well with each other. Whether students were allowed to choose their own groups or grouping was more intentional, teachers would be proactive about teaching their students how to work well with each other and constantly give messages about everyone being capable in the classroom. The last component is that teachers gave messages consistent with building a growth mindset. Teachers would emphasize the importance of learning being a process, that achievement was a result of effort and not innate ability, and teachers would even raise the status of low-status students in the classroom by acknowledging their contributions publicly in the classroom.

These three components serve as a great starting point to teach to teachers who are interested in being successful teachers of detracked classrooms. By combining these with the emphasis on differentiation and the other properties of experienced teachers such as high expectations and a push for deep conceptual understanding, districts can build instructional coaching curriculum that empowers teachers to execute detracked classrooms to their fullest potential.

== Conclusion ==
In the future, I cannot exactly “detrack” my own classroom by myself. Even so, there are many things to take away from this research that will make me a better teacher for my future students within my own classrooms. For starters, using differentiated instruction to the fullest of my abilities to maintain high expectations of all my students and ensure they all engage in deep mathematical thinking will be one of the most important things for me to take away. As important as it is, this is also one of the most difficult ones and will require a lot of practice and trial and error on my part. I will also need to be conscious about what words I use and what messages I send out during the year with my students. I need to communicate to them the values of a growth mindset and its benefits, and how we are all capable of learning math so there is no room for anyone to lower their own expectations of themselves or of others. This is important as it is related to them being able to work together, regardless of their partner’s ability level. If a high achieving student is paired with a low achieving student, it is imperative that both parties maintain high expectations for each other, and that I have spent time teaching the class how to work with one another so that even such an “ability gap” between students does not prevent them from working together successfully. In the distant future in administration, I could maybe put more of the policy takeaways of this paper to use. For now, I can enact what I can in my own classroom.

== References ==
Boaler, Jo. 2011. “Changing Students’ Lives through the De-Tracking of Urban Mathematics Classrooms.” Journal of Urban Mathematics Education 4 (1): 7–14.

Bush, S. B., Huinker, D., &amp; Graham, K. J. (2024). ''Dismantling inequitable structures''. ''Mathematics Teacher: Learning and Teaching PK–12, 117''(4), 262–267. &lt;nowiki&gt;https://doi.org/10.5951/MTLT.2023.0287&lt;/nowiki&gt;

Dee, Thomas S, and Elizabeth Huffaker. (2024). Accelerating Opportunity: The Effects of Instructionally Supported Detracking. (EdWorkingPaper: 24-986). Retrieved from Annenberg Institute at Brown University: &lt;nowiki&gt;https://doi.org/10.26300/v492-1p91&lt;/nowiki&gt;

Dwyer, K., &amp; Kelly, A. M. (2025). U.S. District-Level Algebra Acceleration in Eighth Grade and Longitudinal Mathematics Outcomes. Journal for Research in Mathematics Education, 56(3), 107-124. Retrieved Apr 16, 2026, from &lt;nowiki&gt;https://doi.org/10.5951/jresematheduc-2023-0138&lt;/nowiki&gt;

Hackenberg, A. J., Jones, R., &amp; Borowski, R. (2020). Tiering instruction for seventh-grade students. Mathematics Teacher: Learning &amp; Teaching PK-12, 113 (2), 124–131. &lt;nowiki&gt;https://doi.org/10.5951/MTLT.2018.0048&lt;/nowiki&gt;

Huinker, D., Bush, S. B., &amp; Graham, K. J. (2020). ''Catalyzing change in school mathematics: Creating the opportunities our students deserve''. ''Mathematics Teacher: Learning and Teaching PK–12, 113''(10), 780–790. &lt;nowiki&gt;https://doi.org/10.5951/MTLT.2020.0053&lt;/nowiki&gt;

Nirode, W., &amp; Boyd, B. (2023). The Prevalence of Teacher Tracking in High School Mathematics Departments. ''Journal for Research in Mathematics Education'', ''54''(1), 7-23. Retrieved Apr 16, 2026, from &lt;nowiki&gt;https://doi.org/10.5951/jresematheduc-2020-0296&lt;/nowiki&gt;

Vargas Castaño, Daniel, Dareem K. Antoine, and Trey Miller. (2026). Closing the Gaps: An Examination of Early Impacts of Dallas ISD’s Opt-out Policy on Advanced Course Enrollment. (EdWorkingPaper: 25-1184). Retrieved from Annenberg Institute at Brown University: &lt;nowiki&gt;https://doi.org/10.26300/s914-9n20&lt;/nowiki&gt;</text>
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