User talk:Jtneill

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

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

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

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

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

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

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

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

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

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

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

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

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

== VDT - U3126684 chapter ==

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

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

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

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

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

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

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

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

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

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

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

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

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

== Problem with curator ==

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

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

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

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

== Peer review ==

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

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

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

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

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

== Quantum ==

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

== ShakespeareFan00 ==

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

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

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

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

== Publishing transcripts ==

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

== User:Dan Polansky ==

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

== Moving to personal namespace ==

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

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

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

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

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

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

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

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

== Vandalism ==

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

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

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

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

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

== User Dan Polansky ==

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

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

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

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

== DELETE request ==

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

== Archiving ==

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

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

== Email ==

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

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

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

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

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

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

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

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

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

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

== Thoughts about Wikinews closure ==

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

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

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

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

== Resources suitable for the main namespace ==

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

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

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

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

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

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

Metadata: Since edit summaries are not present on talk pages like this one, I'll link to the LLM interaction history that I saved in my Wikiversity user space here: [[User:ThinkingScience/All_General_AI_Prompt_History_Archive#Goal:_Interact_with_User:Jtneill_at_July_5,_2026]] so that I follow [[Wikiversity:Artificial intelligence]]. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 04:38, 5 July 2026 (UTC)</text>
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  <page>
    <title>Understanding Arithmetic Circuits</title>
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      <id>2818451</id>
      <parentid>2818412</parentid>
      <timestamp>2026-07-17T13:59:49Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Adder */</comment>
      <origin>2818451</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3088" sha1="sp5arbapjkute2tr8c3olnfmzw4j960" xml:space="preserve">
== Adder ==

* Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]]  )
 
{| class=&quot;wikitable&quot;
|-
!  Adder type      !! Overview !! Analysis !! VHDL Level Design  !! CMOS Level Design  
|-
| '''1. Ripple Carry Adder'''
|| [[Media:VLSI.Arith.1A.RCA.20250522.pdf|A]]||
|| [[Media:Adder.rca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]]
|-
| '''2. Carry Lookahead Adder'''
|| [[Media:VLSI.Arith.2A.CLA.20260717.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20260717.pdf|B]] ||
|| [[Media:Adder.cla.20140313.pdf|pdf]]||
|-
| '''3. Carry Save Adder''' 
|| [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]||
|| ||
|-
|| '''4. Carry Select Adder''' 
|| [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]||
|| || 
|-
|| '''5. Carry Skip Adder'''
|| [[Media:VLSI.Arith.5A.CSkip.20250405.pdf|A]]||
|| 
|| [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]]
|- 
|| '''6. Carry Chain Adder'''
|| [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]||
|| [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]],  [[Media:Adder.cca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]]
|- 
|| '''7. Kogge-Stone Adder'''
|| [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]||
|| [[Media:Adder.ksa.20140409.pdf|pdf]]||
|-
|| '''8. Prefix Adder'''
|| [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]||
|| || 
|-
|| '''9.1 Variable Block Adder'''
|| [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]], [[Media:VLSI.Arith.1C.VBA.20250218.pdf|D]]||
|| || 
|-
|| '''9.2 Multi-Level Variable Block Adder'''
|| [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]||
|| || 
|}
&lt;/br&gt;
 
=== Adder Architectures Suitable for FPGA === 
* FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]])
* FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]])
* FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]])
* FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]])
* Carry-Skip Adder
&lt;/br&gt;

== Barrel Shifter ==
* Barrel Shifter Architecture Exploration ([[Media:Bshift.20131105.pdf|bshfit.vhdl]], [[Media:Bshift.makefile.20131109.pdf|bshfit.makefile]])
&lt;/br&gt;

'''Mux Based Barrel Shifter'''
* Analysis  ([[Media:Arith.BShfiter.20151207.pdf|pdf]]) 
* Implementation 
&lt;/br&gt;

== Multiplier == 
=== Array Multipliers ===
* Analysis  ([[Media:VLSI.Arith.1.A.Mult.20151209.pdf|pdf]])
&lt;/br&gt;

=== Tree Mulltipliers ===
* Lattice Multiplication  ([[Media:VLSI.Arith.LatticeMult.20170204.pdf|pdf]])
* Wallace Tree  ([[Media:VLSI.Arith.WallaceTree.20170204.pdf|pdf]])
* Dadda Tree  ([[Media:VLSI.Arith.DaddaTree.20170701.pdf|pdf]])
&lt;/br&gt;

=== Booth Multipliers ===
* [[Media:RNS4.BoothEncode.20161005.pdf|Booth Encoding Note]]
* Booth Multiplier Note ([[Media:BoothMult.20160929.pdf|H1.pdf]])
&lt;/br&gt;

== Divider == 
* Binary Divider ([[Media:VLSI.Arith.1.A.Divider.20131217.pdf|pdf]])&lt;/br&gt;

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

[[Category:Digital Circuit Design]]
[[Category:FPGA]]</text>
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  <page>
    <title>Complex analysis in plain view</title>
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      <timestamp>2026-07-17T14:15:54Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Geometric Series Examples */</comment>
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      <text bytes="5296" sha1="ho2dy75iqmsouw4nleqsp2uz5g8yve6" xml:space="preserve">Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers &lt;math&gt;x+iy&lt;/math&gt;, where &lt;math&gt;i=\sqrt{-1}&lt;/math&gt;, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other &lt;math&gt;L&lt;/math&gt;-functions, modular forms, elliptic functions, etc. &lt;blockquote&gt;The shortest path between two truths in the real domain passes through the complex domain.  — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]&lt;/blockquote&gt;In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}}

==''' Complex Functions '''== 
* Complex Functions ([[Media:CAnal.1.A.CFunction.20140222.Basic.pdf|1.A.pdf]], [[Media:CAnal.1.B.CFunction.20140111.Octave.pdf|1.B.pdf]], [[Media:CAnal.1.C.CFunction.20140111.Extend.pdf|1.C.pdf]])

* Complex Exponential and Logarithm ([[Media:CAnal.5.A.CLog.20131017.pdf|5.A.pdf]], [[Media:CAnal.5.A.Octave.pdf|5.B.pdf]])

* Complex Trigonometric and Hyperbolic ([[Media:CAnal.7.A.CTrigHyper..pdf|7.A.pdf]], [[Media:CAnal.7.A.Octave..pdf|7.B.pdf]])

'''Complex Function Note'''
: 1. Exp and Log Function Note ([[Media:ComplexExp.29160721.pdf|H1.pdf]])

: 2. Trig and TrigH Function Note ([[Media:CAnal.Trig-H.29160901.pdf|H1.pdf]])

: 3. Inverse Trig and TrigH Functions Note ([[Media:CAnal.Hyper.29160829.pdf|H1.pdf]])

==''' Complex Integrals '''== 
* Complex Integrals ([[Media:CAnal.2.A.CIntegral.20140224.Basic.pdf|2.A.pdf]], [[Media:CAnal.2.B.CIntegral.20140117.Octave.pdf|2.B.pdf]], [[Media:CAnal.2.C.CIntegral.20140117.Extend.pdf|2.C.pdf]])

==''' Complex Series '''== 
* Complex Series ([[Media:CPX.Series.20150226.2.Basic.pdf|3.A.pdf]], [[Media:CAnal.3.B.CSeries.20140121.Octave.pdf|3.B.pdf]], [[Media:CAnal.3.C.CSeries.20140303.Extend.pdf|3.C.pdf]])

==''' Residue Integrals '''==
* Residue Integrals ([[Media:CAnal.4.A.Residue.20140227.Basic.pdf|4.A.pdf]], [[Media:CAnal.4.B.pdf|4.B.pdf]], [[Media:CAnal.4.C.Residue.20140423.Extend.pdf|4.C.pdf]]) 

==='''Residue Integrals Note'''===
* Laurent Series with the Residue Theorem Note ([[Media:Laurent.1.Residue.20170713.pdf|H1.pdf]])
* Laurent Series with Applications Note ([[Media:Laurent.2.Applications.20170327.pdf|H1.pdf]])
* Laurent Series and the z-Transform Note ([[Media:Laurent.3.z-Trans.20170831.pdf|H1.pdf]])
* Laurent Series as a Geometric Series Note ([[Media:Laurent.4.GSeries.20170802.pdf|H1.pdf]])

=== Laurent Series and the z-Transform Example Note === 
* Overview ([[Media:Laurent.4.z-Example.20170926.pdf|H1.pdf]])

====Geometric Series Examples====

* Causality ([[Media:Laurent.5.Causality.1.A.20191026n.pdf|A.pdf]], [[Media:Laurent.5.Causality.1.B.20191026.pdf|B.pdf]]) 

* Time Shift ([[Media:Laurent.5.TimeShift.2.A.20191028.pdf|A.pdf]], [[Media:Laurent.5.TimeShift.2.B.20191029.pdf|B.pdf]]) 

* Reciprocity ([[Media:Laurent.5.Reciprocity.3A.20191030.pdf|A.pdf]], [[Media:Laurent.5.Reciprocity.3B.20191031.pdf|B.pdf]]) 

* Combinations ([[Media:Laurent.5.Combination.4A.20200702.pdf|A.pdf]], [[Media:Laurent.5.Combination.4B.20201002.pdf|B.pdf]]) 

* Properties ([[Media:Laurent.5.Property.5A.20220105.pdf|A.pdf]], [[Media:Laurent.5.Property.5B.20220126.pdf|B.pdf]]) 

* Permutations ([[Media:Laurent.6.Permutation.6A.20230711.pdf|A.pdf]], [[Media:Laurent.5.Permutation.6B.20251225.pdf|B.pdf]], [[Media:Laurent.5.Permutation.6C.20260717.pdf|C.pdf]], [[Media:Laurent.5.Permutation.6C.20240528.pdf|D.pdf]]) 

* Applications ([[Media:Laurent.5.Application.6B.20220723.pdf|A.pdf]]) 

* Double Pole Case

:- Examples ([[Media:Laurent.5.DPoleEx.7A.20220722.pdf|A.pdf]], [[Media:Laurent.5.DPoleEx.7B.20220720.pdf|B.pdf]]) 

:- Properties ([[Media:Laurent.5.DPoleProp.5A.20190226.pdf|A.pdf]], [[Media:Laurent.5.DPoleProp.5B.20190228.pdf|B.pdf]])

====The Case Examples====
* Example Overview : ([[Media:Laurent.4.Example.0.A.20171208.pdf|0A.pdf]], [[Media:Laurent.6.CaseExample.0.B.20180205.pdf|0B.pdf]])
* Example Case 1 : ([[Media:Laurent.4.Example.1.A.20171107.pdf|1A.pdf]], [[Media:Laurent.4.Example.1.B.20171227.pdf|1B.pdf]])
* Example Case 2 : ([[Media:Laurent.4.Example.2.A.20171107.pdf|2A.pdf]], [[Media:Laurent.4.Example.2.B.20171227.pdf|2B.pdf]])
* Example Case 3 : ([[Media:Laurent.4.Example.3.A.20171017.pdf|3A.pdf]], [[Media:Laurent.4.Example.3.B.20171226.pdf|3B.pdf]])
* Example Case 4 : ([[Media:Laurent.4.Example.4.A.20171017.pdf|4A.pdf]], [[Media:Laurent.4.Example.4.B.20171228.pdf|4B.pdf]])
* Example Summary : ([[Media:Laurent.4.Example.5.A.20171212.pdf|5A.pdf]], [[Media:Laurent.4.Example.5.B.20171230.pdf|5B.pdf]])

==''' Conformal Mapping '''== 
* Conformal Mapping ([[Media:CAnal.6.A.Conformal.20131224.pdf|6.A.pdf]], [[Media:CAnal.6.A.Octave..pdf|6.B.pdf]])

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

[[Category:Complex analysis]]</text>
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  </page>
  <page>
    <title>Python programming in plain view</title>
    <ns>0</ns>
    <id>212733</id>
    <revision>
      <id>2818463</id>
      <parentid>2818300</parentid>
      <timestamp>2026-07-17T17:27:28Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Using Libraries */</comment>
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      <text bytes="2647" sha1="o91fozprqal576ww8qym0udmo6g3ptf" 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.20260715.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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  <page>
    <title>Social Victorians/Timeline/1870s</title>
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      <text bytes="86055" sha1="j8o9y6kmw9fo8b432z3ctgujtjxe29v" xml:space="preserve">==Time Line==
[[Social Victorians/Timeline/1840s|1840s]]  [[Social Victorians/Timeline/1850s |1850s]] [[Social Victorians/Timeline/1860s | 1860s]] 1870s [[Social Victorians/Timeline/1880s | 1880s]] [[Social Victorians/Timeline/1890s | 1890s]] [[Social Victorians/Timeline/1900s|1900s]] [[Social Victorians/Timeline/1910s|1910s]] [[Social Victorians/Timeline/1920s-30s|1920s-30s]]

==1870==
&quot;Until 1870 all of the money women earned belonged to their husbands, and until 1882 their property did too, even after a divorce or separation.&quot;&lt;ref name=&quot;:4&quot; /&gt; (698 of 1203)

In 1870 Parliament debated and defeated the first bill for women's suffrage, but allowed &quot;women who owned property ... to stand for election to school boards.&quot;&lt;ref name=&quot;:4&quot; /&gt; (698–699 of 1203)

&quot;The bulk of Irish farmers did not own their land, and instead leased it from landlords, the majority of whom lived in England. In 1870, only 3 percent of agricultural holdings were occupied by owners.&quot;&lt;ref name=&quot;:4&quot; /&gt; (742 of 1203)

Dante Gabriel Rossetti and Arthur Sullivan were at the same dinner party in 1870?

Another dinner party had as guests Charles Dickens, Dante Gabriel Rossetti, John Tenniel and George Du Maurier.

January
February
March
April
May
June
July
August
September
October
November
December

==1871==
Although Queen Victoria had opened Parliament for the first time in February 1866, when people saw her for the first time in years as her open carriage made its way, she was unpopular because it seemed she was not working. Gladstone was Prime Minister.&lt;blockquote&gt;Between 1871 and 1874, eighty-five Republican Clubs were founded in Britain, protesting, among other things, the &quot;expensiveness and uselessness of the monarchy&quot; and Bertie's &quot;immoral example.&quot;&lt;ref name=&quot;:4&quot;&gt;Baird, Julia. ''Victoria the Queen, an Intimate Biography of the Woman Who Ruled an Empire''. Random House, 2016. Apple Books: https://books.apple.com/us/book/victoria-the-queen/id953835024.&lt;/ref&gt; (617 of 1203)&lt;/blockquote&gt;&quot;The 1871 Royal Commission on the Contagious Diseases Acts ... declared there was no comparison to be made between prostitutes and their clients: 'With the one sex the offence is committed as a matter of gain, with the other it is an irregular indulgence of a natural impulse.'&quot;&lt;ref name=&quot;:4&quot; /&gt; (704 of 1203)

=== January ===
Germany is united under King William I of Prussia. Julia Baird says, &quot;At the same time, Italy captured and annexed the Papal States, which had been under the direct rule of the Pope since the 700s and had lost their protector in Napoleon III.&quot;&lt;ref name=&quot;:4&quot; /&gt; (646 of 1203)

==== 4 January 1871, Wednesday ====
&lt;blockquote&gt;INVITATION BALL.
&lt;p&gt;On Wednesday evening last Major Goodman and the Officers of the 5th Dragoon Guards gave an invitation ball, which was held in the Drapers’ Hall (kindly placed at their disposal by the Drapers’ Company). The following ladies and gentlemen were amongst those who received invitations The Marquis and Marchioness of Hertford; the Earl and Countess of Aylesford; Lady A. N. Finch, Lord Guernsey, and the Hon. Mr. Finch; Lord and Lady Leigh and Miss Leigh; Lord and Lady Henley and Miss Henley, Miss Elwes, Lord and Lady Wrottealey, Lord and Lady Manners; C. N. Newdegate, Esq., M.P.; Captain, Mrs., and Miss Adams; E. Petre, Esq., and Lady Gwendoline Petre; J. Beech, Esq., Mrs. and Miss Beech, and Mr. Beech, jun.; Mr. and Mrs. Turner; Mr. and Mrs. Fetherstone Dilke, Mrs. and the Misses Fetherstone, Mr. Fetherstone, and Mr. Beaumont Fetherstone; Mr. and Mrs. P. A. Muntz; Captain and Mrs. Boultbee, of Knowle; Mr. C. M. Caldecott, Mrs. Caldecott, and the Misses Caldecott; the Rev. A. Fanshawe and Mrs. Fanshawe; Captain and Mrs. Battine; the Rev. S. C. Spencer Smith; the Rev. R. H. Baynes, M.A., vicar of St. Michael’s; the Rev. H. T. Harris, (Christ Church); General and Mr. Richmond Jones; Colonel F. Chaplin, and the Officers of the 4th Dragoon Guards, stationed at Northampton; Captain Thornelow, and the Officers of the Royal Artillery, at Weedon; the officers of the 4th Royal Regiment at Weedon; Mr. and Mrs. E. Wood; Mr. and Mrs. Herbert Wood; the Colonel and officers of the First Warwickshire Militia; Mrs. and Miss Alston, and Mr. Alston, jun., of Elmdon; Mr. and Mrs. F. Paget; Mr. and Mrs. Gulson; Captain Thomson; Captain and Mrs. Raleigh King; Mrs. Phillipson; Lord and Lady Mountgarret; the Honourable Miss Butler; Mr. and Mrs. Courtenay Lord; the Hon. Mrs. Twistleton; Mr. and the Misses Conant; Captain and Mrs. J. Marsland; Major and Mrs. Edlman; Mr. and Mrs. Astley; Mr. T. Lant, Mr. R. Lant and Mr. J. Lant, Mrs. and Miss Lant; Mr. W. T. Cavendish; Mr. and Mrs. A. Rotherham; the Marquis of Ormonde, of the first Life Guards; the Earl of Calludon, of the First Life Guards; Mrs. and the Misses Hobson; Mr P. Hobson, and Mrs. Hobson; Mr. and Mrs. Soames; Mr. and Mrs. Adderley, Sir John Rae Reid; Capt. and Mrs. Townshend, of Caldecote Hall; Lieut.-Colonel Swinfen and the Officers of the 5th Dragoon Guards stationed at Leeds; Capt. Marsden and the Officers of the 5th Dragoon Guards stationed at Birmingham; Colonel, Mrs., and Miss Bourne; Mr. and Mrs. Wyley Lord; Captain and Mrs. Thursby; Mr. and Mrs Morrice; Lieut.-Colonel Wirgman; Mr. and Mrs. J. Rotherham; [[Social Victorians/People/Abercorn|Lady Caroline Howard]]; Mr. and Mrs. Rotherham; Mr and Mrs John Sankey and the Misses Sankey; Mrs. and the Misses Murphy; Mr. Bibby (4th Hussars), Captain Gist (7th Hussars), Mr. Gregg (8th Hussars), Mr. Hamilton (7th Dragoon Guards), Colonel Rattray, Mr and Mrs. R. Boyd, &amp;c, &amp;c.&lt;/p&gt;
&lt;p&gt;The string band of the 5th Dragoon Guards, under the direction of Mr. Sidney Jones, performed the following selection of music:— Quadrille, Barbe Bleue; Valse, Marian; Galop, Bonderbryllup; Lancers, Knight of St. Patrick; Valse, Hydropaten; Galop, Flick and Flock; Quadrille, Princess of Trebizonde; Valse, the Belle of the Ball; Galop, the Fox Hunters; Valse, the Dragoon Guards; Lancers, the Gaiety; Valse, the Beautiful Danube; Valse, Wiener Kinder; Quadrille, the Fest; Galop, the Village Rose; Valse, the Geraldine; Lancers, Merry Tunes; Galop, Barbe Bleue; Valse, Various; Galop, Glorioso.&lt;ref&gt;&quot;Invitation Ball.&quot; ''Coventry Standard'' 6 January 1871, Friday: 4 [of 4], Col. 5b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000683/18710106/100/0004. Same print title, n.p.&lt;/ref&gt;&lt;/p&gt;&lt;/blockquote&gt;

=== February ===

==== Birmingham Tennis Court Club Ball ====
1871 February 17, Friday, the &quot;bachelors of the Tennis Court Club&quot; hosted a ball in Birmingham:&lt;blockquote&gt;LEAMINGTON.&lt;p&gt;

B&lt;small&gt;ACHELORS'&lt;/small&gt; B&lt;small&gt;ALL&lt;/small&gt;.&lt;p&gt;— Last night the bachelors of the Tennis Court Club gave a grand ball at the Royal Assembly Rooms, Regent Street. The ball was one of the most brilliant of the season, nearly four hundred of the ''élite'' of the town and neighbourhood having accepted the invitation of the bachelors. The ballroom was specially fitted up for the occasion, and a splendid supper was served in the adjoining rooms, where refreshments were also provided. Coote and Tiney's band was specially engaged for the occasion, and played a selection of the newest and most popular dance music. Amongst the distinguished guests present were — The High Sheriff and Mrs. J. T. Arkwright, Lady Arbuthnott, Lord and Lady Conyers, [[Social Victorians/People/Abercorn|Lady Caroline Howard]], Viscount and Viscountess Mountgarret and the Hon. Miss Butler, Sir John and Lady Blois, Sir Thomas Biddulph, the Hon. Miss Somerville, Sir William and Lady Fairfax, the Hon. Charles L. Butler, Rev. Sir John Rae, General and Mrs. Richmond Jones, Major Eldman, Major and Mrs. James Ashton, Major and Mrs. Boothby, Colonel Ruttie, Colonel Duberly, Colonel and Mrs. Machen, Colonel Rattray, Capt. and Mrs. Kennedy, Capt. W. J. Hall, Capt. Hodge, Capt. and Mrs. Morgan, Capt. and Mrs. Pearse, Capt. Roberts, Capt. Story, Mr. and Mrs. Featherstone Dilke (Maxstoke Castle) and Miss Dixie, Mr. C. M., Miss, and Miss M. A. Caldecott (Holbrooke Grange), Mr. and Mrs. J. Dugdale (Wroxhall Abbey), Mr. E. Greaves, M.P., Mr. and Mrs. C. L. Adderley (Hams Hall), and Capt. and Mrs. Hatherall. Several of the officers from the dragoons and artillery at Coventry and Birmingham were also present. The bachelors who gave the ball were twenty-eight in number.&lt;ref&gt;&quot;Leamington.&quot; &quot;District News.&quot; ''Birmingham Morning News'' 18 February 1871, Saturday: 7 [of 8, print and digital], Col. 5b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0005826/18710218/114/0007. Print and digital title are the same.&lt;/ref&gt;&lt;/p&gt;&lt;/blockquote&gt;Another description of this same event, the Bachelors' Ball at the Leamington Spa:&lt;blockquote&gt;The bachelors’ ball at Leamington Spa, which took place on the 17th inst., was a greater success than ever. It was held as usual in the Assembly Rooms, which, by the bye, might be better adapted to such purposes. Theyare not so bad as far as the ball room goes, but to reach the supper room you have to make a pilgrimage up one of the steepest and most uncomfortable staircases ever seen; still, however difficult the journey, a safe arrival will repay one. The room was very prettily decorated, and most sumptuous fare provided. The following is a list of the bachelors who gave the ball: Mr Neville Bagot, Mr Ramsay Clarke, Mr Erasmus Galton, Mr C. H. Gregg (8th Hussars), Mr Ralph C. Gregg, Mr William Gillett, Mr Thomlinson Grant, Col. Hammond, R.A., Capt. Hull, Mr Wm. Harrison, Mr Pulsford Hobson, Mr Sydney Hobson, Mr F. C. Lister Kay, Viscount St. Lawrence, M.P., Capt. Maxwell Lyte (7th Dragoon Guards), Mr Richard Lant, Mr John Lant, Mr Oswald Milne, Mr W. W. Moore, Mr Thomas Norman, Mr Hamilton Osborne, Capt. John Paynter, Capt. Pullin, Mr George Rennie, Mr Alex. G. Stuart, Mr J. H. Sanders, Mr Edmund Vyner, Captain Vandeleur; and nothing that they could do was wanting to make it a most complete success. The frequenters of the subscription balls could scarcely recognise the rendezvous of their fortnightly meetings. A porch had been erected over the entrance in the parade, and the corridors all round the dancing room carpeted with crimson and prettily decorated. Banks of flowers had been arranged in every available corner of the ball room, and a number of mirrors hung against the wall reflected the gay scene. Coote and Tinney’s band played a charming selection, and dancing was kept up with much spirit to a late hour. The company was a large one, the toilettes exceedingly pretty. Among those present were Lord and Lady Conyers, Sir William and Lady Fairfax, [[Social Victorians/People/Abercorn|Lady Caroline Howard]], Viscount and Viscountess Mount-Garrett, [[Social Victorians/People/Ormonde|Hon. Miss Butler]], Sir John Rae Reid, Hon. Mary Somerville, &amp;c.&lt;ref&gt;&quot;Fashionable Entertainments.&quot; ''The Queen'' 25 February 1871, Saturday: 19 [of 24], Col. 3b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002627/18710225/121/0019. Print title: The Queen, ''The Lady's Newspaper'', p. 133.&lt;/ref&gt;&lt;/blockquote&gt;The ''Warwick and Warwickshire Advertiser'' has a more detailed account, especially of the people invited and attending (or not):&lt;blockquote&gt;THE BACHELORS' BALL.

This fashionable ''réunion'' of the ''élite'' of the town and neighbourhood took place the Assembly Rooms last evening The large room was beautifully decorated by Mr. Abotta, of Lower Bedford-street, who had the entire management of the preparations. Coote and Tinney's band occupied the orchestra, and played an admirable selection of first-class dance music. Mr. Wheal, of the Lower-parade, supplied the supper. The following gentlemen constituted the committee of management:— Mr. Neville Bagot, Mr. Ramsay Clarke, Mr. Erasmus Gallon, Mr. C. H. Gregg (8th Hussars), Mr. Ralph C. Gregg, Mr. W. Gillett, Mr. Thomlinson Grant, Colonel Hammond, R.A., Captain Hull, Mr. Wm. Harrison, Mr. Pulsford Hob- [Col. 5c–6a] son [Hobson], Mr. Sydney Hobson. Mr. F. C. Lister Kay. Viscount St. Lawrence, M.P., Captain Maxwell Lyte (7th Dragoon Guards), Mr. R. Lant, Mr. J. Lant, Mr. Oswald Milne, Mr. W. W. Moore, Mr. Thos. Norman, Mr. Hamilton Osborne, Captain John Paynter, Captain Pullin, Mr. George Rennie, Mr. Alexander G. Stuart, Mr. J. H. Sanders, Mr. Edmund Vyuer, and Captain Vandeleur. 

The following is a list of the company, alphabetically arranged:— Mr. Mrs. and Miss Andrew, Moseley Lodge; Major Ashton and Mr. James, 28, Lansdowne-place; Miss Ellen Andrew, Moseley Lodge; Mr. and Mrs. J. T. Arkwright, Hatton House, Hatton; Mr. and Mrs. Frank Ashton, Beech-croft, Kenilworth-road; Mr. and Mrs. Adderley, Hams HalI, Warwick; Mr. and Miss Alston, Elmdon Hall, Solihull; Mr. W. and Mrs. T. Alston, Elmdon Hall, Solihull; Mrs. and Miss Ackers, ''chez'' Mountgarrett [?], 34, Lansdowne-place; Lady Arbuthnott, Shenton Hall, Nuneston; Mr. Augustus Arkwright, Hatton House; Miss Adams, 3, Warwick-place; Mr. J. Angerstein, ''chez'' Paynter Denby Villa; Mr. Astley, Hamilton-place; Captain Arthur, George Hotel, Rugby; Sir Theophilus Biddulph, Birdingbury Hall; Mrs. and the Misses (3) Bunowes, 29, Dale-street; Captain and Mrs. Battine, Eathorpe Hall; Captain and Mrs. Charles Blundell, Dun Edin Villa; Mr. George and Miss Brodie, Rowington Vicarage; Sir John and Lady Blois, 31, Clarendon-square; Mr. and Mrs. Barlow, 15, South-parade; Honourable Charles Lennox Butler, Coton House, Rugby; Mr. and Mrs. Boultbee, Springfield, Knowle; Mr. William Blundell, Dun Edin Villa; Miss K. Browne, ''chez'' Beaver Roberts, Thorn Bank; Mr., Mrs., and Miss Beech, Brandon Lodge, Coventry; Mrs. Bame, Clarendon Hotel; Major and Mrs. Boothby, Glencairn; Mr. and Mrs. Rochfort Boyd, ''chez'' Viscountess Mountgarrett; Miss Florence Booth, Huntley Lodge; Major Butter, ''chez'' Majoribanks; Mr. and Mrs. Bowyer, 1, Clarence-crescent; Mr. Philip Bame, Denby Villa; Captain R. Bedford, Knowle Lodge, Lichfield; Mr. T. Beech, jun., Brandon Hall; Miss Boothly [sic], Glencairn; Mr. Mrs, and Miss Brown Clayton, 35, Clarendon Square; Mr.. Mrs., and Miss Chambers, Enstwood [?] Lodge; Captain C. B. Cave, 9th Lancers, Kenilworth; Miss Carles, Leam-terrace; Lord and Lady Conyers, Wellesbourne; Mr. Mrs., and Miss M. A. Caldecott, Holbrook Grange, Rugby; Mr. and Mrs. Aprice Colis, Clarendon-square; Mrs. and Fitzroy Campbell, Wellesbourne; Miss Mary Browne Clayton, Clarendon-square; Captain Stapleton Colton, Kelstone, Southampton; Dr. Collins, 6, Euston-place; Mr. Chamberlayne, Stoney Thorpe, Southam; Mr S. Corbet, Jephson Villa; Mr. J. and Mr. T. Crampton, ''chez'' Knightley, Kineton; Captain and Mrs. Chichester, R.H.A. Coventry Barracks; Mr. M. Campbell, 45, Clarendon-square; Mr. and Mrs. Duppa, 11, Upper-parade; Miss Dixie, Maxstoke Castle; Mr. Beauchamp Downall, 3, Sherbourne-place; Colonel, Mrs. and Miss Duberley, 19, Clarendon-square; Mr. S. Kevill Davies, Darlaston [?] Hall, Coventry; Mr. Paunesfort Duncombe, ''chez'' Viscountess Mountgarrett; Mr. and Mrs. Dugdale, Wroxhall Abbey; Miss Davies, ''chez'' Unett, Castle Froma [?]; Major and Mrs. Edeman, Bentinck House; Miss Edith Featherston, High-street, Warwick; Sir Wm. and Lady Fairfax, 20, Lansdowne-crescent; Captain Minabull [?] Forde, ''chez'' Unett, Castle Froma; Mrs. Fane, Newbold-terrace; Captain W. Featherstone, Warwick; Mr. Beaumont Featherston, Warwick; Mr. and Mrs. G. Greenway. Binswood Cottage; Mr. and Mrs. Newberry George, Grosvenor House; Major, Mr., and Miss Gresley [?], Meriden Lodge; Mr., Mrs., and Miss Grice, Sherbourne; Mrs. and Miss T. Grant, Clarendon-square; Mrs. Georges, Oakfields; Mr. and Mrs. Graham, Oaklands, near Birmingham; Miss Grant, Oakfield, London; Miss Gumson [?], Clarendon-square; Mr. W. Grant, 6th Regiment, ''chez'' Tomlinson Grant, Clarendon-square; Mr. Watson Gooch, Sherboume-place; Miss Grace Granville, ''chez'' Rolfe, Harvey Villa; Captain Georges, Oakfields; Mr. Edward Greaves, M.P., Avonside; Mr. and Mrs. Hunt. Kenilworth-road; Mr. Yates Hunt, Acton Villa; Captain and Mrs. Hatheral, Radford [?] House; Miss Hoey, St. Helen’s; Miss Hope, Milverton Lodge; Mr. and Mrs. Cinton [sic] Henshaw, Lansdowne Villa; Miss Hughes, Newbold-terrace; Mrs. Clement Hoey, St. Helens; Mrs. and the Misses Hobson, Beauchamp-square; Mr. J. T. Hartley, Long Castle, Shiffnal; Mr. T. Harter, The Cedars; Captain Hobson, (3rd Buffs), Avon Lodge; Mr. John Hetherington, Edstone, Henley; Mr. H. Heathfield, Newbold Comyn; [[Social Victorians/People/Abercorn|Lady Caroline Howard]], Waterloo-place; Captain Hodge, ''chez'' Hobson, Beauchamp House; Miss Hurst, ''chez'' Hobson, Beauchamp House; Capt. W. J. Hall, Junior United Service Club; Miss Alice Hartley, Tony Castle, Salop; Mr. and Miss Hodgson, Clopton, Stratford; Mr. Charles Hartley, Tony Castle, Salop; Mr. Edwin Hobson, Beauchamp House-square; Miss Holbech, ''chez'' Hacket, Binswood; Mr. and Mrs. Jeaffresen, Lansdowne-place; General and Mrs. Jones, Clarendon-square; Mr. Cove, Mrs. and Miss Jones, Loxley Hall, Warwick; Mr. Washington Jackson, ''chez'' Harter, The Cedars; Mr. James Jameson, Church-street; the Misses Johnstone, ''chez'' Pigott, Nowbold-terrace; Mrs. King Harman, Ashley Lodge; Miss Lizzie Holliday, Ashley Lodge; Miss Hetherington, Edston Hall; Mr. A. Hillyard, Southam; Mr. Edgar Hibbert, Whitley Abbey; Major Hogge, 16th Regiment, Rugby; Mr. and Mrs. Kay, Lansdowne-place; Mr. Raleigh King, Lillington; Captain and Mrs. Kennedy, 5th Dragoon Guards, Lillington; Rev. Mr. and Mrs. Knightly, Combrooke, Kineton; Mr. Kershaw, United Hotel, Charles-street, St. James; Mr. J. Maxwell Lyte, Magdalen College, Oxford; Misa C. Lyon, Bankfield; Miss Lowes, Clarendon-square; Miss S. Lowndes, Rugby; Mrs. Lockwood, St. Helen's; Mr. Webb Lindsay, Birmingham; Mr. and Mrs. Lucy, Charlecote Hall; Miss Catharine Lyon, Bankfield; Mr. R. Lancaster, Bilton Grange; Mr. T. H. Lowe, Oxford; the Misses Ley (2) Clarendon-square; Viscount and Viscountess Mountgarrett, Lansdowne-place; Hon. Miss Butler, Lansdowne-place; Mr. and Mrs. Majoribanks, Newbold Firs; Mr. and Mrs. W. H. Milne, Beauchamp-square; Mr and Mrs. Male, Euston-place; Mr. Herbert Molyneux, Tennis Court Club; Capt. and Mrs. Morgan Wellington-street; Mr. H. M. McCalmont, Grosvenor-place, London; Mr. and Miss Moore, Knightcott House, Milverton; Mr. J. M. Middleton, Clarendon-square; Mr. and Mrs. Marsland, Huntley Lodge; Mr. A. Myers, Coldstream Guards, ''chez'' Machen, Lillington Lodge; Mr. J. Middleton, Walton-place; Mr. McLeon, Binswood; Mr. MacGregor, Clarendon-square; Miss Miller, Kenilworth House; Miss Majendie, Newbold-terrace; Mr. and Mrs. Tertius Molliet, Lansdowne-circus; Colonel and Mrs. Machen, Lillington; Miss Newbie, Beechcroft; Mr. and Mrs. Philip Pewman, Warwick-road; Miss Newton, ''chez'' Unett, Castle Froma; Captain Norton, 3rd Dragoon Guards, Beauchamp-square; Dr. and Mrs. O'Callaghan, Clarendon-square; head officers of the 2nd and 5th Dragoon Guards, Leeds, Barracks; ditto, detachment of the 5th Dragoon Guards, Birmingham Barracks; ditto, ditto, Coventry Barracks; Miss Osborne, Clarendon-square; Mr. and Mrs. Osborne, Clarendon-square; Mr. and Mrs. Oldham, Castle Froma; Miss Ommancy, Warwick-place; Miss Emily Owen, and Miss Owen, Coleshill House; Mr. F. Osborne, Clarendon-square; Mr. and Mrs. Billingsley Parrey, Newbold Terrace; Mr. and Mrs. Palmer, Clarendon-square; Mr. Mrs. and Miss Paynter, Denby Villa; Mr. Mrs. and Miss Pigott, Newbold-terrace; Captain and Mrs. Pearce, ''chez'' Marjorbanks [sic], Miss and Miss L. Pritchard, Upper-parade; Miss Pixell, South-bank; Mrs. and the Misses Pullin, Waterloo-place; Miss Louisa Passy, Beauchamp-walk; Miss and Miss Ada Pennington, Thickthom, Kenilworth; Mr. Mrs. and Miss Perry, Bitham House, Avon Dassett; Mr. H. K Pullin, Junior, St. James Club; Miss Penny, Warwick-place; Miss Phillips, Clarendon-square; Mr. Pennington, Thickthorn; Miss Henrietta Passy, Beauchamp-walk; General and Mrs. Potter, Holly-walk; Mr. Mrs. and Miss Beaver Roberts, Thorn-bank; Mr. Stewart Roberts, Thorn-bank; Mr. and Mrs. Roundell, Fulham Villa; Colonel and Mrs. Ruthe, Clarence-terrace; Miss Raymond, Douglas House; Mr. Rowley Robertson, South Lodge; Mr. and Mrs. Russell, Newbold-terrace; Miss Ryland, Barford Hall; Sir John Rae Reid, Rugby; Mr. and Mrs. Worley Roberts, Oakley House; Mr. Percy Robertson and Mr. D. Robertson, Newbold-terrace; Miss Neville Rolfe, Dale-street; Colonel Clerk Rattray, Lansdowne-place; Mr. Maurice Raymond, Douglass House; Captain Roberts, Binswood; Mr. Andrew Robertson, Banbury; Miss Read, Clarendon-square; Mr. R. M. Russell, Leek Wootton; Mr. A. P. Roberts, Brazenose [?] College, Oxford; Mr., Mrs., and Miss Scholes, Zelam Lodge; Mon. Mary Somerville, Riber House: Miss Stuart, Clarendon-square; Mr. E. Sanders, Omskirk, Lancashire; Miss Palgrave Simpson, Princes Park, Liverpool; Miss Smythe, Solihull Rectory; Mr. J. F. Starkey. Stratford; Mr. and Mrs. George Stratton, Husband’s Bosworth, Rugby; Mr. Hamilton Stuart, Clarendon-square; Miss Sinclair, Dalestreet; Mr. Sedgwick, Warwick-place; Mr. Spencer Smith, Clarendon-square; Captain Starry; Miss Stallard, Warneford Villa; Mr. W. Stancombe, Magdalen College, Oxford; Miss Seymour, Warwick-road; Miss Sankey, Beauchamp-walk; Mr. Spooner, 11th Regiment, Clarendon-square; Mr. Strongitharm, Norton House; Mr. and Mrs. Molyneux Seal, Milton House; Mr. W. Sinclair. Dale-street; Mr. J. Smith, Dale-street; Mr., Mrs., and Miss Turner, Milverton Lodge; Miss Ellen Turner, ditto; Miss Tomkinson, Dale-street; Miss Thompson, Binswood; Miss Tuite, Warwick-place; Miss Temple, Newbold-terrace; Mr. Dudley Tarleton, Leam-terrace; Mis Tucker, Dale-street; Mr. and Mrs. G. Unett, Castle Froma; Mr. Gwinett, ditto; Mr. and Miss Unett, Portland-street; Mr. and Mrs. White, Beauchamp-walk; Miss Wheler, Bertie-terrace; Miss E. and Miss C. Wise, Shrublands; Mr. and Miss Wollaston, Shenton Hall, Nuneaton; Mr. E. G. Wheler, Bertie-terrace; Mr. and Miss West, Alscot Park, Stratford; Mr. and Mrs. Woodmass, Mosely Lodge; Miss Wardrope, Waterloo-place; Miss Wetherall, Woodcote; Miss Lilly and Miss Alice Wise, Cubbington Grange; Mrs. and Miss Wright, Lansdowne-crescent; Mr. H. White, Ashfield House; Miss Wakefield, Castle Froma, Mr. Herbert Wood, Newbold Revel; Mr. Young, Whitnash Rectory. 

Invitations were also sent to the following but declined for family and other reasons:— Lord and Lady Leigh and Miss Leighs (2); Mrs. General Hall, the Misses Collinson, Mr., Mrs. and Miss Hobson, Avon Lodge; Lieut-Colonel and Mrs. Fiennes; Captain and Mrs. Gregg; Captain and Mrs. Vaughton; Mrs. Frederick Gubbins, Mr. J. P. and Mrs. Gubbins; Mr. Stuart; Miss Maconehy; the Misses Staunton; Miss Galton; Mr. Raleigh King; Mr. Edward Wheler; Miss Miller; Mr. Jennings; Dr. and Mrs. Jephson; Dr. and Mrs. Thomson; Mr. and Mrs. Philpot; Mr. H. and the Misses Baker; Mr. R. Read; Sir Robert and Lady Hamilton; Mr. H. C. and Mr. G. Wise; Colonel and Miss Daniel; Mr. and Mrs. Bigland; Mr. and Mrs. Robertson; Miss Stevenson; Mrs. Osborne; Miss Harter; Mr. and Mrs. Lister Kay; Mr. and Mrs. Henry Chance; Mr., Mrs. and the Misses Bradshaw; Lady Eardly; Miss Stevenson; Captain Turquand; Major Paynter; Captain Tomkinson; Lady Hampson; Mr. Bame; Mr. Augustus Wise; Mr. and Mrs. John Mordaunt; Lady Willoughby de Broke; Mr. Caldecott; Mr. Hamilton and Miss Story; Miss Mabel Hurst; Mr. and Mrs. Bolton King; Miss Kate Fetherston; Sir Charles Mordaunt, Lord and Lady Willoughby de Broke; Miss Rigby; Mrs. and Miss Wise, Woodcote; Mr. E. Wheler, Mr., Mrs. and Miss Pennington, Westfield; Mr., Mrs. and Miss Mackenzie; Miss Wilkins; Major Lee, Mr. and Mrs. Mark Hammond, Miss P. Hughes; Lord and Lady James Murray; Mr. and Mrs. Barker, Mr. and Mrs. James West, Miss Hackett, Mr. and Miss Walker, Mrs. Harman King, Mr. Clement Hoey, Mr. Herbert Wood, Mr. Thomas Lant, the Earl of Howth, Mr. Robertson, Mr. Bookeley and Mr. E. Steward.&lt;ref&gt;&quot;The Bachelors' Ball.&quot; ''Warwick and Warwickshire Advertiser'' 18 February 1871, Saturday: 2 [of 6], Cols. 5c–6c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001670/18710218/051/0002. Print title: ''Warwick and Warwickshire Advertiser and Leamington Gazette'', n.p.&lt;/ref&gt; &lt;/blockquote&gt;

=== March ===

=== April ===

==== 18 April 1871 ====
&lt;blockquote&gt;Karl Marx “was commissioned by the General Council of the International to write a pamphlet about the Paris [377–378] Commune.&quot;&lt;ref name=&quot;:3&quot;&gt;Smee, Sebastian. ''Paris in Ruins: Love, War, and the Birth of Impressionism''. W. W. Norton, 2024.&lt;/ref&gt;{{rp|377–378 of 667}}&lt;/blockquote&gt;

===May===

==== 9 May 1871, Tuesday, Queen's Drawing-Room ====
&lt;blockquote&gt;THE QUEEN'S DRAWING-ROOM.

The Queen held a Drawing-room at Buckingham Palace on Tuesday afternoon. The Priuce of Wales, Prince Arthur, Prince Leopold, and Princess Beatrice were present. Her Majesty, accompanied by the Prince of Wales and the other members of the royal family, entered the Throne Room shortly after three o'clock. The Queen wore a black moire antique dress with a train, long white tulle veil with a coronet of diamonds. Her Majesty also wore a necklace of diamonds and amethysts, the Riband and Star of the Order of the Garter, the Orders of Victoria and Albert and Louise of Prussia, and the Saxe Coburg and Gotha Family Order. Princess Beatrice wore a dress of white tulle over a rich white silk petticoat looped up with lilies of the valley and apple blossom; ornaments — pearls and diamonds.

The presentations to Her Majesty were about 280 in number, and included the following:— Mrs Atlay, by the Countess Grey; Miss Backhouse, by her mother, Mrs Backhouse; Miss Charlesworth, by her aunt, Frances Lady Hawke; Miss Backhouse Fox, by her aunt, Mrs Backhouse; [[Social Victorians/People/Abercorn|Lady Caroline Howard]], by her mother, [[Social Victorians/People/Abercorn|the Hon. Mrs Howard]]; the Hon. Gwendoline Fitz-Alan Howard, by the Duchess of Sutherland; [[Social Victorians/People/Abercorn|Lady Alice Howard]], by her mother, Hon. Mrs Howard; [[Social Victorians/People/Abercorn|Lady Louisa Howard]], by her mother, Hon. Mrs Howard; Miss Howard (of Corby), by the Hon. Mrs Philip Stourton; Miss Agnes Howard (of Corby), by the Hon. Mrs Philip Stourton; Sir Henry Ingilby, Bart., by Earl Russell; Mrs Frank Lascelles, by Lady Edward Cavendish; Mrs Gerald Liddell, marriage, by the Countess of Normanby.&lt;ref&gt;&quot;Court and Official News.&quot; ''Yorkshire Post and Leeds Intelligencer'' 11 May 1871, Thursday: 3 [of 4], Col. 4c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000686/18710511/074/0003. Same print title and p.n.&lt;/ref&gt;&lt;/blockquote&gt;The ''London Evening Standard'' has a more detailed report, as we would expect from a London paper. (The long lists have been set as bulleted lists to save space.) The Countess of Granville was sponsoring quite a few people in this drawing room — perhaps she was one of the aristocrats who did it for a fee.&lt;blockquote&gt;HER MAJESTY'S DRAWING ROOM.

The following is the list of Presentations to her Majesty at the Drawing Room held on Tuesday:—

The Foreign Ambassadors and Ministers having been introduced in the order of precedence, the following presentations were made in the diplomatic circle:—

* By her Excellency the Countess de Bernstorff. — The Countess Sagn-Wittgenstein, the Countess Eleanora Sagn-Wittgenstein, and the Countess Elisabeth Sagn-Wittgenstein.
* By Madame Balcarce. — Madame Gutierrez de Estrada, daughter of the Argentine Envoy.
* By the Duchess de Saldanha. — The Marchioness de Penafiel.
* By the Countess Granville. — The Princess Eulalie de Solms Braunfels, daughter of a General in the Austrian army; the Countess Kielmansegge, an Austrian lady; the Countess de Canclaux, wife of the Secretary of the French Embassy; the Countess de Behagne, a French lady; the Duchess de Caraniolo, an Italian lady; the Duchess de Osuna, and the Countess Fernandina, Spanish Ladies; Miss Newbold, a lady of New York, U.S.; Miss Clara Carlisle, and Miss Florence Carlisle, ladies of Cincinnati, U.S.; and Miss Constance Kinney, a lady of Washington City, U.S.
* By his Excellency the Turkish Ambassador. — The Count Alexander Kielmansegge, a Captain of the Austro-Imperial Navy.
* By his Excellency the Russian Ambassador. — M. P. Monkhanow, Lieutenant of Marine, and Naval Attache of the Embassy.
* By the Argentine Envoy. — M. Gutierrez de Estrada, formerly First Secretary of the Legation.
* By the Belgian Envoy. — M. le Baron Van den Boschen.
* By the Italian Knvoy. — M. le Duc de Caranio'o.
* By the Spanish Envoy. — Mons. le Duc de Osuna and Mons. le Comte de Fernandina.
* By the Portuguese Envoy. — The Marquis le Penaflel, and the Count de Carnota, brother-in-law of the Envoy.
* By the Charge d'Affaires of the United States. — Commander W. G. Whiting, United States Navy, and Lieutenant Commander F. Pearson, United States Navy.

The following presentations to her Majesty were made (about 280 in number), the names having been previously left at the Lord Chamberlain's Office, and submitted for her Majesty's approval:—

* Mrs. Adair, by the Countess Granville, in absence of Frances Countess Waldegrave.
* Viscountess Adare, on her marriage, by the Duchess of Buccleuch.
* The Hon. Lady Adderley, by Lady Leigh.
* The Hon. Mrs. Acheson, by the Lady Gertrude Foljambe.
* Miss Ackers, by her mother, Mrs. Ackers.
* Hon. Evelyn Addington, by her mother, Viscountess Sidmouth.
* Miss Elizabeth Alsopp, by her Mother.
* Mrs. Ince Anderton, by the Lady Stafford.
* Miss Ince Anderton, by the Lady Stafford. [repetition sic]
* Lady Antrobus, by the Marchioness of Ely.
* Mrs. William Rae Arthur, wife of the Lord Provost of Glasgow, by the Lady Emily Foley.
* Mrs. Atlay, by the Countess Grey.
* Miss Backhouse, by her mother, Mrs. Backhouse.
* Miss Alice Bagot, by her mother, Mrs. Charles Bagot.
* Mrs. Arthur Baird, by Lady Helen Macgregor.
* Mrs. Hervey Bathurst, by the Countess of Sefton.
* Miss Laura Hicks-Beach, by her mother, the Dowager Lady Hicks-Beach.
* Miss Mary Hicks-Beach, by her mother, the Dowager Lady Hicks-Beach.
* Miss Constance Beresford, by her mother, Mrs. Marcus Beresford.

* Sir Edward Hunter Blair, by Rear Admiral Sir John Dalrymple Hay.
* Lady Hunter Blair, by the Duchess of Sutherland.
* Miss Hunter Blair, by the Duchess of Sutherland.
* Miss Alice Mary Hunter Blair, by the Duchess of Sutherland.
* Mrs. Gore Booth, by the Countess of Scarborough.
* Mrs. R. Vicars Boyle, by the Lady Rayleigh.
* Miss Edith Brewer, by her mother, Mrs. Brewer.
* Miss Selina Brewer, by her mother, Mrs. Brewer.
* Lord Brougham and Vaux (on succeeding to the title), by Viscount Sidmouth.
* The Hon. Adela Brougham, by Viscountess Sidmouth, in the absence of her mother through illness.
* Mrs. James Clifton Brown, by her mother-in-law, Mrs. Alexander Brown.
* Mrs. Stewart Brown, by Mrs. Alexander Brown.
* Miss Lucile Brooke, by her mother, Mrs. Brooke.
* Mrs. John Brooks, by the Marchioness of Huntly.
* Miss Margaret Brooks, by her mother, Mrs. John Brooks.
* Viscount Bury, on being made K.C.M.G. by the Secretary of State.
* Mrs. Walter Byles, by Lady Byles.
* The Hon. Mrs. Arthur Cadogan, on her marriage, by the Countess of Craven.
* Lady Campbell of Dunstaffnage, by the Duchess of Argyll.
* Miss F. Julia Pole-Carew, by Mrs. Pole-Carew.
* Mrs. Carruthers of Dormont, by the Marchioness of Queensberry.
* The Hon. Mary Cavendish, by her mother, Lady Chesham.
* Lady Chapman, by the Hon. Mrs. Mostyn.
* Miss Chapman, by her mother, Lady Chapman.
* Miss Charlesworth, by her aunt, Frances Lady Hawke.
* Miss Evelyn Chichester, by her mother, the Hon. Mrs. Frederick Chichester.
* Lady Chute, by the Hon. Mrs. Claughton.
* Miss Hyde Clarke, by the Marchioness of Queensberry.
* Miss Lucy Claughton, by her mother.
* Lady Margaret Coke, by Viscountess Powerscourt.
* Viscountess Cole, on her marriage, by the Marchioness of Ormonde.
* Mrs. Collingwood, by Countess Percy.
* Miss Collingwood, by her mother, Mrs. Collingwood.
* Miss Adelaide Collingwood, by her mother, Mrs. Collingwood.
* Miss Annie Colthurst, by her mother, Lady Colthurst.
* Mrs. Cookson, by the Duchess of Sutherland.
* Miss Cookson, by Mrs. Cookson.
* Miss Gibson Craig, by her mother, Lady Gibson Craig.
* The Countess of Crawford and Balcarres, by the Countess of Caledon.
* Viscountess Crichton, on her marriage, by the Countess of Dartrey.
* Miss Gertrude Creyke, by the Duchess of Buckingham.
* Lady Cunliffe, on her marriage, by the Marchioness of Westminster.
* Mrs. Robert Capel Cure, on her mamage, by Lady Rayleigh.
* Sir Benjamin Chapman, by Lord Lurgan.
* Miss Gwendoline Irving-Davies, by her mother, Mrs. Irving-Davies.
* Miss Mary Dalzell, by Lady Helen Stewart.
* Miss Laura Day, by her mother, Mrs. John Day.
* Lady Mary Dalrymple, by the Countess of Stair.
* Miss Davison, by the Countess of Limerick.
* Miss Dora Davison, by the Countess of Limerick.
* Mrs. Harold Arthur Dillon, on her marriage, by Viscountess Dillon.
* Mrs. George Ashley-Dodd, on ber marriage, by her mother, Mrs. Edwards.
* Mrs. Douglas, by the Hon. Mrs. Speir.
* Mrs. W. E. Dowdeswell, by the Countess Beauchamp.
* Mrs. Dowse, by Lady Katherine Coke.
* Miss Dowse, by her mother, Mrs. Dowse.
* Lady Eliott-Drake, by Lady Hylton.
* Lady Edith Drummond, by the Countess of Perth.
* Miss Dundas, by her mother, the Hon. Mrs. Dundas.
* Miss Mary Dundas, by her mother, the Hon. Mrs. Dundas.
* Lord Dunglass, on his marriage, by the Duke of Buccleuch.
* Lady Dunglass, on her marriage, by the Countess of Home.
* Lady Dunbar of Northfield, by the Hon. Mrs. Grant of Grant.
* Mrs. Dugdale, by Lady William Wynn.
* Miss Dugdale, by her mother, Mrs. Dugdale.
* Miss Durham, by Mrs. Grenfell.
* Lady William Godolphin Osborne Elphinstone, by Lady Blanche Morris.
* Mrs. Elrington, by the Hon. Mrs. Edward Coke.
* Miss Elrington, by her mother, Mrs. Elrington.
* Miss Susan Elwes, by her mother, Mrs. Robert Elwes.
* Mrs. William Everett, by Lady Charles Wellesley.
* Miss Louisa Ewart, by her mother, Mrs. Ewart.
* Mrs. William Fairbairn, by the Duchess of Buckingham and Chandos.
* Miss Emily Fairbairn, by her mother, Mrs. William Fairbairn.
* Miss Georgia Fellows, of New York, by Lady Granville.
* Countess Ferrers, by the Countess of Bradford.
* Miss Fitzherbert, by the Lady Waterpark.
* Miss Eleanor Fitzroy, by the Dowager Duchess of Grafton.
* Mrs. Cuddon-Fletcher, by the Duchess of Argyll.
* Miss Laura Fletcher, by her mother, Mrs. Fletcher.
* Mrs. Foljambe, on her marriage, by Lady Catherine Vernon Harcourt.
* Mrs. William Fowler, by Mrs. Backhouse.
* Lady Georgiana Fortescue, by Lady Camilla Fortescue.
* Miss Mary Fothergill, by her mother, Mrs. Fothergill.
* Miss Backhouse Fox, by her aunt, Mrs. Backhouse.
* Miss Georgiana Fullerton, by her mother, Mrs. David Fullerton.
* The Hon. Georgina Evans Freke, by her mother, Lady Carberry.
* Mrs. John Tudor Frere, on her marriage, by her mother, Mrs. Forbes Winslow.
* Mr. William Fowler, M.P., by Mr. W. E. Forster.
* Lady Alice Gaisford, by the Countess Brownlow.
* Mrs. Gaussen, on her marriage, by Viscountess Cole.
* Miss Clara Gervis, by her mother, Lady Gervis.
* The Hon. Eleanor Gifford, by her sister, Hon. Mrs. A. Douglas Pennant.
* Mrs. Maxwell Goad, by Lady William Godolphin Osborne Elphinstone.
* Miss Georginna Goodford by Mrs. Goodford.
* Mrs. Gerold Goodlake, on her marriage, by Lady Louisa Spencer. ['''Col. 3c–4a''']
* Miss Francis Goodwin, by her mother, Mrs. Harvey Goodwin.
* Mrs. J. H. Gordon, on her marriage, by the Duchess of Richmond.
* Mrs. James Augustus Grant, by the Hon. Mrs. Grant of Grant.
* Miss Grenfell, by Mrs. Grenfell.
* Miss Greenwood, by her mother, Mrs. Greenwood.
* The Lady Anne Grenville, by her mother, the Duchess of Buckingham and Chandos.
* The Lady Mary Grenville, by her mother, the Duchess of Buckingham and Chandos.
* Miss Grey, by the Countess Grey.
* Miss Emily Hardcastle, by the Hon. Mrs. Hardcastle.
* Miss Constance Harford, by her mother, Mrs. Harford.
* Miss Harford, by her mother, Mrs. Harford.
* Mrs. Cecil Haflenden Hall, by the Marchioness of Queensberry.
* Miss A . J. Harris, by Mrs. Charles Hardy.
* Lady Louisa Hastings, by the Marchioness of Waterford.
* Miss Hargreaves, by her aunt, Lady Gervis.
* Miss Marguerite Henry, by her mother, Mrs. Mitchell Henry.
* Miss Hemming, by her mother, Mrs. Hemming.
* Miss Constance Hesketh, by Lady Palk.
* Miss Constance Hemming, by her mother, Mrs. Walter Hemming.
* Miss Heygate, by her mother, Lady Heygate.
* Mrs. H. W. Hitchins, by the Lady Mary Phipps.
* [[Social Victorians/People/Abercorn|Lady Caroline Howard]], by her mother, the [[Social Victorians/People/Abercorn|Hon. Mrs. Howard]].
* The Hon. Gwendoline Fitzalan Howard, by the Duchess of Sutherland.
* [[Social Victorians/People/Abercorn|Lady Alice Howard]], by her mother, the Hon. Mrs. Howard.
* [[Social Victorians/People/Abercorn|Lady Louisa Howard]], by her mother, the [[Social Victorians/People/Abercorn|Hon. Mrs. Howard]].
* Miss Howard (of Corby), by the Hon. Mrs. Philip Stourton.
* Miss Agnes Howard (of Corby), by the Hon. Mrs. Philip Stourton.
* Mrs. Charles Hoare, by the Countess of Morley.
* Miss Hopton, by the Lady Emily Foley.
* Mrs. Cecil Hughes, by Lady Waterpark.
* Sir Henry Ingilby, Bart., by Earl Russell.
* Lady Ingilby, by her mother, Mrs. Robertson, of Ladykirk.
* Lady Jackson, by the Countess Granville, in the absence of Mrs. Gladstone.
* Miss Miriam Bertha Jackson, by her mother, Lady Jackson.
* Miss Jarvis, by Mrs. Jarvis.
* Miss Jerome, by Countess Granville.
* Mrs. Johnston, by Mrs. Clarke.
* The Hon. Mrs. Sydney Hylton-Jolliffe, on her marriage, by Lady Hylton.
* Miss Maude Kekewich, by the Hon. Mrs. Walrond.
* Miss Shaw Kennedy, by the Hon. Mrs. Walrond.
* Miss Eleanor Shaw-Kennedy, by the Hon. Mrs. Walrond.
* Mrs. Alfred Ker, on her marriage, by her mother, the Hon. Lady Bateson.
* Mrs. Francis Kerr, on her marriage, by the Duchess of Buccleuch.
* Miss Mary D'Arcy [D'Arey?] Kerr, by the Duchess of Buccleuch.
* Mrs. Montagu Knight, by her mother, Mrs. Charles Hardy.
* Mrs. Rowley Lambert, by Mrs. Montgomery.
* Mrs. Stephen Gore Langton, on her marriage, by Lady Anna Gore Langton.
* Mrs. Frank Lascelles, by Lady Edward Cavendish.
* Lady Lawrence, on her marriage, by Lady Amelius-Beauclerk.
* Mrs. Lawrence, by Mrs. Tyssen-Amhurst.
* Miss Meta Leader, by Mrs. Leader.
* Miss Florence Lees, by Countess Russell.
* Miss Ellen Lempriere, by the Countess of Morley.
* Mrs. Macalpine Leny, by Mrs. Halsey.
* Miss Macalpine Leny, by her mother, Mrs. Macalpine Leny.
* Miss Rosa Macalpine Leny, by her mother, Mrs. Macalpine Leny.
* Miss Amy Leslie, by her mother, Mrs. Leslie (of Warthill).
* Miss Rose Leslie, by her mother, Mrs. Leslie (of Warthill).
* Mrs. Edward Levy, by the Lady Caroline Barrington.
* Mrs. Gerald Liddell, on her marriage, by the Countess of Normanton.
* The Lady Lindsay, on her marriage, by the Countess of Crawford.
* The Lady Alice Lindsay, by her mother, the Countess of Crawford.
* The Lady Mary Lindsay, by her mother, the Countess of Crawford.
* Miss Luttrell, by her mother, Mrs. Luttrell.
* Mrs. Mackenzie, of Findon, by Lady Matheson.
* Miss Mackenzie, of Kintail, by Lady Cecilia Bingham.
* Miss Alice Mackenzie, of Kintail, by Lady Cecilia Bingham.
* Mrs. Mackenzie, of Portmore, on her marriage, by the Lady Anna Gore Langton.
* Mrs. K. D. Mackenzie, by Lady Claud Hamilton.
* Lady Muir Mackenzie, on her marriage, by the Duchess of Buccleuch.
* Mrs. Mackarness, by Lady Coleridge.
* Miss Mackarness, by Lady Coleridge.
* Lady Sophia Macnamara, on her appointment as Lady of the Bedchamber to Princess Louise, by the Countess of Yarborough.
* Mr. Alfred George Marten, by Mr. John J. Horsley.
* Mrs. Alfred George Marten, on her marriage, by the Countess of Limerick.
* Miss Reid Martin, by the Lady Waterpark.
* Miss Edith Heron Maxwell, by her mother, Lady Heron Maxwell.
* Miss Helen Meade, by her mother, Mrs. Ed. Meade.
* Mrs. Mitchell, by the Countess Dowager of Belmore.
* Miss Henrietta St. John-Mildmay, by Mrs. Edmond St. John-Mildmay.
* Miss Milne, by her mother, Lady Milne.
* Hon. Mrs. Caryl Molyneux, by Viscountess Downe.
* Miss Edith Montgomery, by her mother, Lady Charlotte Montgomery.
* Miss Morrieson, by her aunt, Mrs. H. W. Hitchins.
* Miss Massingberd Mundy, by Mrs. Francis Dawkins.
* Miss Fanny Massingberd Mundy, by Mrs. Francis Dawkins.
* Hon. Mrs. Newdigate, by Mrs. Lynedoch Gardiner.
* Vicountess Newport, by Countess of Bradford.
* The Countess of Normanton, by Countess Nelson.
* Miss Judith Savill-Onley, by her mother, Mrs. Savill-Onley.
* Hon. Mary Onslow, by her mother, Viscountess Cranley.
* Mrs. Charles M. Palmer, by Countess of Yarborough.
* Miss Gambier Parry, by Mrs. Gambier Parry.
* Mrs. Florence Parsons, by Hon. Mrs. Parsons.
* Miss Patton, by her mother, Mrs. Patton.
* Mrs. Peploe Peploe, by the Marchioness of Ormonde.
* Miss Anna Maria Perry, by Mrs. Fitzherbert.
* Lady Peyton, by Lady Leconfield.
* Lady Emily Pierrepont, by Countess Manvers.
* The Countess de Pomar, by Countess Granville.
* Miss Quick, by Lady Palk.
* M. de la Quintana, Peruvian Consul General, by the Peruvian Minister.
* Madame de la Quintana, by Madame Galvez.
* Mrs. D. Gano Ray, of Cincinnati, United States, by Lady Granville.
* Mr. W. H. Rennie, on appointment as Lieutenant Governor of St. Vincent, by the Secretary of State for the Colonies.
* Mrs. W. H. Rennie, on her marriage, by Countess Granville, in the absence of the Countess of Kimberley.
* Miss Ricketts, by her mother, Lady Caroline Ricketts.
* Miss Rolleston, by her mother, Mrs. Rolleston.
* The Countess of Rosse, on her marriage, by her mother, Frances, Lady Hawke.
* Mrs. Round, on her marriage, by Lady Rayleigh.
* Miss Katharine Rowley, by her mother, Hon. Lady Rowley.
* Lady Agatha Russell, by Countess Russell.
* Lady Emily Russell, on her marriage, by Countess Russell.
* Miss Alberta Russell, by the Duchess of Sutherland.
* Mrs. Sandwith, by Mrs. Hamilton.
* Mrs. William Sandwith, by the Countess of Limerick.
* The Lady Sandhurst, by the Countess Granville.
* The Hon. Violet Sandys, by her mother, Lady Sandys.
* Miss Alice Schenley, by her mother, Mrs. Schenley.
* Miss Richmond Schenley, by her mother, Mrs. Schenley.
* Mrs. George Salis Schwabe, on her marriage, by Mrs. Salis Schwabe.
* Lady Edith Scott, by her mother, the Countess of Clonmell.
* Lady Augusta Shirley, by her mother, the Countess Ferrers.
* Miss Caroline Bridgeman Simpson, by her mother, Lady Francis Bridgeman Simpson.
* Miss Slade, by her mother, Mrs. Marcus Slade.
* Mrs. Sneyd of Keele, by Mrs. Bromley Davenport.
* Miss Ina Spencer, by Lady Louisa Spencer.
* Lady St. George, by Lady Gervis.
* Miss Edith Stephenson, by her mother, Lady Mary Whitbread.
* Lady Isabel Taylour, by the Countess of Bective.
* Mrs. Charles Tennant, by Lady Gervis.
* Miss Elsie Tennant, by her mother, Mrs. Charles Tennant.
* Miss Matilda Thomas, by Lady Macarthur.
* Miss Isabel Thomson, by her aunt, Mrs. Thomson.
* Mrs. Acton Tindal, by Lady Chesham.
* Mrs. Raymond Cely Trevilian, on her marriage, by her mother, Lady Vincent.
* Lady Edith Tudway, on her marriage, by the Countess of Normanton.
* Mrs. Edward Winterton Turnour, by her cousin, Lady Charlotte Heard.
* Miss Augusta Twining, by her mother, Mrs. Thomas Twining.
* Miss Verelst, by Mrs. Henming.
* Lady Verner, by Viscountess Sudley.
* Miss Edith Verner, by her mother, Lady Verner.
* The Hon. Susan Verney, by her mother, (Georgiana) Lady Willoughby De Broke.
* Mrs. Julius Vogel, by the Countess Granville, in the absence of the Countess of Kimberley.
* Miss Wadsworth, by the Countess of Granville.
* Mrs. Waithman, by Countess Ferrers.
* Miss Waithman, by her mother, Mrs. Waithman.
* Mrs. Waldy [?], by Lady Matheson.
* Miss Emily Mary Walker, by her sister-in-law, Mrs. Walker.
* Mrs. William Hood Walrond, on her marriage, by Hon. Mrs. Wallrond.
* Miss Gertrude Walrond, by Hon. Mrs. Walrond.
* Mrs Watts, by Lady Pauncefort Duncombe.
* Mrs. Henry West, on her marriage, by Mrs. Algernon West.
* The Marchioness of Westminster, by Lady Leigh.
* Lady Ina White, by her mother, Countess of Bantry.
* Lady Elizabeth White, by her mother, Countess of Bantry.
* Lady Whitworth, by her aunt, Mrs. Tootal.
* Mrs. G. Hampden Wilkieson, from Canada, by Lady Rendlesham.
* Miss Wilcox, by her aunt, Mrs. Milne-Redhead.
* Miss Williams, by her mother, Lady Williams.
* Mrs. Charles Williamson, on her marriage, by the Countess of Normanton.
* Miss Wilson, by Mrs. F. Maitland Wilson.
* Colonel Sir Garnet Wolseley, by H.R.H. the Duke of Cambridge.
* Lady Wolseley, by Lady Sarah Lindsay.
* Miss Ellen Wrottesley, by her mother, Hon. Mrs. Edward Wrottesley.
* Mrs. Wynne, on her marriage, by the Marchioness of Ormonde.&lt;ref&gt;&quot;Her Majesty's Drawing Room.&quot; ''London Evening Standard'' 11 May 1871, Thursday: 3 [of 8], Cols. 3b–4c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000183/18710511/014/0003. Print title: ''The Standard'', same p.&lt;/ref&gt;
&lt;/blockquote&gt;

==== 24 May 1871, Wednesday: Derby Day ====
Baron Rothschild's Favonius won. The Prince of Wales attended.

==== 25 May 1871, Thursday, Dinner Party Hosted by Mr. and Mrs. Charltons ====
&lt;blockquote&gt;Mr. and Mrs. Charlton, of Hesleyside, entertained at dinner, on Thursday evening, at 47, Princesgate — his Excellency the Spanish Minister, Count de Beaufort Spontin, Lord and Lady Houghton and the Hon. Miss Milnes, Lord and Lady Acton, the Hon. Lady Williamson, [[Social Victorians/People/Abercorn|Lady Caroline Howard]], Mrs. and Miss Milner Gibson, Viscount Burke, Lord Beaumont, Lord Campbell, the Master of Herries, Major Fife, &amp;c.&lt;ref&gt;&quot;Fashionable World.&quot; ''Morning Post'' 27 May 1871, Saturday: 5 [of 8], Col. 6c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18710527/019/0005. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;June
July
August
September
===October===
'''October 1871'''&lt;blockquote&gt;At Londesborough Lodge near Scarborough, where Lady Londesborough gave a royal house party in October 1871, not only [ 41/42 ] were the bathrooms few but the drains seeped into the drinking water. Several guests, including the Prince [of Wales] and his groom and Lord Chesterfield, contracted typhoid fever. When Chesterfield and the groom died, the doctors abandoned hope for the Prince.&lt;ref name=&quot;:1&quot;&gt;Leslie, Anita. ''The Marlborough House Set''. New York: Doubleday, 1973. Print.&lt;/ref&gt;{{rp|41–42}}&lt;/blockquote&gt;

The Prince of Wales recovered on 14 December 1871.

November
December

==1872==
January
February
March
April
===May===
'''29 May 1872, Wednesday''': Derby Day

June
July
===August===
'''August 1872''': The &quot;dance on the cruiser Ariadne&quot; probably occurred in August 1872:&lt;blockquote&gt;When his [the Prince of Wales'] brother, the Duke of Edinburgh, married the attractive Grand Duchess Marie, daughter of Tsar Alexander II of Russia, her family made a fuss because she was not granted precedence above the Princess of Wales. Albert Edward soothed ruffled feelings by inviting the Tsarevitch and his wife Marie Feodorovna (who was Alexandra's sister) to stay for two months and be entertained at Cowes. ...&lt;p&gt;&lt;/p&gt;

... At the dance on the cruiser Ariadne which the Prince gave in honour of the Tsarevitch and his Grand Duchess,&quot; Lord Randolph Churchill met the 19-year-old &quot;Miss Jennie Jerome of New York.&quot;&lt;ref name=&quot;:1&quot; /&gt;{{rp|42–43}}&lt;/blockquote&gt;

September
October
November
December

==1873==

=== January ===

==== 13 January 1873, Monday ====

==== Ball at the Chief Secretary's Lodge ====
On Tuesday, 14 January 1873, the Dublin Evening Telegraph reported that the Marquis of Hartington's ball had taken place the evening before.&lt;blockquote&gt;The Marquis of Hartington gave a ball last evening at the Chief Secretary's Lodge, to their Excellencies the Lord Lieutenant and the Countess Spencer, who were accompanied by the Dowager Countess Spencer, the Ladies Sarah and Victoria Spencer and the Hon Robert Spencer, Lord and Lady Charles Bruce, and Major Stirling, A D C.&lt;p&gt;

The following had the honour of receiving invitations to meet their Excellencies — The Duke of Leinster, the Marquis and Marchioness of Kildare, the Ladies Fitzgerald, the Marquis and Marchioness of Drogheda, the Earl and Countess of Listowel, Lord and Lady Edward Cavendish, the Earl of Charleville, the Lord Chancellor and Lady O'Hagan, Viscount, Viscountess, the Hon Misses, and Hon Henry Monck; the Archbishop of Dublin, the Hon Mrs and the Misses Trench; Lord Talbot de Malahide and the Hon Francis Talbot, Lord and Lady Sandhurst and Captain Bang, A D C; Lady Cloncurry, Hon Emily and Hon Mary Lawless, Viscount, Viscountess, Hon Georgiana, and Hon Beatrice [de?] Vesci; Lord and Lady Kilmaize [?], Hon Gertrude [?] Browze, Lord and Lady Ventry, Hon Norah Westenra, Lord and Lady Athlumney, Lord, Lady, and Hon D Plunket, M P; Viscountess and the Hon. Miss Netterivlle, Capt the Hon Mrs Vesey, Captain and Lady Julia Follett, Sir Arthur and Lady Olive Guiness and the Ladies White, the Hon H W L Corry, Lord and Lady and the Hon Miss O'Neill, Viscount Hawarden, the Hon Florence Maude, the Hon. Clementina Maude, the Hon Jenico and Mrs Preston, the Hon Henry Leeson, Colonel and the Hon Mrs Caulfield, Mr and the Hon Mrs Robert Hobart, Captain, Lady Mary and Miss Lindsay; Mr Ion [?] Trent Hamilton, M P; Mr Bagwell; the Hon Mrs and the Misses Bagwell, and Mr Bagwell; Colonel the Hon L and Mrs Curzon Smyth, Mr, Lady Margaret, and the Misses Stronge [?]; Mr and the Hon Mrs O'Hagan, Hon Charles Bourke, Hon Mrs Alfred and Lady Kathleen Bury, [[Social Victorians/People/Abercorn|Hon Mrs, Lady Alice, and Lady Louisa Howard]]; Captain, the Hon Mrs, and Miss Donaldson; Dr and Miss Bans, Mrs Grattan Bellew, Sir Edward and Miss Borough, Mr Arthur Cane, Sir Dominic, Lady, and Miss Corrigan; Mr Corrigan, Mr and Mrs Gustavus Cornwall and Miss Cornwall, Mr D'Arcy, M P, and Mrs D'Arcy; Mr Baron Dowse [?], and Mrs and Miss Dowse, Mr Baron Deasy and Mrs Deasy, Dr, Mrs, and Miss de Ricci; Dr and Miss Hatchell, Sir George and Lady Hudson, Mr, Mrs, and the Misses Huband; Mr Arthur Huband, Miss Caroline Huband, Mr and Mrs Arthur Hume, Dr Hughes, Mr Henry Jephsen and Miss Jephsen, Mr Kearney and the Misses Kearney, Captain Kearney, A D C; Captain Lascelles, A D C; Mr, Mrs, and Miss Kirwan; Mr Justice Lawson and Mrs Lawson, Mr and Mrs W Le Fanu, Mr, Mrs, and Miss Lentaigne; Sir George L'Estrange and the Misses L'Estrange, the Lord and Lady Mayoress, and the Misses Mackey; the Lord Chief Justice Monahan, Mrs and Miss Monahan; Sir J, Lady, and Miss Power; Mr John Talbot Power, M P; Col, Mrs, and Miss Radcliffe; the Master of the Rolls, Mrs and Miss Sullivan; Capt and Mrs Moorsom, A D C; General Sir Thomas and Lady Steel, Captain and Mrs Brownrigg, A D C, Mr Granville Milner, Capt, Mrs and Miss Talbot, Colonel, Mrs, and the Misses White; Sir John Stewart Wood, Lady and the Misses Wood; Mrs and the Misses Williams, Mr Justice Fitzgerald and the Hon Mrs Fitzgerald, Mr Fitzgerald, Mr Justice Barry and Mrs Barry, Mr Sergeant Sherlock, M P, Mrs and Miss Sherlock; Mr Sheriock, the Right Hon W H Conan, M P, and Mrs Cogan; Mr Justice Keogh and Mrs Keogh, Mr Keogh, Capt Keogh, R N; Lord Chief Baron and Miss Pigott, Dr, Mrs, and Miss Nugent; General Wardlaw, Colonel M'Kerlie, Mr Sergeant and Mrs and Miss Armstrong; Col, Mrs, and the Misses Maude; Col, Mrs, and Miss Hillier; Mr Heron, M P; Mr and Mrs Watters, Col and Mrs Wynyard, Dr and the Misses Kennedy, the Attorney General and Mrs Palles, the Solicitor General and Mrs Law, Col, Mrs, and Miss Lake; Lady and the Misses Butler, Mr Butler, Col and Mrs Colthurst Vesey, and Miss Walton; Mr, Lady Fanny and Miss Lambert; Mr E C Guinness, Mr and Mrs MMorer O'Ferrall, Mr and Mrs Leonard Morrogh, Sir Bernard and Lady Burke, Mr G and Mrs G Brooke and Miss Brooke, Mr and Mrs Roe, Mr Vance, M P, Mrs and Miss Vance; Col and Mrs Primrose, Lieut Col Ferdall [?], Col and Mrs Goodlake and Miss Alexander, Mr Alison, Mr, Mrs, and Miss Barton, Mr Justice Flanagan, Mrs and Miss Flanagan, Mer J. N. Lentaigne, Mr Johnson, Captain Harrison, Mr, Mrs, and the Misses Maturin; Mr Justice Morris and Mrs Morris, Mr and Mrs Mazlere [?] Brady, Major, Mrs, and Miss Wilkinson; Mr, Mrs, and Miss Donnelly; Mr and Mrs Cruise, Mrs Power, Mr Braon Fitzgerald and Mrs Fitzgerald, Mr Henry Yates Thompson, Mr Courtenay Boyle, Colonel Forster, Mr, Mrs, and Miss Taylor, Mr Bland and Mrs Godfrey Bland, Mr and Miss Dillon, Mr and Mrs Wallace, Mr M'Kenna, Mr Cullinane, Mr Armstrong, Mr C E [?] Dobbin, Mr J A Blake, Major and Mrs Papillon, Capt and Mrs Keane, Mr E Pretty, Mr, Mrs John L O Ferrall and Miss O'Ferrall, Mrs and Miss Walsh, Mr and Mrs R Howard Brook, Mrs and Miss Brook, Mrs and the Misses Blake, Mr and Mrs J Warren, Sir John Gray, M P, Lady, and Miss Gray; Colonel and Mrs Frank Chaplin, Mr, Mrs, and Miss Hemphill; Sir R, Lady and Miss Kane, Mrs and Miss Courtenay, Mr Arthur Courtenay, Mr G Courtenay, Mr E Hardtop, A D C; Mr Bellew, Dr and Mrs Nedley, Dr and Mrs Newell, Mr and Mrs Freeman, Mr and Mrs Geale, Captain Hutten, A D C; Mr and Mrs Adair and Miss Wadsworth, Captain and Mrs J M Benthall, Sir R, Lady, and the Misses M'Causlend [?]; Mr, Mrs, and the Misses Newell Barron; Mr Hawkins, Colonel Goodlake and the Officers of the Coldstream Guards; Captain Spain, R N, and the Officers (4) of her Majesty's ship Vanguard; Colonel Radcliffe and Officers (4), Royal Artillery; Colonel Spade and Officers (4) 1st King's Dragoon Guards; Colonel Ainslie and Officers (4), 1st Royal Dragoons; Colonel Thompson and Officers (4), 14th Hussars; Colonel Ross and Officers (4), 4th Battalion Rifle Brigade; Colonel Hawkins and Officers (4), Royal Engineers; Colonel Gloster and Officers (4), 97th Regiment; Lieutenant-Colonel Maunsell and Officers (4), 13th Regiment.&lt;ref&gt;&quot;Fashionable.&quot; ''Dublin Evening Telegraph'' 14 January 1873, Tuesday: 4 [of 4], Col. 7a–b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002093/18730114/044/0004. Print title ''The Evening Telegraph'', n.p.&lt;/ref&gt; &lt;/blockquote&gt;

==== 29 January 1873, Wednesday ====

==== Drawingroom at Dublin Castle ====
The women listed in the 2nd paragraph, about the members of the Household who were present, were listed as accompanying their husband or father, not as working members of the Household.&lt;blockquote&gt;DRWNINGROOM [sic] AT DUBLIN CASTLE.

His Excellency the Lord Lieutenant and the Countess Spencer held the first Drawingroom for the season at Dublin Castle on Wednesday evening.

Shortly after nine o’clock their Excellencies entered the Throne Room, attended by the following members of the Household:— The under Secretary — Thomas H. Burke, Esq. The Private Secretary — Henry Y. Thompson, Esq; Miss Thompson. The State Steward — Colonel the Hon. Luke White. Comptroller — Lieutenant-Colonel Caulfield; Hon. Mrs. Caulfield. Gentleman Usher — Major the Hon. E. Boyle; Hon. Mrs E. Boyle. Chamberlain — Hon. H. Leeson. Master of the Horse — Lieutenant-Colonel Forster. The Gentleman in Waiting — Lieutenant-Colonel J. M'Donnell and Hon. Mrs. M‘Donnell. The Gentlemen at Large — Lowery Balfour, Esq, Captain Donaldson, and Hon. Mr. Donaldson. Aides-de-Camp — Major Sterling, Lieutenant the Hon. V. Lyttelton, Captain Lascelles, Captain Bridges, Capt. F. Seymour, Captain Kearney, Captain Chaplain, V. C; Lieutenant Hartopp, Lieutenant Wynne Finch, Lieutenant A. Egerton, Captain Hutton, Captain Wood.

The Physician in Ordinary — Thomas Nedley, Esq, M.D. The Surgeon in Ordinary — George Hatchell, Esq., M.D., and Miss Hatohell. The Surgeon to the Household — James S. Hughes, Esq. MD. Her Excellency’s Pages of Honour — Hon. J. Somerville, and Mr. Charles White.

There was very large company present among them being the Lord Mayor and the Lady Mayoress; [sic]

The Lord Chancellor and Lady O’Hagan. The Lord Chief Justice, and Mrs. Whithside. The Lord Chief Baron and Mrs. Pigot, the Attorney-General and Mrs. Palles, the Solicitor-General and Mrs. Law. Major-General Sir Thomas Steele, K.C.B., and Lady Steele (presented.) Captain Brownrigg, A.D.C., and Mrs. Studholm Brownrigg. Colonel Primrose, C.S.I., Deputy Adjutant-General. Colonel the Hon. Leicester Smith, C.B., Deputy Quartermaster-General, and the Hon. Mrs. Leicester Smith. Mr. Porter, Surgeon in Ordinary to the Qneen in Ireland, and Mrs. Porter. Marquis and Marchioness of Kildare, Lady Alice Fitzgerald, and Lady Eva Fitzgerald. Marquis of Headfort, Lady Adelaide Taylour, Lady Florence Taylour. Marquis of Drogheda and Marchioness of Drogheda. Earl and Countess of Shannon, Earl of Kenmare, Countess of Charlemont, Anna Countess of Kingston, Dowager Countess Spencer and Lady Victoria Spencer, Viscount and the Viscountess Monck, and the Hon. Frances Monck, Viscountess Gormanstnwn, Viscountess Netterville, Lord Talbot de Malahide and Hon. Frances Talbot, Lord and Lady Lisgar, Lord Crofton, Lord and Lady Plunket, Lady Sandhurst, Lady Athlumney, Lady Hastings, Lady Cloncurry, Lady Colthurst, Lady Louisa Tenison and Lieutenant-Colonel Tenison, Lady Barbara Chetwynd Stapylton, [[Social Victorians/People/Abercorn|Lady Louisa Howard]], [[Social Victorians/People/Abercorn|Lady Caroline Howard]], Lady Julia Follett and Captain Follett, Lady Georgina Croker, Lady Catherine Bury, Lady Steward Wood, ['''Col. 3c–4a'''] Miss Stewart Wood, and Miss Elvyn Stewart Wood, The Right Hon. J. D. Fitzgerald and the Hon. Mrs. Fitzgerald, the Right Hon. Mr. Justice Morris, the Right Hon. Mr. Justice Barry, and Mrs. Barry, the Right Hon. Baron Dowse, Mrs. Dowse, and Miss Dowse, Judge Woulfe Flanagan, Mrs. and Miss Woulfe Flanagan.

The Provost of Trinity College and Mrs. Lloyd, the Moderator of the General Assembly.

Colonfel Frederick Maude, V.C., C.B., Deputy Inspector General of Auxiliary Forces; Mrs. Frederick Maude, and Miss Ada Cecil Maude (presented). Colonel Lake, C. B. Commissioner of Police, and Miss Lake.

LADIES’ DRESSES.

Her Excellency the Countess Spencer — Train and corsage of rich Lyons peon velvet, lined poult de foie, trimmed bouillones of tulle illusion to match, nœuds of satin and plumes of peacock, and ostrich feathers, same shade; corsage, Raphael, trimmed band of peon velvet, beautifully embroidered in self colours, plumes of ostrich and peon to correspond; petticoat of richest satin antique, with jupes of tulle, beautifully trimmed three broad plisses, with plumes of peacock's tail, headed with shells of velvet all to match train in colour; at sides and backs stoles and broad sashes of peon velvet, beautifully embroidered in self colour; across body of dress was band of velvet, worn like sash; studded with the most magnificent brilliants. Headdress a tiara of diamonds and peon plume; ornaments, diamonds.

The Lady Mayoress, Mansion House — Train and corsage of richest black satin raye, lined blue glace, and trimmed plisses of blue poult de soie; corsage, trimmed a draperie of tulle, with fall of very fine Irish point lace; petticoat of rich blue poult de joie, with volants of Irish point lace, and tulle plaitings, headed blue satin. Head-dress, coart plume, Irish point lace; ornaments, diamonds.

Hon. Mrs. Caulfield, Dublin Castle — Train and corsage of the richest black gros de Suez, lined black taffeta, tastefully trimmed; bouillones of tulle and silver wheat; corsage, trimmed a draperie of tulle, silver wheat, and silver bullion fringe, with a fall fine Brussels point; petticoat of rich black glace under jupe of chantilly; trimmed tablier tulle and satin shells, tunic to correspond, looped black velvet bows, and bouquets of silver wheat. Headdress, court plume, point lappets and diamonds; ornaments, diamonds.

Mrs. Whiteside, Mountjoy-square — Train and corsage of rich pink satin antique, lined with white Florence, beautifully trimmed with bias and nœuds of satin, and a volant of very fine Brussels point; corsage, trimmed draperie of tulle and satin, with fall point lace; petticoat of white satin antique, with jupe of Alencon tulle, tulle plaitings edged with folds of pink satin, and volant fine Brussels point. Headdress.

Lady Butler, Ballintemple, county Carlow — Train and corsage of richest white satin, trimmed bouillones, and pouffs of white tulle de chene, festooned with bouquets of pink laburnum, set rosettes of white tulle de chene; petticoat of white Bruxelles net, trimmed with roulleax of white satin, and bouilloned the waist en pompadour. Headdress, court plume, lappets, and feathers; ornaments, diamonds and pearls.

Miss Wynn, Wynstay, Roebuck — Train and corsage of rouleaux satin, trimmed with pouffs and bouillones of white tulle de chene, and edged with richest blonde lace; petticoat lavender glace, trimmed with rings and frillings of tulle de chene and rich flounce of blonde lace. Headdress, Court plume and lappets ; ornaments, tiara of diamonds.

The Countess of Shannon, Castlemartyr, county Cork — Train of richest white satin, lined marceline, &amp;c., trimmed with white tulle, studded with pearls, and volantes of real Brussels lace; jupe of richest white satin, with tunic of finest real Brussels lace, looped up with chatelaine of pink roses; corsage, a la gracque trimmed en suite. Headdress, plumes of feathers with lappets ; ornaments, diamonds.

Mrs. Murphy, Mount Loftus — Train and corsage of rich mauve gros grain, lined with white satin, and trimmed with Carrickmacross lace and bias folds of silk; petticoat of mauve glace, with mauve tulle, jupe, trimmed en tablier with Carrickmacross lace, and flounce and buillons of tulle. Headdress — Lappets, feathers, and tiara of diamonds. Ornaments, pearls and diamonds.

Mrs. Maxwell, Cruiserath, Clonsilla — Train and corsage of rich ruby velvet, lined with rich white silk, and trimmed with Brussels lace, centre of train trimmed with bows of moire ribbon, the train looped at the side with an echarpe of wide ribbon; corsage to correspond; of rich gros de Suez silk, trimmed with white Brussels lace, flounces headed with ruche of green tulle illusion, studded with green flowers, front trimmed en tablier. Headdeess [sic] — Court plume, Brussels lace lappets, and diadem of diamonds. 

Miss Pigot, 15, Merrion-square, East — Train with pouffe of magnificent black silk, lined with white marcelline, beautifully trimmed with broad bias of lavender satin, ruching of lavender net and Spanish blonde; sash of lavender satin, fastening side under pouffe; corsage, Louis Quinze; petticoat of white poult de soie, with overskirt of white Brussels net, trimmed en tablier, with platings of lavender net and satin, fastening at side, with nœuds of lavender satin. Coiffure — Court plume and tulle veil. Ornaments — Diamonds.

Miss Jackson, Ahanesk, Midleton, Co. Cork — Presentation train, with pouffe and sash of richest white faye silk, lined with marcelline, tastefully trimmed with fluffed plaiting of white silk and satin; corsage, Pompadour style, trimmed with white satin and tulle; jupon of white poult de soie, with overskirt of white tulle, trimmed with alternate plaitings of tulle and white satin. Coiffure — Court plume and tulle veil. Ornaments — Diamonds and pearls.

Mrs. Safford, 97th Regiment — Train and corsage of rich maize satin, lined, richly trimmed with tulle ruche, true-lover’s knots, and nœuds de velour noir, from agrafe; corsage, garnier richment de danlette ancienne; jupe, tulle, maize ruche, richly trimmed to match train. Headdress — Ostrich feather and tulle lappets. Ornaments — Diamonds and pearls.

Miss Mackey—Train and corsage of the richest maize poult de soi, lined with Florence silk, and elegantly trimmed with bouffants of tulle, illusion, and guirlands of cherita leaves; corsage trimmed to correspond; jupe of white tarlatane buillonee and wreaths of cherita leaves. Coiffure — Maize feather and long tulle veil. Ornaments — Silver.&lt;ref&gt;&quot;Drawingroom at Dublin Castle.&quot; ''Cork Constitution'' 31 January 1873, Friday: 3 [of 4], Col. 3c–4b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001646/18730131/056/0003. Print title: ''The Constitution; Or Cork Advertiser'', n.p.&lt;/ref&gt;&lt;/blockquote&gt;February
March
April

===May===
'''28 May 1873, Wednesday''': Derby Day

=== June ===

==== 19 June 1873, Thursday, Polo Match Between Officers of the Royal Horse Guards and Officers of the 9th Lancers ====
&lt;blockquote&gt;THE POLO CLUB.

Although the weather was dull and gloomy yesterday, there was a large company at the club grounds to witness the match between the officers of the Royal Horse Guards (Blue) and the officers of the 9th Lancers. A number of carriages surrounded the enclosure, and many ladies were present, among whom were the Marchioness of Waterford, Viscountess Middelton, Lady Philippa Stanhope, the Countess of Mayo, the Hon. Miss Brodrick, Lady Little, [[Social Victorians/People/Abercorn|Lady Louisa Howard]], [[Social Victorians/People/Abercorn|Lady Caroline Howard]], Lady Harriet Duncombe, Miss Duncoinbe and Miss E. Duncombe, the Hon. Mrs. O'Grady and Miss O'Grady, Lady Knollys and Miss Knollys, the Dowager Lady Craven, Lady Grey de Wilton, Lady Fanny Fitzwigram, Lady Petre, Lady M. Egerton, Misses E. and G. Egerton, the Countess of Gleichen, Lady C. Brineman, Lady Campbell, Lady Emily Ormsby Gore, the Countess of Coventry, Lady Maria Ponsonby, and Lady Henry Somerset. Just before 4 o'clock the competitors took up their stations at the goals, the Hon. H. Boscawen and Sir Beach Cunard being the judges. The Guards, having choice of stations, elected to play from the Pavilion goal, although there was a strong wind blowing against them. Play was called for the first &quot;bully,&quot; and when the ball was tossed into the centre of the ground the advanced guard of both sides missed their blows; and, this brought the others close up, and after some spirited hitting the Guards got the ball nearly to the bottom goal, where it was knocked out of bounds three or four times. Each time it was returned into play some severe rallies ensued, and the scientific hitting and stopping of the Marquis of Worcester, the Hon. C. W. Fitzwilliam, and Lord Kilmarnock met with loud applause, while the play of the whole of the Lancers was so determined and vigorous that the Guards could not break through their defence, but in a good ''mêlée'' [sic] close to the goal the ball was hit just outside the bottom posts. They then had a rest, and the ponies were attended to and carefully watered, and when the ball was hit off the Lancers, playing well together, drove the ball nearly to the top goal, but just missed getting it through the post. The rain now came down and made the turf heavy and slippery, and the play was rather wild, many well-intended hits being lost by the little &quot;tits&quot; slipping when turning sharply at their best speed. Both sides were doing their utmost to obtain the honours; but, although the ball was sent to all parts of the enclosure, and rally after rally came off, each goal being assaulted in its turn, no goal was made. The Guards now got the ball to the bottom end of the ground, and the Marquis of Worcester made a fine drive for victory; the ball, however, did not quite reach the goal, but his Lordship was well backed up by the Hon. C. Fitzwilliam, who, in the midst of a rattling ''mélée'' [sic] close on the posts, cleverly &quot;pushed&quot; the ball through the goal, and scored the first to the Guards, after playing lh. 20min., being the longest time that as [sic] occurred this season. After a rest and a change of ponies the second &quot;bully&quot; was commenced, but, after a short time, during which some fine play was exhibited by both sides, &quot;time&quot; was called by the judges, and the Guards won the game by one goal. Appended will be found the sides:
{| class=&quot;wikitable&quot;
|+
!The Royal Horse Guards
!The Lancers
|-
|Marquis of Worcester,
|Capt. Grissell.
|-
|Lord C. Somerset.
|Lord W. Beresford.
|-
|Hon. C. W. Fitzwilliam.
|Mr. Moore.
|-
|Mr. Egerton.
|Capt. Polaret.
|-
|Lord Kilmarnock.
|Hon. E. Willoughby.
|}
Sides were then chosen by Viscount amentia and Mr. C. de Murrietta, and after some exciting play a goal was got by each.
{| class=&quot;wikitable&quot;
|+Sides
|Lord Valentia.
|Mr. C. de Murietta
|-
|Capt. Middelton.
|Marquis of Queensberry.
|-
|Hon. H. C. Needham.
|Sir Beach Cunard.
|-
|Mr. Green.
|Sir W. Gordon Cumming.
|-
|Hon. R. Neville-Nugent.
|Hon. C. W. Fitzwilliam.
|-
|Mr. A. de Murietta.
|Lord Aberdour.
|-
|
|Mr. Powell.
|}
&lt;ref&gt;&quot;The Polo Club.&quot; ''Hour'' 20 June 1873, Friday: 7 [of 8], Col. 6a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002814/18730620/078/0007. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

July
August
September

=== October ===

==== 18 October 1873, Saturday, Orange Order Events at Govan ====
This festival seems to have included some speeches and the laying of a foundation stone for an Orange Hall. The speeches were extremely anti-Catholic and bigoted.&lt;blockquote&gt;ORANGE FESTIVAL AT GOVAN.

The third annual festival of the Govan Orangemen and their friends was held in the Govan Hall on Friday night — Br. H. A. Long [?] in the chair. After a service of tea and sake,

The C&lt;small&gt;HAIRMAN&lt;/small&gt; delivered an address, in which he stated, after a few preliminary remarks, that Orangeism had to be looked at from two points of view — one political and the other religious. The political one looked at the Pope and grasped the sword, while the other looked at Christ and opened its arms. One of them was for offence — that was fighting against Popery in all its varied forms, while the other was for the adoption and union of the great system of thrice-blessed Christianity. He congratulated them on living in comparatively happy days, and seeing the complete destruction of the Court of Rome and the Pope's temporal power. Not many years ago, he said, diplomatists came from all parts of the world to the Quirinal or the Vatican, but all that had now passed away, and not left a shadow behind. The chairmen then reviewed at some length the events of Italian history since 1846, and the great contrast in the treatment of priests in Rome at that time and at the present day. It must have been a bitter pill, he went on to say, for the Vatican to swallow when they heard the shouts of triumph of 25,000 Romans rejoicing that they had got free from priestly influence. Mr. Long next referred to the late visit of Victor Emmanuel to the Emperors of Austria and Germany, which he is garded as a pledge of defence against the French nation's interference in Italian affairs. The chairman referred to the immense treasures stored in the Vatican, amounting to eight hundred millions of sovereigns, and to the cramping of the power of the priesthood in Germany by Bismarck[.] The Rev. C. A. M'Kenzie, after apologising for not having any text, gave an interesting sketch of the connection of the North of Ireland with the Western Highlands of Scotland, from the middle of the sixth century, when St. Columba crossed over with his twelve followers, till the perversion of the early Culdee Church by the wife of Malcolm Canmore and her son King David. Popery, he asserted, was an invasion of comparatively recent origin, and the Roman Catholics had no right to the ancient abbeys, to which they seemed inclined to lay claim. In conclusion, he urged upon them, as good Orange-men and followers of the famous King William, of glorious memory, who inscribed on his banner &quot;the liberties of England and the Protestant religion,&quot; never to forget that noble man; and to beware of Puseyism, which was only Popery in disguise. The meeting was afterwards addressed by Mr. Martin, and the proceedings were enlivened with songs by a number of the brethren and their lady friends. After the soiree an assembly took place, and dowering was kept up till an early hour.— ''Glasgow News''.

N&lt;small&gt;EW&lt;/small&gt; O&lt;small&gt;RANGE&lt;/small&gt; H&lt;small&gt;ALL&lt;/small&gt;. — The foundation stone of Staffordstown [?] Orange Hall has been laid by Lady Louisa O'Neill, in presence of Lady O'Neill, [[Social Victorians/People/Abercorn|Lady Caroline Howard]], the Hon. Edward O'Neill, and a large assemblage of Orangemen. After the ceremony, the entire party adjourned to a field adjoining, where a platform had been erected. The lodges present were — Staffordstown L.O.L., 504 [?]; Ballydonnall L.O.L., 306 [?]; Tailorstown True Blues, 544; Grange L.OL., 701; Duneane [?] L.O L., 719; Grange L.O.L., 919; Cranfield L.O.L , 705 [?]; Fenton Invincibles, L.O.L., 1104; and the Fenton Invincibles (juveniles), L.O.L., 1104. Amongst those present on the platform were — Lady O'Neil, the Hon. Edward O'Neill, M.P.; the Hon. Louisa O'Neill, Lady Caroline Howard, William J. Gwynne, Esq.; Richard Lilburn, Esq.; J. J. Carson, Esq., Mrs. Carson, and Miss Carson; Rev. J. B. Greer, Rector of Grange; Rev. J. H. Wright, bector [sic] of Portglenone; Rev. A. Gault, Vicar of Antrim; Rev. William Denham, Presbyterian minister, Duncane; Wm. J. Scully, Esq.; Messrs. John Fulton, John M Kelvey, John Nimmons. W.D.M.; Wm. M'Cullough, Hugh Nicholl, Joshua Hume, James Brooks, Charles Richardson, Robert Chesney, Robert Barton, Wm. Allen, Alexander M'Fadden, Hugh Logan. D. S Beekerstaff, Glenavy District; George French, James M'Manus, John Hume Richardson, Wm. J. Senly. Mr. Gwynne was called to the chair, and the meeting having been opened with prayer, appropriate addresses were afterwards delivered by the chairman, the Hon. Edward O'Neill, the Rev. Mr. Wright, Mr. Lilburn, and the Rev. Mr. Greer. The chairman having made a few concluding remarks, the meeting separated after having given three hearty lowly cheers for Lady O'Neill and party.&lt;ref&gt;&quot;Orange Festival at Govan.&quot; ''Belfast Weekly Telegraph'' 18 October 1873, Saturday: 8 [of 8], Col. 3b–c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003434/18731018/044/0005. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;November
December

==1874==
January
February
March
April
===May===

==== 1874 May, Early ====
&lt;blockquote&gt;As monarchists’ hopes flared, the Catholic Church, too, enjoyed a conspicuous revival. The National Assembly approved a design for a new basilica for Paris. Intended as an act of collective atonement, Sacré-Coeur was to perch atop Montmartre, immediately above where Nadar’s balloons had been launched and where the radicals’ insurrection had broken out. Excavations began in early May 1874 ....

But the focus of the penance the basilica was intended to embody gradually shifted from the moral decline of French society in general to the despicable excesses of the Commune. In 1872 Archbishop Darboy’s successor claimed to have had a vision as he climbed the Butte Montmartre. The clouds dispersed, and he realized that it was there, “where the martyrs” were (he meant the murdered generals Lecomte and Clément-Thomas), that a new church should be built. And when the Assembly voted to proceed with the construction, legislators specified that its purpose was to “expiate the crimes of the Commune.”&lt;ref name=&quot;:3&quot; /&gt; (464 of 667)&lt;/blockquote&gt;

===June===
'''3 June 1874, Wednesday''': Derby Day

June
July
August
September

=== October ===
November
===December===
'''8 December 1874, Tuesday''': &quot;CHATSWORTH, Tuesday, December 8th, 1874. — We are come to the last slide of the Chatsworth magic lantern: the Duke of Cambridge and his equerry, a funny little man called Tyrwhitt, of no particular age, in a grey wig; Lord Carlingford and Ly. Waldegrave, the Spencers, Mr. Leveson, Cavendish.&quot;&lt;ref&gt;{{Cite web|url=http://ladylucycavendish.blogspot.com/2010/12/08dec1874-chatsworth-magic-lantern.html|title=Lady Lucy Cavendish: 08Dec1874, The Chatsworth Magic Lantern|last=H|first=Denise|date=2010-12-04|website=Lady Lucy Cavendish|access-date=2025-06-18}}&lt;/ref&gt;

==1875==
Disraeli's progressive legislation for labor rights:&lt;blockquote&gt;In 1875, he passed a series of enlightened acts protecting labor rights, arguing they were as important as property rights. Two of the laws ensured that workers would have the same recourse as employers when contracts were breached, and made peaceful picketing legal, protecting unions from charges of conspiracy.&lt;ref name=&quot;:4&quot; /&gt; (578 of 1203)&lt;/blockquote&gt;After women who owned property were allowed by Parliament to stand for local school-board elections in 1870, &quot;Elizabeth Garrett Anderson, the first woman to qualify as a doctor in Britain — in 1865 — stood and was elected to her local board five years later.&quot;&lt;ref name=&quot;:4&quot; /&gt; (199 of 1203)

The relationship between Swinburne and Lord Houghton:&lt;blockquote&gt;...not all Lord Houghton's children appreciated the catholicity of &quot;Papa's&quot; taste in friends: &quot;Swinburne (in a very excited state) came in in the evening,&quot; wrote Florence Milnes to her brother in 1875: &quot;He is madder than ever, to my astonishment he flopped down on one knee in front of me, &amp; announced that my hair had grown darker. This was rather embarrassing, and he is also so deaf now, which does not make it easier to talk to him.&quot;&lt;ref name=&quot;:2&quot;&gt;Pope-Hennessy Lord Crewe.&lt;/ref&gt;{{rp|5}}&lt;/blockquote&gt;

January
February
March
April
===May===
'''26 May 1875, Wednesday''': Derby Day. The Prince and Princess of Wales attended, as did a number of others of the royal family, including Princess Louise and Lorne.

June
July
===August===
'''August through October 1875''' Richard Monckton Milnes (Lord Houghton) and son Robert Milnes toured the U.S. and Canada:&lt;blockquote&gt;They set off in the steamer s.s Sarmatian from Liverpool in August 1875, stopping at Ireland to pick up the usual load of emigrants bound for the U.S.A. The most interesting among the passengers was 'Mr. Butler, author of Erewhon, who is very amusing and clever though infidel,' but, although he played whist with Samuel Butler, the young man was far more interested in the Eustace Smiths (parents of his friend W. H. Smith), and in a Canadian family named Macpherson, the youngest of whose two daughters, the dark-eyed Isobel, caught his fancy: he saw them afterwards in Toronto, and when they parted she gave him two larger than carte-de-visite photographs of herself, he gave her a smaller one of himself together with the inevitable volume of his father's verse.&quot;&lt;ref name=&quot;:2&quot; /&gt;{{rp|10}}&lt;/blockquote&gt;September
October
November
December

==1876==
Disraeli pushed through the Cruelty to Animals Act in order to please Queen Victoria. This act &quot;forced researchers to demonstrate that any experiments with animals involving pain were absolutely necessary, and ensured they would be anesthetized if so.&quot;&lt;ref name=&quot;:4&quot; /&gt; (679 of 1203)

January
February
March
April
===May===
'''11 May 1876''': In the midst of the Aylesford scandal, the Prince of Wales returned from a journey to Egypt and India, etc.:&lt;blockquote&gt;However harassed and exhausted, the Prince and Princess of Wales would put up a good show. Within an hour of their arrival home they set forth to attend a gala performance at Covent Garden Opera House. It was a brave decision to face the public and allow an immediate opportunity for demonstration. The Prince and Princess were rewarded when the audience rose to its feet to give them a standing ovation before the start of every act, as well as at the end, of Verdi's Ballo in Maschera.&lt;ref name=&quot;:1&quot; /&gt;{{rp|63}}&lt;/blockquote&gt;

'''27 May 1877''': Lily Langtry:&lt;blockquote&gt;Her big moment on May 27, 1877, when Sir Allen Young, the arctic explorer, invited her to late supper in his house, where it had been arranged that the Prince of Wales should meet her after the opera. The result was all that could have been expected. Mrs. Langtry became the Prince's first openly recognised mistress.&lt;ref name=&quot;:1&quot; /&gt;{{rp|69}}&lt;/blockquote&gt;'''31 May 1877, Wednesday''': Derby Day. The Prince and Princess of Wales did not attend, as he was ill.

June
July
August
September
October
November
December

==1877==
&quot;In 1877, unemployment was 4.7 percent; by 1879, it had risen to 11.4 percent.&quot;&lt;ref name=&quot;:4&quot; /&gt; (690 of 1203)

January
February
March
April
===May===
'''30 May 1877, Wednesday''': Derby Day.

June
July
August
September
October
November
===December===
'''15 December 1877'''&lt;blockquote&gt;On Dec. 15, 1877, the Queen honoured Lord Beaconsfield, the Premier, with a visit at Hughenden Manor. Her Majesty, accompanied by Princess Beatrice and attended by General Ponsonby and the Marchioness of Ely, left Windsor at 12.40 and proceeded by special train to High Wycombe, which was reached at 1.15. The Premier received the Queen at the station. A lofty triumphal arch spanned the entrance to the station-yard, and beneath this the royal party drove into the gaily decorated little town. The reception along the route was of the heartiest, and the drive of two miles to Hughenden was one long triumph. Lord Beaconsfield, who had preceded the party, welcomed the Queen at his own door. Lunch was served, and her Majesty remained about two hours. Before leaving she planted a memorial tree.&lt;ref&gt;&quot;The Queen's Glorious Reign.&quot; ''Illustrated London News'' (London, England), Saturday, May 27, 1899; pp. 757–765?; Issue 3136. Queen's Glorious Reign [Supplement]: 762?&lt;/ref&gt;&lt;/blockquote&gt;

==1878==
January
February
March
April
May
===June===
'''5 June 1878, Wednesday''': Derby Day.

July
August
September
October
===November===
'''8 November 1878''': from the journal of George, Duke of Cambridge:&lt;blockquote&gt;''November'' 8. — Gave farewell diner to the Lornes; Louise and Lorne, Augusta, Mary and Francis, Arthur, Leopold, Gleichens, J. Macdonald and self, and played at Nap afterwards. It was a good and nice little dinner.&quot;&lt;ref&gt;Sheppard, Edgar, Ed. ''George, Duke of Cambridge: A Memoir of His Private Life, Based on the Journals and Correspondence of His Royal Highness''. Vol. 2, 1871–1904. New York: Longmans, Green, 1906. http://books.google.com/books?id=dFoMAAAAYAAJ.&lt;/ref&gt;&lt;/blockquote&gt;December

==1879==
===January===
'''12 January 1879'''&lt;blockquote&gt;On 12 January 1879 Robert Milnes came of age, an event celebrated at Fryston by a tenants' ball.&lt;ref name=&quot;:2&quot; /&gt;{{rp|18}}&lt;/blockquote&gt;

'''28 January 1879''': Brett &quot;Harte kicked off his tour at the Crystal Palace in Sydenham on January 28, 1879.&quot;&lt;ref&gt;Nissen, Alex. ''Brett Harte: Prince and Pauper''. Jackson, MS: University Press of Mississippi, 2000.&lt;/ref&gt;{{rp|174}}

February
March

===April===
'''Early April 1879''' or so, probably, Bret Harte got &quot;an invitation to dine the same evening with Arthur Sullivan and the Prince of Wales&quot; as a dinner in Birmingham where Harte met T. Edgar Pemberton.&lt;ref&gt;Scharnhorst, Gary. ''Bret Harte: Opening the American Literary West''. Norman, OK: Univ. of Oklahoma Press, 2000.&lt;/ref&gt;{{rp|152}}
===May===
'''28 May 1879, Wednesday''': Derby Day; the Prince and Princess of Wales attended.

===June===
'''June 1879''', Robert Milnes became engaged to &quot;Sibyl Marcia, a daughter of a North-country baronet, Sir Frederick Graham of Netherby.&quot;&lt;ref name=&quot;:2&quot; /&gt;{{rp|18}} Parties must have followed.

July
August
September
October
November
===December===
'''28 December 1879''': The Tay Bridge Disaster: The Tay Bridge collapsed with a train on it. The weather was very bad, with gale-force winds and rain.

The ''Times'' reported that the average high temperature for the week ending December 31, 1879, was 53° F. and the low was 20° F.

In his column &quot;What the World Says&quot; in the 21 January 1880 World, Edmund Yates writes the following:&lt;blockquote&gt;How am I to describe better the magnificence of the Earl and Countess of Rosslyn’s ball at Euston Lodge last month, than by calling attention to the fact that M. Carlo, the eminent Knightsbridge coiffeur, arrived early in the day to crimp and powder the lacqueys? My informant adds, however, that the curled darlings were rather the worse for the festivities towards night. Was it not enough to turn their heads in every sense of the word?&lt;ref name=&quot;:0&quot;&gt;Edmund Yates, &quot;What the World Says,&quot; ''The World: A Journal for Men and Women''.&lt;/ref&gt;{{rp|21 Jan. 1880, p. 8, col. b.}}&lt;/blockquote&gt;

'''31 December 1879''': Edmund Yates, editor of The World: A Journal for Men and Women, in his column &quot;What the World Says,&quot; describes a private viewing at the Grosvenor Gallery:&lt;blockquote&gt;The private view at the Grosvenor on the last day of the year gave people something to do on a desperately wet afternoon. The artistic dresses were perhaps in greater force than ever; indeed the faces and the hair and the attitudes pursued me to my bed, and gave me many a nightmare. I suppose the plain woman of all time has had the ambition to be looked at: centuries of failure have at last been crowned with a real success. Besides the Cimabue Browns there was an interesting menagerie of real lions, artistic, literary, and clerical. The artists were numerous, and their host and hostess seemed to enjoy themselves very thoroughly.

Frequenters of the picture private views have a new sensation this winter. Last season they mobbed beauty: now hideously-attired unkempt dowdiness provokes the stare. The prize for the new style seems generally awarded to a rhubarb coloured flannel Ulster and a cart-wheel beaver hat, which pervaded both the private views last week. [2 private views last week, one at the Grosvenor]&lt;ref name=&quot;:0&quot; /&gt;{{rp|7 Jan. 1880, p. 9}}&lt;/blockquote&gt;

The official premiere of ''The Pirates of Penzance'' occurred in New York City on 31 December 1879 at the Fifth Avenue Theatre, to establish international copyright. Gilbert and Sullivan were there with the cast. The performance was a social event: attending were Mrs. Vanderbilt and Mrs. Astor.

==Works Cited==
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      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Applications */</comment>
      <origin>2818454</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9878" sha1="8xrg3m8am561uexbto0ag6rquznkx5m" xml:space="preserve">=== Introduction ===
* Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]])
* Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]])
* Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]])

=== Handling Repetition ===
* Control ([[Media:C02.Repeat1.Control.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat1.Control.1.B.20170918.pdf |B.pdf]], [[Media:C02.Repeat1.Control.1.C.20170926.pdf |C.pdf]])
* Loop ([[Media:C02.Repeat2.Loop.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat2.Loop.1.B.20170918.pdf |B.pdf]])

=== Handling a Big Work ===
* Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]])
* Functions &amp; Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]])
* Functions &amp; Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]])
* Functions &amp; Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]])

=== Handling Series of Data ===

==== Background ==== 
* Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]])
 
==== Basics ====
* Pointers ([[Media:C04.S1.Pointer.1A.20240524.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]])
* Arrays ([[Media:C04.S2.Array.1A.20240514.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]])
* Array Pointers ([[Media:C04.S3.ArrayPointer.1A.20240208.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20221130.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.1111.pdf |B.pdf]])
* Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]])

==== Examples ==== 
* Spreadsheet Example Programs
:: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]])
:: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]])
:: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]])
:: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]])

==== Applications ==== 
* Address-of and de-reference operators ([[Media:C04.SA0.PtrOperator.1A.20260717.pdf |A.pdf]])
* Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20241121.pdf |A.pdf]])
* Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240715.pdf |A.pdf]])
* Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.pdf |A.pdf]])
* Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]])
* Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]])
* Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]])

=== Handling Various Kinds of Data ===
* Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]])
* Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]])
* Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]])
* Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]])

=== Handling Low Level Operations ===
* Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]])
* Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]])
* Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]])
* Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]])

=== Declarations ===
* Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]])
* Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]])
* Scope 


=== Class Notes ===

* TOC ([[Media:TOC.20171007.pdf |TOC.pdf]])
* Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library
* Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements
* Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers
* Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts
* Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops
* Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control
* Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions
* Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope
* Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion
* Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions
* Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications
* Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions
* Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications
* Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) 
* Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2)
* Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO
* Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions
* Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications
* Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum
* Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List 
* Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing
* Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing


&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;
See also https://cprogramex.wordpress.com/

== '''Old Materials '''==

until 201201
* Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]])
* Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]])
* Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]])
* Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]])
* Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]])
* Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]])
* Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]])
* Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]])
* Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]])
* Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]])
* Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]])
* Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]])
* Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]])
 
&lt;br&gt;

until 201107
* Intro.1.A ([[Media:Intro.1.A.pdf |pdf]])
* Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]])
* Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]])
* Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]])
* Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]])
* Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]])
* Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]])
* Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]])
* Array.1.A ([[Media:Array.1.A.pdf |pdf]])
* Type.1.A ([[Media:Type.1.A.pdf |pdf]])
* Structure.1.A ([[Media:Structure.1.A.pdf |pdf]])


go to [ [[C programming in plain view]] ]

[[Category:C programming language]]

&lt;/br&gt;</text>
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  </page>
  <page>
    <title>User:Dc.samizdat/Golden chords of the 120-cell</title>
    <ns>2</ns>
    <id>326765</id>
    <revision>
      <id>2818467</id>
      <parentid>2818444</parentid>
      <timestamp>2026-07-18T00:26:20Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 24-cell */</comment>
      <origin>2818467</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="89055" sha1="bsmylqku4hcu94wcw40k2ani5cw6m8f" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. Their procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]

The &lt;math&gt;r_1&lt;/math&gt; chords of the 24-cell form a Petrie polygon {12/1} which zig-zags back and forth, in the left and right rotational directions, between two Clifford parallel great hexagons formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of the 24-cell form an ''edge polygon'' {24/4}=4{6}. The four great hexagons lie Clifford parallel to each other. A ''simple'' rotation of the 24-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

We can rotate the 24-cell isoclinically in Clifford parallel invariant planes containing 16-cell edges, for example in the characteristic rotation of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords.

We can also rotate the 24-cell isoclinically in Clifford parallel invariant planes containing 24-cell edges. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in any invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the invariant planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit, and its entire orbit traces an isocline circle in 4-space over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords. There are two chiral ways we can rotate the 24-cell isoclinically in invariant great hexagon planes containing its edges, called the ''great hexagon left rotation'' and the ''great hexagon right rotation,'' respectively.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords in the 24-cell]]
We can rotate the 24-cell isoclinically in 12 Clifford parallel invariant planes containing two &lt;math&gt;r_{1}&lt;/math&gt; edges each, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon left rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; 2{12/5} star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. The orbit of each vertex traces an isocline circle in 4-space over 12 &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. 

[[File:Regular_star_figure_8(3,1).svg|thumb|left|150px|{24/8}=8{3}&lt;small&gt; &lt;/small&gt;shows 8 of 32&lt;small&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; triangles in the 24-cell]]
We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing six &lt;math&gt;r_{2}&lt;/math&gt; edges each, over &lt;math&gt;r_{4}&lt;/math&gt; isocline chords. This is the  ''great hexagon right rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_4&lt;/math&gt; 8{3} star polygon which constructs &lt;math&gt;1/r_4&lt;/math&gt;. The rotational curve over each 120° &lt;math&gt;r_4&lt;/math&gt; chord makes four 30° turns.  Eight Clifford parallel triangle geodesic isoclines of circumference &lt;math&gt;2\pi&lt;/math&gt; over &lt;math&gt;r_4&lt;/math&gt; chords form a circular fibration of 8 twisted parallel strands {24/8}=8{3} that intersects each 24-cell vertex once. In three successive 60° isoclinic displacements each vertex circles a triangle and returns to its original position, but the 24-cell returns to its original orientation only after each vertex has completed circuits of the four distinct triangles which intersect at the vertex. The isocline curves over a self-intersecting dodecagram of 12 &lt;math&gt;r_4&lt;/math&gt; chords.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;6&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Edge chord
! colspan=&quot;3&quot; |Isocline chord
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|15°
|165°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|30°
|150°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|45°
|135°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|75°
|105°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: gainsboro;&quot; |
|1.414~
|1.414~
|- style=&quot;background: gainsboro;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we found three distinct isoclinic rotations. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these and also three other distinct isoclinic rotations.  

Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords that sum to 180°. The edge chords form the rotation's edge {24}-gon, and lie in invariant planes of the rotation. The isocline chords form the rotation's Clifford {24}-gon and lie in the invariant planes completely orthogonal to the edge planes. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°.  

We can rotate the 24-cell isoclinically in Clifford parallel invariant planes containing 16-cell edges in 6 Clifford parallel invariant great square planes containing four &lt;math&gt;t_{6}&lt;/math&gt; edges each, over &lt;math&gt;t_{6}&lt;/math&gt; isocline chords. The &lt;math&gt;t_6&lt;/math&gt; chord is the 16-cell-&lt;math&gt;r_2&lt;/math&gt; chord. The edge polygon and the Clifford polygon are both {24/6}=6{4}. This is the ''characteristic right rotation of the 24-cell''.  

The rotational curve over each 90° &lt;math&gt;t_6&lt;/math&gt; chord makes six 15° turns.  Six Clifford parallel skew triangle geodesic isoclines of circumference &lt;math&gt;2\pi&lt;/math&gt; over &lt;math&gt;t_6&lt;/math&gt; chords form a circular fibration of six twisted parallel strands that intersects each 24-cell vertex once. &lt;s&gt;In every 360° of isoclinic rotation each vertex circles a skew great square and returns to its original position, but the 24-cell returns to its original orientation only after each vertex has completed circuits of the three distinct skew squares which intersect at the vertex and the three distinct skew squares which intersect at its antipodal vertex. The isocline curves over a self-intersecting {24}-gon of &lt;math&gt;t_6&lt;/math&gt; chords.&lt;/s&gt;  

...  

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short edge chord
! Section
! colspan=&quot;3&quot; |Long isocline chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|0°
|180°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|12°
|168°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|24°
|156°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|36°
|144°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|48°
|132°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|72°
|108°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|84°
|96°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell,  with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== The 5-cell 4-simplex ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|138.6~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|130.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|124°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|113.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|110.2~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|98.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|95.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

{{Clear}}
== Finally the 120-cell ==
The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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  <page>
    <title>Athena problem</title>
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        <username>Athene241</username>
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      <text bytes="52522" sha1="c3pohhblgo3t6c4suwso9q6pedz58up" xml:space="preserve">{{mathematics}}
'''Athena problem''' is an [[:w:List of unsolved problems in mathematics|unsolved problem]] in [[:w:Number theory|number theory]] and [[:w:Formal language theory|formal language theory]] and [[:w:Order theory|order theory]], this problem is named after the ancient Greek goddess [[:w:Athena|Athena]] (which is associated with [[:w:Wisdom|wisdom]]). Athena problem is: Give a [[:w:Natural number|natural number]] ''b'' &gt; 1, find the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the set of the &quot;[[:w:Prime number|prime number]] [[:w:Greater than|&gt;]] ''b''&quot; [[:w:Numerical digit|digit]] [[:w:String (computer science)|string]]s in the [[:w:Positional numeral system|positional numeral system]] with [[:w:Radix|base]] ''b'' for the [[:w:Subsequence|subsequence]] [[:w:Partially ordered set|ordering]]. (A string ''x'' is a subsequence of another string ''y'', if ''x'' can be obtained from ''y'' by deleting zero or more of the [[:w:Character (computing)|character]]s in ''y''. For example, 514 is a subsequence of 352148, &quot;string&quot; is a subsequence of &quot;meistersinger&quot;. In contrast, 758 is not a subsequence of 378259, &quot;abc&quot; is not a subsequence of &quot;cbacacba&quot;, since the characters must be in the same order) (Unlike [[:w:Substring|substring]], subsequence is not required to occupy consecutive positions within the original sequences, e.g. the [[:w:Longest common subsequence|longest common subsequence problem]] is different from the [[:w:Longest common substring|longest common substring problem]])

Using [[:w:Formal language theory|formal language theory]] terminology, Athena problem is finding the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the [[:w:Formal language|language]] of base-''b'' [[:w:Representation (mathematics)|representation]]s of the [[:w:Prime number|prime number]]s [[:w:Greater than|&gt;]] ''b'' (which is a set of [[:w:String (computer science)|string]]s of [[:w:Symbol|symbol]]s over the [[:w:Alphabet (formal languages)|alphabet]] ''Σ''&lt;sub&gt;''b''&lt;/sub&gt; := {0, 1, ..., ''b''−1}), under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), for a given natural number ''b'' &gt; 1. (You can draw this partial ordering as a [[:w:Hasse diagram|Hasse diagram]] to find all [[:w:Minimal element|minimal element]]s)

By [[:w:Higman's lemma|Higman's lemma]], there are no [[:w:Infinite set|infinite]] [[:w:Antichain|antichain]]s for the subsequence ordering (i.e. the subsequence ordering is always a [[:w:Well-quasi-ordering|well quasi order]]) (i.e. under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), every set of pairwise incomparable (i.e. not [[:w:Comparability|comparable]]) strings is finite), thus there must be only finitely many such minimal elements. In other words, the set of such minimal elements must be a [[:w:Finite set|finite set]], e.g. in [[:w:Decimal|decimal]] (base ''b'' = 10), this set has exactly 77 [[:w:Element of a set|element]]s: {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}.

For bases 2 ≤ ''b'' ≤ 36, Athena problem is fully solved in bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24, and also solved in bases ''b'' = 11, 13, 16, 22, 30 if [[:w:Probable prime|probable prime]]s are allowed. For the unsolved bases ''b'' = 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, Athena problem is solved (if probable primes are allowed) except 771 [[:w:Indexed family|families]] of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be [[:w:Empty string|empty]]) of digits in base ''b'', ''y'' is a digit in base ''b'') = sequence {''xz'', ''xyz'', ''xyyz'', ''xyyyz'', ''xyyyyz'', ''xyyyyyz'', ...} (i.e. &quot;''xy''&lt;sup&gt;+&lt;/sup&gt;''z''&quot; in [[:w:Regular expression|regular expression]]), all of these 771 families contain no primes &gt; ''b'' or probable primes &gt; ''b'' with length ≤ 100000.

== Solve the problem ==

To solve the Athena problem for a given base ''b'', we must [[:w:Computing|compute]] the elements up to families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), and find the smallest prime &gt; ''b'' in all such families.

We call families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') &quot;linear&quot; families, and we reduce these families by removing all trailing digits ''y'' from ''x'', and removing all leading digits ''y'' from ''z'', to make the families be easier, e.g. family 12333{3}33345 in base ''b'' is reduced to family 12{3}45 in base ''b'', since they are in fact the same family. Our [[:w:Algorithm|algorithm]] then proceeds as follows:

* 1. ''M'' := {minimal primes in base ''b'' of length 2 or 3}, ''L'' := union of all ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'') such that ''x'' ≠ 0 and ''gcd''(''z'', ''b'') = 1 and ''Y'' is the set of digits ''y'' in base ''b'' such that ''xyz'' has no subsequence in ''M''.
* 2. While ''L'' contains nonlinear families (families which are not linear families): Explore each family of ''L'', and update ''L''. Examine each family of ''L'' by:
* 2.1. Let ''w'' be the shortest string in the family. If ''w'' has a subsequence in ''M'', then remove the family from ''L''. If ''w'' represents a prime, then add ''w'' to ''M'' and remove the family from ''L''.
* 2.2. If possible, simplify the family.
* 2.3. Using the techniques below (covering congruence, algebraic factorization, or combine of them), check if the family can be proven to only contain composites (only count the numbers &gt; ''b''), and if so then remove the family from ''L''.
* 3. Update ''L'', after each split examine the new families as in step 2.

e.g. in decimal (base ''b'' = 10):

''M'' := {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991}

''L'' := {2{0,2}1, 2{0,8}7, 3{0,3,6,9}3, 3{0,3,6,9}9, 4{6}9, 5{0,5,8}1, 5{0,2}7, 6{0,3,6,9}3, 6{0,3,4,6,9}9, 7{0,7}7, 8{0,5}1, 8{0}7, 9{0,2,5,8}1, 9{0,3,6,9}3, 9{0,3,4,6,9}9}

and since 2221 is prime, it follows that the family 2{0,2}1 splits into the families 2{0}1 and 2{0}2{0}1

and since the family 2{0}1 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

and since 20201 is prime, it follows that the family 2{0}2{0}1 splits into the families 2{0}21 and 22{0}1

221 and 2021 are composites, but 20021 is prime, thus add 20021 to ''L''

none of 221, 2201, 22001, 220001, 2200001 are primes, but 22000001 is prime, thus add 22000001 to ''L''

and since the family 3{0,3,6,9}3 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

etc.

Since the number of possible (first digit,last digit) (also called (initial digit,final digit)) combos ([[:w:Ordered pair|ordered pair]]s) of a prime &gt; ''b'' in base ''b'' is (''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(''b'') (all digits except 0 can be the first digit of a prime &gt; ''b'' in base ''b'' (thus ''b''−1 possible digits), but only the digits coprime to ''b'' can be the last digit of a prime &gt; ''b'' in base ''b'' (thus ''eulerphi''(''b'') possible digits), and by the [[:w:Rule of product|rule of product]], there are (''b''−1)×''eulerphi''(''b'') choices of the (first digit,last digit) combo, also, both &quot;numbers of primes in the set of the Athena problem in base ''b''&quot; and &quot;length of the largest prime in the set of the Athena problem in base ''b''&quot; are [[:w:Asymptotic analysis|roughly]] ''[[:w:E (mathematical_constant)|e]]''&lt;sup&gt;''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)&lt;/sup&gt;.

Shrinking the family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'')

* If ''y'' ∈ ''Y'' and the string ''xyyz'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''}''z'' ∪ ''x''{''Y'' \ ''y''}''y''{''Y'' \ ''y''}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and the string ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and both the strings ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' and ''xy''&lt;sub&gt;2&lt;/sub&gt;''y''&lt;sub&gt;1&lt;/sub&gt;''z'' represent a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or have a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}''z'' ∪ ''x''{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.

e.g. in decimal (base ''b'' = 10):

* 2221 is a prime &gt; 10, thus the family 2{0,2}1 splits into the two families 2{0}1 and 2{0}2{0}1.
* 227 is a prime &gt; 10, and it is a subsequence of 5227, thus the family 5{0,2}7 splits into the two families 5{0}7 and 5{0}2{0}7.
* 449 is a prime &gt; 10, and it is a subsequence of 6449, thus the family 6{0,3,4,6,9}9 splits into the two families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9.
* Both 5051 and 5501 are primes &gt; 10, thus the family 5{0,5}1 splits into the two families 5{0}1 and 5{5}1 = {5}1.
* 8501 is a prime &gt; 10, thus the family 8{0,5}1 splits into the family 8{0}{5}1.
* 887 is a prime &gt; 10, and it is a subsequence of 2887, also 2087 is a prime &gt; 10, thus the family 2{0,8}7 splits into the two families 2{0}7 and 28{0}7.
* 349 and 449 are primes &gt; 10, and they are subsequences of 9349 and 9449, respectively, also 9049, 9649, 9949 are primes &gt; 10, thus the family 9{0,3,4,6,9}9 splits into the two families 9{0,3,6,9}9 and 94{0,3,6,9}9.
* 251, 281, 521, 821, 881 are primes &gt; 10, and they are subsequences of 9251, 9281, 9521, 9821, 9881, respectively, also 9001, 9221, 9551, 9851 are primes &gt; 10, thus the family 9{0,2,5,8}1 splits into the numbers {91, 901, 921, 951, 981, 9021, 9051, 9081, 9201, 9501, 9581, 9801, 90581, 95081, 95801}.

If the methods we have discussed cannot be used to rule out or shrink ''x''{''Y''}''z'' where ''Y'' = {''y''&lt;sub&gt;1&lt;/sub&gt;, ''y''&lt;sub&gt;2&lt;/sub&gt;, ..., ''y''&lt;sub&gt;''n''&lt;/sub&gt;}, then we can replace ''x''{''Y''}''z'' by ''xy''&lt;sub&gt;1&lt;/sub&gt;{''Y''}''z'' ∪ ''xy''&lt;sub&gt;2&lt;/sub&gt;{''Y''}''z'' ∪ ... ∪ ''xy''&lt;sub&gt;''n''&lt;/sub&gt;{''Y''}''z'' and re-run the methods on this new [[:w:Formal language|language]].

If all remain families are linear families (i.e. of the form ''x''{''y''}''z'', where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), then we search the smallest (probable) primes in these families and add these primes to the list.

e.g. in decimal (base ''b'' = 10):

* The smallest prime in the family 5{0}27 is 5000000000000000000000000000027.
* The smallest prime in the family {5}1 is 555555555551.
* The smallest prime in the family 8{5}1 is 8555555555555555555551, but 8555555555555555555551 is not a minimal element since 555555555551 is a subsequence of 8555555555555555555551.

There is no guarantee that the techniques discussed will ever terminate, but in practice they often do. They are able to determine the set of the minimal elements in base ''b'' for 2 ≤ ''b'' ≤ 16 and ''b'' = 18, 20, 22, 24, 30. The bases ''b'' = 17, 19, 21, 23, 25 ≤ ''b'' ≤ 29, 31 ≤ ''b'' ≤ 36 are solved with the exception of 771 families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'').

The following is a &quot;[[:w:Semi-algorithm|semi-algorithm]]&quot; that is guaranteed to solve the Athena problem for a given base ''b'', but it is not so easy to implement:

# ''M'' = ''[[:w:Empty string|∅]]''
# while (''L'' ≠ ''∅'') do
# choose ''x'', a shortest string in ''L''
# ''M'' := ''M'' ∪ {''x''}
# ''L'' := ''L'' − ''sup''({''x''})

In practice, for arbitrary ''L'', we cannot feasibly carry out step 5. Instead, we work with ''L''&amp;#39;, some regular overapproximation to ''L'', until we can show ''L''&amp;#39; = ''∅'' (which implies ''L'' = ''∅''). In practice, ''L''&amp;#39; is usually chosen to be a finite [[:w:Union (set theory)|union]] of sets of the form ''L''&lt;sub&gt;1&lt;/sub&gt;{''L''&lt;sub&gt;2&lt;/sub&gt;}''L''&lt;sub&gt;3&lt;/sub&gt;, where each of ''L''&lt;sub&gt;1&lt;/sub&gt;, ''L''&lt;sub&gt;2&lt;/sub&gt;, ''L''&lt;sub&gt;3&lt;/sub&gt; is finite. In the case we consider in this project, we then have to determine whether such a family contains a prime or not.

Thus, the [[:w:Time complexity|time complexity]] of the Athena problem in base ''b'' may be ''[[:w:Big O notation|O]]''(''[[:w:E (mathematical_constant)|e]]''&lt;sup&gt;''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)&lt;/sup&gt;), and the [[:w:CPU time|CPU time]] of the Athena problem in base ''b'' may be longer than [[:w:Age of the universe|the age of the universe]] for bases ''b'' = 19, 23, 25, 27, 29, 31, 32, 33, 34, 35, also, Athena problem in bases ''b'' around 500 may be [[:w:NP-complete|NP-complete]] or [[:w:NP-hard|NP-hard]], or an [[:w:Undecidable problem|undecidable problem]], or an example of [[:w:Gödel's incompleteness theorems|Gödel's incompleteness theorems]] (like the [[:w:Continuum hypothesis|continuum hypothesis]] and the [[:w:Halting problem|halting problem]]).

To solve the Athena problem, we need to determine whether a given family contains a prime. In practice, if family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') could not be ruled out as only containing composites and ''Y'' contains two or more digits, then a relatively small prime &gt; ''b'' could always be found in this family. Intuitively, this is because there are a large number of small strings in such a family, and at least one is likely to be prime (e.g. there are 2&lt;sup&gt;''n''−2&lt;/sup&gt; strings of length ''n'' in the family 1{3,7}9, and there are over a thousand strings of length 12 in the family 1{3,7}9, thus it is very impossible that these numbers are all composite). In the case ''Y'' contains only one digit, this family is of the form ''x''{''y''}''z'', and there is only a single string of each length &gt; (the length of ''x'' + the length of ''z''), and it is not known if the following [[:w:Decision problem|decision problem]] is recursively solvable (just like [[:w:Sierpiński number|Sierpiński problem]] and [[:w:Riesel number|Riesel problem]], Sierpiński problem and Riesel problem can be generalized to other bases ''b'', in fact, Athena problem in base ''b'' covers the Sierpiński problem in base ''b'' and the Riesel problem in base ''b'' with ''k'' &lt; ''b'', i.e. finding the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (or prove such prime does not exist) with ''k'' &lt; ''b'' (specially, for bases ''b'' such that the conjectured smallest Sierpiński number or the conjectured smallest Riesel number is &lt; ''b'', Athena problem base ''b'' covers the Sierpiński problem base ''b'' or the Riesel problem base ''b'', respectively), since the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (if exists) must be a minimal element in base ''b'', also, Athena problem base ''b'' covers finding the smallest prime of these forms in base ''b'' (or proving that such prime does not exist): (''b''&lt;sup&gt;''n''&lt;/sup&gt;−1)/(''b''−1) (for this form, ''n'' must be prime, and we want ''n'' ≥ 2), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 1), (''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/2 (for odd ''b'') (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 2), (''sqrt''(''b'')×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''sqrt''(''b'')+1) (for square ''b'') (for this form, 2×''n''+1 must be prime, and we want ''n'' ≥ 2), ((''b''−2)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''b''−1) (''n'' ≥ 2), 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (''n'' ≥ 1), 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (''n'' ≥ 1), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+2 (''n'' ≥ 1), ''b''&lt;sup&gt;''n''&lt;/sup&gt;−2 (''n'' ≥ 2), (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (''n'' ≥ 1), (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (''n'' ≥ 1), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+(''b''−1) (''n'' ≥ 1), ''b''&lt;sup&gt;''n''&lt;/sup&gt;−(''b''−1) (''n'' ≥ 2)):

Problem: Given strings ''x'', ''z'' (may be empty), a digit ''y'', and a base ''b'' (''x'' does not [[:w:Leading zero|start with the digit 0]], ''z'' ends with a digit which [[:w:Coprime integers|coprime]] to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty), does there exist a prime number whose base-''b'' expansion is of the form ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' for some ''n'' ≥ 0?

Some families can be ruled out to contain no prime &gt; ''b'' by [[:w:Covering set|covering congruence]], [[:w:Factorization of polynomials|algebraic factorization]] (e.g. [[:w:Difference of two squares|difference of two squares]], [[:w:Sum of two cubes|sum of two cubes]], [[:w:Sophie Germain's identity|Sophie Germain's identity of ''x''&lt;sup&gt;4&lt;/sup&gt;+4×''y''&lt;sup&gt;4&lt;/sup&gt;]]), or combine of them, e.g.

* The base 9 family 2{7}: Always divisible by 2 or 5
* The base 16 family {8}F: Always divisible by 3, 7, or 13
* The base 21 family {7}D: Always divisible by 2, 13, or 17
* The base 23 family {D}GA: Always divisible by 2, 5, 7, 37, or 79
* The base 9 family 3{8}: Can be written as 4×9&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 8 family 1{0}1: Can be written as 8&lt;sup&gt;''n''&lt;/sup&gt;+1 and can be factored as (2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (4&lt;sup&gt;''n''&lt;/sup&gt;−2&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 16 family {4}1: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−49)/15 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−7) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+7) / 15
* The base 16 family {C}D: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;+1)/5 and can be factored as (2×4&lt;sup&gt;''n''&lt;/sup&gt;−2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (2×4&lt;sup&gt;''n''&lt;/sup&gt;+2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) / 5
* The base 14 family 8{D}: Can be written as 9×14&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even
* The base 12 family {B}9B: Can be written as 12&lt;sup&gt;''n''&lt;/sup&gt;−25, it is divisible by 13 if ''n'' is odd and can be factored as (12&lt;sup&gt;''n''/2&lt;/sup&gt;−5) × (12&lt;sup&gt;''n''/2&lt;/sup&gt;+5) if ''n'' is even
* The base 17 family 1{9}: Can be written as (25×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 19 family 1{6}: Can be written as (4×19&lt;sup&gt;''n''&lt;/sup&gt;−1)/3, it is divisible by 5 if ''n'' is odd and can be factored as (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;+1) / 3 if ''n'' is even

By the [[:w:Prime number theorem|prime number theorem]], the [[:w:Probability|chance]] that a [[:w:Random number|random]] ''n''-digit base ''b'' number is prime is [[:w:Asymptotic analysis|approximately]] 1/''n'' (more accurately, the chance is approximately 1/(''n''×''ln''(''b'')), where ''ln'' is the [[:w:Natural logarithm|natural logarithm]]). If one conjectures the numbers ''x''{''y''}''z'' behave similarly (i.e. the numbers ''x''{''y''}''z'' is a [[:w:Pseudorandomness|pseudorandom sequence]]) you would expect [[:w:Harmonic_series (mathematics)|1/1 + 1/2 + 1/3 + 1/4 + ... = ∞]] primes of the form ''x''{''y''}''z'' (of course, this does not always happen, since some ''x''{''y''}''z'' families can be ruled out to contain no prime &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them), but it is at least a reasonable conjecture in the absence of evidence to the contrary. Hence, the [[:w:Heuristic argument|heuristic argument]] suggests there are always infinitely many primes in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') if it cannot be ruled out to contain no prime or only contain finitely many primes, by covering congruence, algebraic factorization, or combine of them. However, some families ''x''{''y''}''z'' could not be proven to contain no primes &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them) but no primes &gt; ''b'' could be found in the family, even after searching through numbers with over 100000 digits. In such a case, the only way to proceed is to [[:w:Primality test|test the primality]] of larger and larger numbers of such form and hope a prime is eventually discovered. e.g. the smallest (probable) prime in the family A{3}A in base ''b'' = 13 is A3&lt;sub&gt;592197&lt;/sub&gt;A, its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4, when written in decimal contains 659677 digits (it is only probable prime, i.e. not definitely prime).

== Data ==

These are the results of the Athena problem in bases 2 ≤ ''b'' ≤ 36 (we stop at base 36 since this base is the maximum base for which it is possible to write the numbers with the [[:w:Symbol|symbol]]s 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 [[:w:Arabic numerals|Arabic numerals]] and the 26 [[:w:Latin script|Latin letters]]): (some large primes are only probable primes, i.e. not definitely primes, since they are too large to be [[:w:Elliptic curve primality|ECPP proved]] and [[:w:Pocklington primality test#Extensions and variants|neither ''N''−1 nor ''N''+1 can be ≥ 1/3 factored]], all of them pass the [[:w:Baillie–PSW primality test|Baillie–PSW primality test]] and the [[:w:Strong pseudoprime|strong primality test]] (i.e. the [[:w:Miller–Rabin primality test|Miller–Rabin primality test]]) with all prime bases ''p'' ≤ 61, however, all primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 are definitely primes, most of them &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Elliptic curve primality|ECPP proving]], others &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Pocklington primality test#Extensions and variants|''N''−1 or ''N''+1 proving]])

All numbers are written in base ''b'', [[:w:Senary#Base 36 as senary compression|using A to Z to represent digit values 10 to 35]], &quot;{}&quot; means repeating, e.g. family 12{3}45 means the sequence {1245, 12345, 123345, 1233345, 12333345, 123333345, ...} (where the members are expressed as base ''b'' strings), subscripts are used to indicate repetitions of digits, e.g. 123&lt;sub&gt;4&lt;/sub&gt;567 means 123333567 (all subscripts are written in decimal).

Base 2: 1 prime (the largest of which has 2 digits (it is 11, and its value is 3 in decimal)): {11}

Base 3: 3 primes (the largest of which has 3 digits (it is 111, and its value is 13 in decimal)): {12, 21, 111}

Base 4: 5 primes (the largest of which has 3 digits (it is 221, and its value is 41 in decimal)): {11, 13, 23, 31, 221}

Base 5: 22 primes (the largest of which has 96 digits (it is 10&lt;sub&gt;93&lt;/sub&gt;13, and its algebraic form is 5&lt;sup&gt;95&lt;/sup&gt;+8)): {12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013}

Base 6: 11 primes (the largest of which has 5 digits (it is 40041, and its value is 5209 in decimal)): {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041}

Base 7: 71 primes (the largest of which has 17 digits (it is 3&lt;sub&gt;16&lt;/sub&gt;1, and its algebraic form is (7&lt;sup&gt;17&lt;/sup&gt;−5)/2)): {14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331}

Base 8: 75 primes (the largest of which has 221 digits (it is 4&lt;sub&gt;220&lt;/sub&gt;7, and its algebraic form is (4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7)): {13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447}

Base 9: 151 primes (the largest of which has 1161 digits (it is 30&lt;sub&gt;1158&lt;/sub&gt;11, and its algebraic form is 3×9&lt;sup&gt;1160&lt;/sup&gt;+10)): {12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011}

Base 10: 77 primes (the largest of which has 31 digits (it is 50&lt;sub&gt;28&lt;/sub&gt;27, and its algebraic form is 5×10&lt;sup&gt;30&lt;/sup&gt;+27)): {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}

Base 11: 1068 primes (including 1 unproven probable prime: 57&lt;sub&gt;62668&lt;/sub&gt;), the largest of which has 62669 digits (it is 57&lt;sub&gt;62668&lt;/sub&gt;, and its algebraic form is (57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel11 Data of Athena problem base 11]

Base 12: 106 primes (the largest of which has 42 digits (it is 40&lt;sub&gt;39&lt;/sub&gt;77, and its algebraic form is 4×12&lt;sup&gt;41&lt;/sup&gt;+91)): {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077}

Base 13: 3197 primes (including 4 unproven probable primes: C5&lt;sub&gt;23755&lt;/sub&gt;C, 80&lt;sub&gt;32017&lt;/sub&gt;111, 95&lt;sub&gt;197420&lt;/sub&gt;, A3&lt;sub&gt;592197&lt;/sub&gt;A), the largest of which has 592199 digits (it is A3&lt;sub&gt;592197&lt;/sub&gt;A, and its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel13 Data of Athena problem base 13]

Base 14: 650 primes, the largest of which has 19699 digits (it is 4D&lt;sub&gt;19698&lt;/sub&gt;, and its algebraic form is 5×14&lt;sup&gt;19698&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel14 Data of Athena problem base 14]

Base 15: 1284 primes, the largest of which has 157 digits (it is 7&lt;sub&gt;155&lt;/sub&gt;97, and its algebraic form is (15&lt;sup&gt;157&lt;/sup&gt;+59)/2), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel15 Data of Athena problem base 15]

Base 16: 2347 primes (including 3 unproven probable primes: DB&lt;sub&gt;32234&lt;/sub&gt;, 4&lt;sub&gt;72785&lt;/sub&gt;DD, 3&lt;sub&gt;116137&lt;/sub&gt;AF), the largest of which has 116139 digits (it is 3&lt;sub&gt;116137&lt;/sub&gt;AF, and its algebraic form is (16&lt;sup&gt;116139&lt;/sup&gt;+619)/5), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel16 Data of Athena problem base 16]

Base 17: 10415 known primes (including many unproven probable primes) and 12 unsolved families (1{7}, 1F{0}7, 4{7}A, 70F{0}D, 8{B}9, 9{5}9, A{D}F, B{0}B3, {B}E9, {B}EE, F1{9}, FD0{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel17 Data of Athena problem base 17] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left17 Data of unsolved families for base 17]

Base 18: 549 primes, the largest of which has 6271 digits (it is C0&lt;sub&gt;6268&lt;/sub&gt;C5, and its algebraic form is 12×18&lt;sup&gt;6270&lt;/sup&gt;+221), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel18 Data of Athena problem base 18]

Base 19: 31417 known primes (including many unproven probable primes) and 17 unsolved families (4B5{0}H, {5}3, 5{H}05, 5{H}0H, 5{H}5, 66{B}, 71{0}177, 7AF{0}H, 97{0}3, C{H}C, EE1{6}, F{7}5, F{B}G, F{D}F, H0F{0}7A, HB{0}5B5, II{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel19 Data of Athena problem base 19] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left19 Data of unsolved families for base 19]

Base 20: 3314 primes, the largest of which has 6271 digits (it is G0&lt;sub&gt;6269&lt;/sub&gt;D, and its algebraic form is 16×20&lt;sup&gt;6270&lt;/sup&gt;+13), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel20 Data of Athena problem base 20]

Base 21: 13386 known primes (including many unproven probable primes) and 8 unsolved families (5{0}DJ, {9}D, B3{0}EB, B{H}6H, C{F}0K, {F}35, G{0}FK, H{0}7771, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel21 Data of Athena problem base 21] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left21 Data of unsolved families for base 21]

Base 22: 8003 primes (including 1 unproven probable prime: BK&lt;sub&gt;22001&lt;/sub&gt;5), the largest of which has 22003 digits (it is BK&lt;sub&gt;22001&lt;/sub&gt;5, and its algebraic form is (251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel22 Data of Athena problem base 22]

Base 23: 65178 known primes (including many unproven probable primes) and 87 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel23 Data of Athena problem base 23] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left23 Data of unsolved families for base 23]

Base 24: 3409 primes, the largest of which has 8134 digits (it is N00N&lt;sub&gt;8129&lt;/sub&gt;LN, and its algebraic form is 13249×24&lt;sup&gt;8131&lt;/sup&gt;−49), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel24 Data of Athena problem base 24]

Base 25: 133639 known primes (including many unproven probable primes) and 85 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel25 Data of Athena problem base 25] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left25 Data of unsolved families for base 25]

Base 26: 25256 known primes (including 7 unproven probable primes: 5&lt;sub&gt;19391&lt;/sub&gt;6F, 7&lt;sub&gt;20279&lt;/sub&gt;OL, LD0&lt;sub&gt;20975&lt;/sub&gt;7, 6K&lt;sub&gt;23300&lt;/sub&gt;5, J0&lt;sub&gt;44303&lt;/sub&gt;KCB, M0&lt;sub&gt;61186&lt;/sub&gt;2BB, 85M&lt;sub&gt;197060&lt;/sub&gt;B) and 3 unsolved families ({A}6F, {H}MH, {I}GL, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel26 Data of Athena problem base 26]

Base 27: 102852 known primes (including many unproven probable primes) and 44 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel27 Data of Athena problem base 27] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left27 Data of unsolved families for base 27]

Base 28: 25528 known primes (including 3 unproven probable primes: N6&lt;sub&gt;24051&lt;/sub&gt;LR, 5OA&lt;sub&gt;31238&lt;/sub&gt;F, O4O&lt;sub&gt;94535&lt;/sub&gt;9) and 1 unsolved family (O{A}F, no primes or probable primes with length ≤ 900000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel28 Data of Athena problem base 28]

Base 29: 355242 known primes (including many unproven probable primes) and 125 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel29 Data of Athena problem base 29] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left29 Data of unsolved families for base 29]

Base 30: 2619 primes (including 1 unproven probable prime: I0&lt;sub&gt;24608&lt;/sub&gt;D), the largest of which has 34206 digits (it is OT&lt;sub&gt;34205&lt;/sub&gt;, and its algebraic form is 25×30&lt;sup&gt;34205&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel30 Data of Athena problem base 30]

Base 31: 569323 known primes (including many unproven probable primes) and 77 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel31 Data of Athena problem base 31] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left31 Data of unsolved families for base 31]

Base 32: 168882 known primes (including many unproven probable primes) and 120 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel32 Data of Athena problem base 32] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left32 Data of unsolved families for base 32]

Base 33: 280012 known primes (including many unproven probable primes) and 81 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel33 Data of Athena problem base 33] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left33 Data of unsolved families for base 33]

Base 34: 184785 known primes (including many unproven probable primes) and 47 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel34 Data of Athena problem base 34] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left34 Data of unsolved families for base 34]

Base 35: 720002 known primes (including many unproven probable primes) and 60 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel35 Data of Athena problem base 35] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left35 Data of unsolved families for base 35]

Base 36: 35286 known primes (including 3 unproven probable primes: 7K&lt;sub&gt;26567&lt;/sub&gt;Z, S0&lt;sub&gt;75007&lt;/sub&gt;8H, P&lt;sub&gt;81993&lt;/sub&gt;SZ) and 4 unsolved families (B{0}EUV, HM{0}N, N{0}YYN, O{L}Z, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel36 Data of Athena problem base 36]

== The fully proof of Athena problem in decimal (base ''b'' = 10) ==

'''Bold''' for the minimal elements, ''x'' ◁ ''y'' means ''x'' is a subsequence of ''y''.

Assume ''p'' is a prime &gt; 10, and the last digit of ''p'' must lie in {1,3,7,9}.

Case 1: ''p'' ends with 1.

In this case we can write ''p'' = ''x''1. If ''x'' contains 1, 3, 4, 6, or 7, then (respectively) '''11''' ◁ ''p'', '''31''' ◁ ''p'', '''41''' ◁ ''p'', '''61''' ◁ ''p'', or '''71''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 8, or 9.

Case 1.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''1. If 5 ◁ ''y'', then '''251''' ◁ ''p''. If 8 ◁ ''y'', then '''281''' ◁ ''p''. If 9 ◁ ''y'', then 29 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then '''2221''' ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 2{0}1. But then, since the sum of the digits of ''p'' is 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 2''z''2''w''1, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''20201''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 22{0}1, and the smallest prime ''p'' ∈ 22{0}1 is '''22000001'''.

If ''w'' is empty, then ''p'' ∈ 2{0}21, and the smallest prime ''p'' ∈ 2{0}21 is '''20021'''.

Case 1.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''1. If 2 ◁ ''y'', then '''521''' ◁ ''p''. If 9 ◁ ''y'', then 59 ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 5, or 8.

If 05 ◁ ''y'', then '''5051''' ◁ ''p''. If 08 ◁ ''y'', then '''5081''' ◁ ''p''. If 50 ◁ ''y'', then '''5501''' ◁ ''p''. If 58 ◁ ''y'', then '''5581''' ◁ ''p''. If 80 ◁ ''y'', then '''5801''' ◁ ''p''. If 85 ◁ ''y'', then '''5851''' ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ {5} ∪ {8}.

If ''y'' ∈ {0}, then ''p'' ∈ 5{0}1. But then, since the sum of the digits of ''p'' is 6, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' ∈ {5}, then ''p'' ∈ 5{5}1, and the smallest prime ''p'' ∈ 5{5}1 is '''555555555551'''.

If ''y'' ∈ {8}, since if 88 ◁ ''y'', then 881 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'',8}, and thus ''p'' ∈ {51,581}, but 51 and 581 are both composite.

Case 1.3: ''p'' begins with 8.

In this case we can write p = 8''y''1. If 2 ◁ ''y'', then '''821''' ◁ ''p''. If 8 ◁ ''y'', then '''881''' ◁ ''p''. If 9 ◁ ''y'', then 89 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 5.

If 50 ◁ ''y'', then '''8501''' ◁ ''p''. Hence we may assume y ∈ {0}{5}.

If 005 ◁ ''y'', then '''80051''' ◁ p. Hence we may assume y ∈ {0} ∪ {5} ∪ 0{5}.

If y ∈ {0}, then ''p'' ∈ 8{0}1. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ {5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'', 5, 55, 555, 5555, 55555, 555555, 5555555, 55555555, 555555555, 5555555555}, and thus ''p'' ∈ {81, 851, 8551, 85551, 855551, 8555551, 85555551, 855555551, 8555555551, 85555555551, 855555555551}, but all of these numbers are composite.

If y ∈ 0{5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {0, 05, 055, 0555, 05555, 055555, 0555555, 05555555, 055555555, 0555555555, 05555555555}, and thus ''p'' ∈ {801, 8051, 80551, 805551, 8055551, 80555551, 805555551, 8055555551, 80555555551, 805555555551, 8055555555551}, and of these numbers only 80555551 and 8055555551 are primes, but 80555551 ◁ 8055555551, thus only '''80555551''' is a minimal element.

Case 1.4: ''p'' begins with 9.

In this case we can write p = 9''y''1. If 9 ◁ ''y'', then '''991''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 2, 5, or 8.

If 00 ◁ ''y'', then '''9001''' ◁ ''p''. If 22 ◁ ''y'', then '''9221''' ◁ ''p''. If 55 ◁ ''y'', then '''9551''' ◁ ''p''. If 88 ◁ ''y'', then 881 ◁ ''p''. Hence we may assume ''y'' contains at most one 0, at most one 2, at most one 5, and at most one 8.

If ''y'' only contains at most one 0 and does not contain any of {2,5,8}, then ''y'' ∈ {''𝜆'',0}, and thus ''p'' ∈ {91,901}, but 91 and 901 are both composite. If ''y'' only contains at most one 0 and only one of {2,5,8}, then the sum of the digits of ''p'' is divisible by 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume ''y'' contains at least two of {2,5,8}.

If 25 ◁ ''y'', then 251 ◁ ''p''. If 28 ◁ ''y'', then 281 ◁ ''p''. If 52 ◁ ''y'', then 521 ◁ ''p''. If 82 ◁ ''y'', then 821 ◁ ''p''. Hence we may assume ''y'' contains no 2's (since if ''y'' contains 2, then ''y'' cannot contain either 5's or 8's, which is a contradiction).

If 85 ◁ ''y'', then '''9851''' ◁ ''p''. Hence we may assume ''y'' ∈ {58,580,508,058}, and thus ''p'' ∈ {9581,95801,95081,90581}, and of these numbers only 95801 is prime, but 95801 is not a minimal element since 5801 ◁ 95801.

Case 2: ''p'' ends with 3.

In this case we can write p = ''x''3. If ''x'' contains 1, 2, 4, 5, 7, or 8, then (respectively) '''13''' ◁ ''p'', '''23''' ◁ ''p'', '''43''' ◁ ''p'', '''53''' ◁ ''p'', '''73''' ◁ ''p'', or '''83''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 3: ''p'' ends with 7.

In this case we can write ''p'' = ''x''7. If ''x'' contains 1, 3, 4, 6, or 9, then (respectively) '''17''' ◁ ''p'', '''37''' ◁ ''p'', '''47''' ◁ ''p'', '''67''' ◁ ''p'', or '''97''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 7, or 8.

Case 3.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''7. If 2 ◁ ''y'', then '''227''' ◁ ''p''. If 5 ◁ ''y'', then '''257''' ◁ ''p''. If 7 ◁ ''y'', then '''277''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 8.

If 08 ◁ ''y'', then '''2087''' ◁ ''p''. If 88 ◁ ''y'', then 887 ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ 8{0}.

If ''y'' ∈ {0}, then ''p'' ∈ 2{0}7. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ 8{0}, then ''p'' ∈ 28{0}7. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 40&lt;sub&gt;''n''&lt;/sub&gt;1 = 280&lt;sub&gt;''n''&lt;/sub&gt;7.

Case 3.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''7. If 5 ◁ ''y'', then '''557''' ◁ ''p''. If 7 ◁ ''y'', then '''577''' ◁ ''p''. If 8 ◁ ''y'', then '''587''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then 227 ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 5{0}7. But then, since the sum of the digits of ''p'' is 12, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 5''z''2''w''7, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''50207''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 52{0}7, and the smallest prime ''p'' ∈ 52{0}7 is '''5200007'''.

If ''w'' is empty, then ''p'' ∈ 5{0}27, and the smallest prime ''p'' ∈ 5{0}27 is '''5000000000000000000000000000027'''.

Case 3.3: ''p'' begins with 7.

In this case we can write ''p'' = 7''y''7. If 2 ◁ ''y'', then '''727''' ◁ ''p''. If 5 ◁ ''y'', then '''757''' ◁ ''p''. If 8 ◁ ''y'', then '''787''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 7, and thus all digits of ''p'' are 0 or 7. But then, since the digits of ''p'' all have a common factor 7, ''p'' is divisible by 7, so ''p'' cannot be prime.

Case 3.4: ''p'' begins with 8.

In this case we can write ''p'' = 8''y''7. If 2 ◁ ''y'', then '''827''' ◁ ''p''. If 5 ◁ ''y'', then '''857''' ◁ ''p''. If 7 ◁ ''y'', then '''877''' ◁ ''p''. If 8 ◁ ''y'', then '''887''' ◁ ''p''. Hence we may assume ''y'' ∈ {0}, and thus ''p'' ∈ 8{0}7. But then, since the sum of the digits of ''p'' is 15, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 4: ''p'' ends with 9.

In this case we can write ''p'' = ''x''9. If ''x'' contains 1, 2, 5, 7, or 8, then (respectively) '''19''' ◁ ''p'', '''29''' ◁ ''p'', '''59''' ◁ ''p'', '''79''' ◁ ''p'', or '''89''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 4, 6, or 9.

If 44 ◁ ''x'', then '''449''' ◁ ''p''. Hence we may assume ''x'' contains zero or one 4's.

If x contains no 4's, then all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume that ''x'' contains exactly one 4.

Case 4.1: ''p'' begins with 3.

In this case we can write ''p'' = 3''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. We must have '''349''' ◁ ''p''.

Case 4.2: ''p'' begins with 4.

In this case we can write ''p'' = 4''y''9, where all digits of ''y'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''409''' ◁ ''p''. If 3 ◁ ''y'', then 43 ◁ ''p''. If 9 ◁ ''y'', then '''499''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}, and thus ''p'' ∈ 4{6}9. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 6&lt;sub&gt;''n''&lt;/sub&gt;7 = 46&lt;sub&gt;''n''&lt;/sub&gt;9.

Case 4.3: ''p'' begins with 6.

In this case we can write p = 6''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 6 ◁ ''z'', then '''6469''' ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' is empty.

If 3 ◁ ''y'', then 349 ◁ ''p''. If 9 ◁ ''y'', then '''6949''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 6.

If 06 ◁ ''y'', then '''60649''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}{0}.

If 666 ◁ ''y'', then '''666649''' ◁ ''p''. If 00000 ◁ ''y'', then '''60000049''' ◁ ''p''. Hence we may assume ''y'' ∈ {''𝜆'', 0, 00, 000, 0000, 6, 60, 600, 6000, 60000, 66, 660, 6600, 66000, 660000}, and thus ''p'' ∈ {649, 6049, 60049, 600049, 6000049, 6649, 66049, 660049, 6600049, 66000049, 66649, 666049, 6660049, 66600049, 666000049}, and of these numbers only '''66000049''' and '''66600049''' are primes.

Case 4.4: ''p'' begins with 9.

In this case we can write p = 9''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''9049''' ◁ ''p''. If 3 ◁ ''y'', then 349 ◁ ''p''. If 6 ◁ ''y'', then '''9649''' ◁ ''p''. If 9 ◁ ''y'', then '''9949''' ◁ ''p''. Hence we may assume ''y'' is empty.

If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' ∈ {6}, and thus ''p'' ∈ 94{6}9, and the smallest prime ''p'' ∈ 94{6}9 is 946669.

[[Category:Number theory]]</text>
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      <text bytes="52522" sha1="mwg1oxzockplpgxuayvfc6z26dkko01" xml:space="preserve">{{mathematics}}
'''Athena problem''' is an [[:w:List of unsolved problems in mathematics|unsolved problem]] in [[:w:Number theory|number theory]] and [[:w:Formal language theory|formal language theory]] and [[:w:Order theory|order theory]], this problem is named after the ancient Greek goddess [[:w:Athena|Athena]] (which is associated with [[:w:Wisdom|wisdom]]). Athena problem is: Give a [[:w:Natural number|natural number]] ''b'' &gt; 1, find the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the set of the &quot;[[:w:Prime number|prime number]] [[:w:Greater than|&gt;]] ''b''&quot; [[:w:Numerical digit|digit]] [[:w:String (computer science)|string]]s in the [[:w:Positional numeral system|positional numeral system]] with [[:w:Radix|base]] ''b'' for the [[:w:Subsequence|subsequence]] [[:w:Partially ordered set|ordering]]. (A string ''x'' is a subsequence of another string ''y'', if ''x'' can be obtained from ''y'' by deleting zero or more of the [[:w:Character (computing)|character]]s in ''y''. For example, 514 is a subsequence of 352148, &quot;string&quot; is a subsequence of &quot;meistersinger&quot;. In contrast, 758 is not a subsequence of 378259, &quot;abc&quot; is not a subsequence of &quot;cbacacba&quot;, since the characters must be in the same order) (Unlike [[:w:Substring|substring]], subsequence is not required to occupy consecutive positions within the original sequences, e.g. the [[:w:Longest common subsequence|longest common subsequence problem]] is different from the [[:w:Longest common substring|longest common substring problem]])

Using [[:w:Formal language theory|formal language theory]] terminology, Athena problem is finding the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the [[:w:Formal language|language]] of base-''b'' [[:w:Representation (mathematics)|representation]]s of the [[:w:Prime number|prime number]]s [[:w:Greater than|&gt;]] ''b'' (which is a set of [[:w:String (computer science)|string]]s of [[:w:Symbol|symbol]]s over the [[:w:Alphabet (formal languages)|alphabet]] ''Σ''&lt;sub&gt;''b''&lt;/sub&gt; := {0, 1, ..., ''b''−1}), under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), for a given natural number ''b'' &gt; 1. (You can draw this partial ordering as a [[:w:Hasse diagram|Hasse diagram]] to find all [[:w:Minimal element|minimal element]]s)

By [[:w:Higman's lemma|Higman's lemma]], there are no [[:w:Infinite set|infinite]] [[:w:Antichain|antichain]]s for the subsequence ordering (i.e. the subsequence ordering is always a [[:w:Well-quasi-ordering|well quasi order]]) (i.e. under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), every set of pairwise incomparable (i.e. not [[:w:Comparability|comparable]]) strings is finite), thus there must be only finitely many such minimal elements. In other words, the set of such minimal elements must be a [[:w:Finite set|finite set]], e.g. in [[:w:Decimal|decimal]] (base ''b'' = 10), this set has exactly 77 [[:w:Element of a set|element]]s: {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}.

For bases 2 ≤ ''b'' ≤ 36, Athena problem is fully solved in bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24, and also solved in bases ''b'' = 11, 13, 16, 22, 30 if [[:w:Probable prime|probable prime]]s are allowed. For the unsolved bases ''b'' = 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, Athena problem is solved (if probable primes are allowed) except 771 [[:w:Indexed family|families]] of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be [[:w:Empty string|empty]]) of digits in base ''b'', ''y'' is a digit in base ''b'') = sequence {''xz'', ''xyz'', ''xyyz'', ''xyyyz'', ''xyyyyz'', ''xyyyyyz'', ...} (i.e. &quot;''xy''&lt;sup&gt;+&lt;/sup&gt;''z''&quot; in [[:w:Regular expression|regular expression]]), all of these 771 families contain no primes &gt; ''b'' or probable primes &gt; ''b'' with length ≤ 100000.

== Solve the problem ==

To solve the Athena problem for a given base ''b'', we must [[:w:Computing|compute]] the elements up to families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), and find the smallest prime &gt; ''b'' in all such families.

We call families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') &quot;linear&quot; families, and we reduce these families by removing all trailing digits ''y'' from ''x'', and removing all leading digits ''y'' from ''z'', to make the families be easier, e.g. family 12333{3}33345 in base ''b'' is reduced to family 12{3}45 in base ''b'', since they are in fact the same family. Our [[:w:Algorithm|algorithm]] then proceeds as follows:

* 1. ''M'' := {minimal primes in base ''b'' of length 2 or 3}, ''L'' := union of all ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'') such that ''x'' ≠ 0 and ''gcd''(''z'', ''b'') = 1 and ''Y'' is the set of digits ''y'' in base ''b'' such that ''xyz'' has no subsequence in ''M''.
* 2. While ''L'' contains nonlinear families (families which are not linear families): Explore each family of ''L'', and update ''L''. Examine each family of ''L'' by:
* 2.1. Let ''w'' be the shortest string in the family. If ''w'' has a subsequence in ''M'', then remove the family from ''L''. If ''w'' represents a prime, then add ''w'' to ''M'' and remove the family from ''L''.
* 2.2. If possible, simplify the family.
* 2.3. Using the techniques below (covering congruence, algebraic factorization, or combine of them), check if the family can be proven to only contain composites (only count the numbers &gt; ''b''), and if so then remove the family from ''L''.
* 3. Update ''L'', after each split examine the new families as in step 2.

e.g. in decimal (base ''b'' = 10):

''M'' := {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991}

''L'' := {2{0,2}1, 2{0,8}7, 3{0,3,6,9}3, 3{0,3,6,9}9, 4{6}9, 5{0,5,8}1, 5{0,2}7, 6{0,3,6,9}3, 6{0,3,4,6,9}9, 7{0,7}7, 8{0,5}1, 8{0}7, 9{0,2,5,8}1, 9{0,3,6,9}3, 9{0,3,4,6,9}9}

and since 2221 is prime, it follows that the family 2{0,2}1 splits into the families 2{0}1 and 2{0}2{0}1

and since the family 2{0}1 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

and since 20201 is prime, it follows that the family 2{0}2{0}1 splits into the families 2{0}21 and 22{0}1

221 and 2021 are composites, but 20021 is prime, thus add 20021 to ''L''

none of 221, 2201, 22001, 220001, 2200001 are primes, but 22000001 is prime, thus add 22000001 to ''L''

and since the family 3{0,3,6,9}3 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

etc.

Since the number of possible (first digit,last digit) (also called (initial digit,final digit)) combos ([[:w:Ordered pair|ordered pair]]s) of a prime &gt; ''b'' in base ''b'' is (''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(''b'') (all digits except 0 can be the first digit of a prime &gt; ''b'' in base ''b'' (thus ''b''−1 possible digits), but only the digits coprime to ''b'' can be the last digit of a prime &gt; ''b'' in base ''b'' (thus ''eulerphi''(''b'') possible digits), and by the [[:w:Rule of product|rule of product]], there are (''b''−1)×''eulerphi''(''b'') choices of the (first digit,last digit) combo, also, both &quot;numbers of primes in the set of the Athena problem in base ''b''&quot; and &quot;length of the largest prime in the set of the Athena problem in base ''b''&quot; are [[:w:Asymptotic analysis|roughly]] ''[[:w:E (mathematical_constant)|e]]''&lt;sup&gt;''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)&lt;/sup&gt;.

Shrinking the family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'')

* If ''y'' ∈ ''Y'' and the string ''xyyz'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''}''z'' ∪ ''x''{''Y'' \ ''y''}''y''{''Y'' \ ''y''}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and the string ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and both the strings ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' and ''xy''&lt;sub&gt;2&lt;/sub&gt;''y''&lt;sub&gt;1&lt;/sub&gt;''z'' represent a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or have a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}''z'' ∪ ''x''{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.

e.g. in decimal (base ''b'' = 10):

* 2221 is a prime &gt; 10, thus the family 2{0,2}1 splits into the two families 2{0}1 and 2{0}2{0}1.
* 227 is a prime &gt; 10, and it is a subsequence of 5227, thus the family 5{0,2}7 splits into the two families 5{0}7 and 5{0}2{0}7.
* 449 is a prime &gt; 10, and it is a subsequence of 6449, thus the family 6{0,3,4,6,9}9 splits into the two families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9.
* Both 5051 and 5501 are primes &gt; 10, thus the family 5{0,5}1 splits into the two families 5{0}1 and 5{5}1 = {5}1.
* 8501 is a prime &gt; 10, thus the family 8{0,5}1 splits into the family 8{0}{5}1.
* 887 is a prime &gt; 10, and it is a subsequence of 2887, also 2087 is a prime &gt; 10, thus the family 2{0,8}7 splits into the two families 2{0}7 and 28{0}7.
* 349 and 449 are primes &gt; 10, and they are subsequences of 9349 and 9449, respectively, also 9049, 9649, 9949 are primes &gt; 10, thus the family 9{0,3,4,6,9}9 splits into the two families 9{0,3,6,9}9 and 94{0,3,6,9}9.
* 251, 281, 521, 821, 881 are primes &gt; 10, and they are subsequences of 9251, 9281, 9521, 9821, 9881, respectively, also 9001, 9221, 9551, 9851 are primes &gt; 10, thus the family 9{0,2,5,8}1 splits into the numbers {91, 901, 921, 951, 981, 9021, 9051, 9081, 9201, 9501, 9581, 9801, 90581, 95081, 95801}.

If the methods we have discussed cannot be used to rule out or shrink ''x''{''Y''}''z'' where ''Y'' = {''y''&lt;sub&gt;1&lt;/sub&gt;, ''y''&lt;sub&gt;2&lt;/sub&gt;, ..., ''y''&lt;sub&gt;''n''&lt;/sub&gt;}, then we can replace ''x''{''Y''}''z'' by ''xy''&lt;sub&gt;1&lt;/sub&gt;{''Y''}''z'' ∪ ''xy''&lt;sub&gt;2&lt;/sub&gt;{''Y''}''z'' ∪ ... ∪ ''xy''&lt;sub&gt;''n''&lt;/sub&gt;{''Y''}''z'' and re-run the methods on this new [[:w:Formal language|language]].

If all remain families are linear families (i.e. of the form ''x''{''y''}''z'', where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), then we search the smallest (probable) primes in these families and add these primes to the list.

e.g. in decimal (base ''b'' = 10):

* The smallest prime in the family 5{0}27 is 5000000000000000000000000000027.
* The smallest prime in the family {5}1 is 555555555551.
* The smallest prime in the family 8{5}1 is 8555555555555555555551, but 8555555555555555555551 is not a minimal element since 555555555551 is a subsequence of 8555555555555555555551.

There is no guarantee that the techniques discussed will ever terminate, but in practice they often do. They are able to determine the set of the minimal elements in base ''b'' for 2 ≤ ''b'' ≤ 16 and ''b'' = 18, 20, 22, 24, 30. The bases ''b'' = 17, 19, 21, 23, 25 ≤ ''b'' ≤ 29, 31 ≤ ''b'' ≤ 36 are solved with the exception of 771 families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'').

The following is a &quot;[[:w:Semi-algorithm|semi-algorithm]]&quot; that is guaranteed to solve the Athena problem for a given base ''b'', but it is not so easy to implement:

# ''M'' = ''[[:w:Empty string|∅]]''
# while (''L'' ≠ ''∅'') do
# choose ''x'', a shortest string in ''L''
# ''M'' := ''M'' ∪ {''x''}
# ''L'' := ''L'' − ''sup''({''x''})

In practice, for arbitrary ''L'', we cannot feasibly carry out step 5. Instead, we work with ''L''&amp;#39;, some regular overapproximation to ''L'', until we can show ''L''&amp;#39; = ''∅'' (which implies ''L'' = ''∅''). In practice, ''L''&amp;#39; is usually chosen to be a finite [[:w:Union (set theory)|union]] of sets of the form ''L''&lt;sub&gt;1&lt;/sub&gt;{''L''&lt;sub&gt;2&lt;/sub&gt;}''L''&lt;sub&gt;3&lt;/sub&gt;, where each of ''L''&lt;sub&gt;1&lt;/sub&gt;, ''L''&lt;sub&gt;2&lt;/sub&gt;, ''L''&lt;sub&gt;3&lt;/sub&gt; is finite. In the case we consider in this project, we then have to determine whether such a family contains a prime or not.

Thus, the [[:w:Time complexity|time complexity]] of the Athena problem in base ''b'' may be ''[[:w:Big O notation|O]]''(''[[:w:E (mathematical_constant)|e]]''&lt;sup&gt;''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)&lt;/sup&gt;), and the [[:w:CPU time|CPU time]] of the Athena problem in base ''b'' may be longer than [[:w:Age of the universe|the age of the universe]] for bases ''b'' = 19, 23, 25, 27, 29, 31, 32, 33, 34, 35, also, Athena problem in bases ''b'' around 500 may be [[:w:NP-complete|NP-complete]] or [[:w:NP-hard|NP-hard]], or an [[:w:Undecidable problem|undecidable problem]], or an example of [[:w:Gödel's incompleteness theorems|Gödel's incompleteness theorems]] (like the [[:w:Continuum hypothesis|continuum hypothesis]] and the [[:w:Halting problem|halting problem]]).

To solve the Athena problem, we need to determine whether a given family contains a prime. In practice, if family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') could not be ruled out as only containing composites and ''Y'' contains two or more digits, then a relatively small prime &gt; ''b'' could always be found in this family. Intuitively, this is because there are a large number of small strings in such a family, and at least one is likely to be prime (e.g. there are 2&lt;sup&gt;''n''−2&lt;/sup&gt; strings of length ''n'' in the family 1{3,7}9, and there are over a thousand strings of length 12 in the family 1{3,7}9, thus it is very impossible that these numbers are all composite). In the case ''Y'' contains only one digit, this family is of the form ''x''{''y''}''z'', and there is only a single string of each length &gt; (the length of ''x'' + the length of ''z''), and it is not known if the following [[:w:Decision problem|decision problem]] is recursively solvable (just like [[:w:Sierpiński number|Sierpiński problem]] and [[:w:Riesel number|Riesel problem]], Sierpiński problem and Riesel problem can be generalized to other bases ''b'', in fact, Athena problem base ''b'' covers the Sierpiński problem base ''b'' and the Riesel problem base ''b'' with ''k'' &lt; ''b'', i.e. finding the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (or prove such prime does not exist) with ''k'' &lt; ''b'' (specially, for bases ''b'' such that the conjectured smallest Sierpiński number or the conjectured smallest Riesel number is &lt; ''b'', Athena problem base ''b'' covers the Sierpiński problem in base ''b'' or the Riesel problem in base ''b'', respectively), since the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (if exists) must be a minimal element in base ''b'', also, Athena problem in base ''b'' covers finding the smallest prime of these forms in base ''b'' (or proving that such prime does not exist): (''b''&lt;sup&gt;''n''&lt;/sup&gt;−1)/(''b''−1) (for this form, ''n'' must be prime, and we want ''n'' ≥ 2), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 1), (''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/2 (for odd ''b'') (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 2), (''sqrt''(''b'')×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''sqrt''(''b'')+1) (for square ''b'') (for this form, 2×''n''+1 must be prime, and we want ''n'' ≥ 2), ((''b''−2)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''b''−1) (''n'' ≥ 2), 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (''n'' ≥ 1), 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (''n'' ≥ 1), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+2 (''n'' ≥ 1), ''b''&lt;sup&gt;''n''&lt;/sup&gt;−2 (''n'' ≥ 2), (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (''n'' ≥ 1), (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (''n'' ≥ 1), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+(''b''−1) (''n'' ≥ 1), ''b''&lt;sup&gt;''n''&lt;/sup&gt;−(''b''−1) (''n'' ≥ 2)):

Problem: Given strings ''x'', ''z'' (may be empty), a digit ''y'', and a base ''b'' (''x'' does not [[:w:Leading zero|start with the digit 0]], ''z'' ends with a digit which [[:w:Coprime integers|coprime]] to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty), does there exist a prime number whose base-''b'' expansion is of the form ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' for some ''n'' ≥ 0?

Some families can be ruled out to contain no prime &gt; ''b'' by [[:w:Covering set|covering congruence]], [[:w:Factorization of polynomials|algebraic factorization]] (e.g. [[:w:Difference of two squares|difference of two squares]], [[:w:Sum of two cubes|sum of two cubes]], [[:w:Sophie Germain's identity|Sophie Germain's identity of ''x''&lt;sup&gt;4&lt;/sup&gt;+4×''y''&lt;sup&gt;4&lt;/sup&gt;]]), or combine of them, e.g.

* The base 9 family 2{7}: Always divisible by 2 or 5
* The base 16 family {8}F: Always divisible by 3, 7, or 13
* The base 21 family {7}D: Always divisible by 2, 13, or 17
* The base 23 family {D}GA: Always divisible by 2, 5, 7, 37, or 79
* The base 9 family 3{8}: Can be written as 4×9&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 8 family 1{0}1: Can be written as 8&lt;sup&gt;''n''&lt;/sup&gt;+1 and can be factored as (2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (4&lt;sup&gt;''n''&lt;/sup&gt;−2&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 16 family {4}1: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−49)/15 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−7) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+7) / 15
* The base 16 family {C}D: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;+1)/5 and can be factored as (2×4&lt;sup&gt;''n''&lt;/sup&gt;−2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (2×4&lt;sup&gt;''n''&lt;/sup&gt;+2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) / 5
* The base 14 family 8{D}: Can be written as 9×14&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even
* The base 12 family {B}9B: Can be written as 12&lt;sup&gt;''n''&lt;/sup&gt;−25, it is divisible by 13 if ''n'' is odd and can be factored as (12&lt;sup&gt;''n''/2&lt;/sup&gt;−5) × (12&lt;sup&gt;''n''/2&lt;/sup&gt;+5) if ''n'' is even
* The base 17 family 1{9}: Can be written as (25×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 19 family 1{6}: Can be written as (4×19&lt;sup&gt;''n''&lt;/sup&gt;−1)/3, it is divisible by 5 if ''n'' is odd and can be factored as (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;+1) / 3 if ''n'' is even

By the [[:w:Prime number theorem|prime number theorem]], the [[:w:Probability|chance]] that a [[:w:Random number|random]] ''n''-digit base ''b'' number is prime is [[:w:Asymptotic analysis|approximately]] 1/''n'' (more accurately, the chance is approximately 1/(''n''×''ln''(''b'')), where ''ln'' is the [[:w:Natural logarithm|natural logarithm]]). If one conjectures the numbers ''x''{''y''}''z'' behave similarly (i.e. the numbers ''x''{''y''}''z'' is a [[:w:Pseudorandomness|pseudorandom sequence]]) you would expect [[:w:Harmonic_series (mathematics)|1/1 + 1/2 + 1/3 + 1/4 + ... = ∞]] primes of the form ''x''{''y''}''z'' (of course, this does not always happen, since some ''x''{''y''}''z'' families can be ruled out to contain no prime &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them), but it is at least a reasonable conjecture in the absence of evidence to the contrary. Hence, the [[:w:Heuristic argument|heuristic argument]] suggests there are always infinitely many primes in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') if it cannot be ruled out to contain no prime or only contain finitely many primes, by covering congruence, algebraic factorization, or combine of them. However, some families ''x''{''y''}''z'' could not be proven to contain no primes &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them) but no primes &gt; ''b'' could be found in the family, even after searching through numbers with over 100000 digits. In such a case, the only way to proceed is to [[:w:Primality test|test the primality]] of larger and larger numbers of such form and hope a prime is eventually discovered. e.g. the smallest (probable) prime in the family A{3}A in base ''b'' = 13 is A3&lt;sub&gt;592197&lt;/sub&gt;A, its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4, when written in decimal contains 659677 digits (it is only probable prime, i.e. not definitely prime).

== Data ==

These are the results of the Athena problem in bases 2 ≤ ''b'' ≤ 36 (we stop at base 36 since this base is the maximum base for which it is possible to write the numbers with the [[:w:Symbol|symbol]]s 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 [[:w:Arabic numerals|Arabic numerals]] and the 26 [[:w:Latin script|Latin letters]]): (some large primes are only probable primes, i.e. not definitely primes, since they are too large to be [[:w:Elliptic curve primality|ECPP proved]] and [[:w:Pocklington primality test#Extensions and variants|neither ''N''−1 nor ''N''+1 can be ≥ 1/3 factored]], all of them pass the [[:w:Baillie–PSW primality test|Baillie–PSW primality test]] and the [[:w:Strong pseudoprime|strong primality test]] (i.e. the [[:w:Miller–Rabin primality test|Miller–Rabin primality test]]) with all prime bases ''p'' ≤ 61, however, all primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 are definitely primes, most of them &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Elliptic curve primality|ECPP proving]], others &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Pocklington primality test#Extensions and variants|''N''−1 or ''N''+1 proving]])

All numbers are written in base ''b'', [[:w:Senary#Base 36 as senary compression|using A to Z to represent digit values 10 to 35]], &quot;{}&quot; means repeating, e.g. family 12{3}45 means the sequence {1245, 12345, 123345, 1233345, 12333345, 123333345, ...} (where the members are expressed as base ''b'' strings), subscripts are used to indicate repetitions of digits, e.g. 123&lt;sub&gt;4&lt;/sub&gt;567 means 123333567 (all subscripts are written in decimal).

Base 2: 1 prime (the largest of which has 2 digits (it is 11, and its value is 3 in decimal)): {11}

Base 3: 3 primes (the largest of which has 3 digits (it is 111, and its value is 13 in decimal)): {12, 21, 111}

Base 4: 5 primes (the largest of which has 3 digits (it is 221, and its value is 41 in decimal)): {11, 13, 23, 31, 221}

Base 5: 22 primes (the largest of which has 96 digits (it is 10&lt;sub&gt;93&lt;/sub&gt;13, and its algebraic form is 5&lt;sup&gt;95&lt;/sup&gt;+8)): {12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013}

Base 6: 11 primes (the largest of which has 5 digits (it is 40041, and its value is 5209 in decimal)): {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041}

Base 7: 71 primes (the largest of which has 17 digits (it is 3&lt;sub&gt;16&lt;/sub&gt;1, and its algebraic form is (7&lt;sup&gt;17&lt;/sup&gt;−5)/2)): {14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331}

Base 8: 75 primes (the largest of which has 221 digits (it is 4&lt;sub&gt;220&lt;/sub&gt;7, and its algebraic form is (4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7)): {13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447}

Base 9: 151 primes (the largest of which has 1161 digits (it is 30&lt;sub&gt;1158&lt;/sub&gt;11, and its algebraic form is 3×9&lt;sup&gt;1160&lt;/sup&gt;+10)): {12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011}

Base 10: 77 primes (the largest of which has 31 digits (it is 50&lt;sub&gt;28&lt;/sub&gt;27, and its algebraic form is 5×10&lt;sup&gt;30&lt;/sup&gt;+27)): {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}

Base 11: 1068 primes (including 1 unproven probable prime: 57&lt;sub&gt;62668&lt;/sub&gt;), the largest of which has 62669 digits (it is 57&lt;sub&gt;62668&lt;/sub&gt;, and its algebraic form is (57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel11 Data of Athena problem base 11]

Base 12: 106 primes (the largest of which has 42 digits (it is 40&lt;sub&gt;39&lt;/sub&gt;77, and its algebraic form is 4×12&lt;sup&gt;41&lt;/sup&gt;+91)): {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077}

Base 13: 3197 primes (including 4 unproven probable primes: C5&lt;sub&gt;23755&lt;/sub&gt;C, 80&lt;sub&gt;32017&lt;/sub&gt;111, 95&lt;sub&gt;197420&lt;/sub&gt;, A3&lt;sub&gt;592197&lt;/sub&gt;A), the largest of which has 592199 digits (it is A3&lt;sub&gt;592197&lt;/sub&gt;A, and its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel13 Data of Athena problem base 13]

Base 14: 650 primes, the largest of which has 19699 digits (it is 4D&lt;sub&gt;19698&lt;/sub&gt;, and its algebraic form is 5×14&lt;sup&gt;19698&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel14 Data of Athena problem base 14]

Base 15: 1284 primes, the largest of which has 157 digits (it is 7&lt;sub&gt;155&lt;/sub&gt;97, and its algebraic form is (15&lt;sup&gt;157&lt;/sup&gt;+59)/2), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel15 Data of Athena problem base 15]

Base 16: 2347 primes (including 3 unproven probable primes: DB&lt;sub&gt;32234&lt;/sub&gt;, 4&lt;sub&gt;72785&lt;/sub&gt;DD, 3&lt;sub&gt;116137&lt;/sub&gt;AF), the largest of which has 116139 digits (it is 3&lt;sub&gt;116137&lt;/sub&gt;AF, and its algebraic form is (16&lt;sup&gt;116139&lt;/sup&gt;+619)/5), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel16 Data of Athena problem base 16]

Base 17: 10415 known primes (including many unproven probable primes) and 12 unsolved families (1{7}, 1F{0}7, 4{7}A, 70F{0}D, 8{B}9, 9{5}9, A{D}F, B{0}B3, {B}E9, {B}EE, F1{9}, FD0{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel17 Data of Athena problem base 17] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left17 Data of unsolved families for base 17]

Base 18: 549 primes, the largest of which has 6271 digits (it is C0&lt;sub&gt;6268&lt;/sub&gt;C5, and its algebraic form is 12×18&lt;sup&gt;6270&lt;/sup&gt;+221), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel18 Data of Athena problem base 18]

Base 19: 31417 known primes (including many unproven probable primes) and 17 unsolved families (4B5{0}H, {5}3, 5{H}05, 5{H}0H, 5{H}5, 66{B}, 71{0}177, 7AF{0}H, 97{0}3, C{H}C, EE1{6}, F{7}5, F{B}G, F{D}F, H0F{0}7A, HB{0}5B5, II{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel19 Data of Athena problem base 19] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left19 Data of unsolved families for base 19]

Base 20: 3314 primes, the largest of which has 6271 digits (it is G0&lt;sub&gt;6269&lt;/sub&gt;D, and its algebraic form is 16×20&lt;sup&gt;6270&lt;/sup&gt;+13), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel20 Data of Athena problem base 20]

Base 21: 13386 known primes (including many unproven probable primes) and 8 unsolved families (5{0}DJ, {9}D, B3{0}EB, B{H}6H, C{F}0K, {F}35, G{0}FK, H{0}7771, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel21 Data of Athena problem base 21] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left21 Data of unsolved families for base 21]

Base 22: 8003 primes (including 1 unproven probable prime: BK&lt;sub&gt;22001&lt;/sub&gt;5), the largest of which has 22003 digits (it is BK&lt;sub&gt;22001&lt;/sub&gt;5, and its algebraic form is (251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel22 Data of Athena problem base 22]

Base 23: 65178 known primes (including many unproven probable primes) and 87 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel23 Data of Athena problem base 23] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left23 Data of unsolved families for base 23]

Base 24: 3409 primes, the largest of which has 8134 digits (it is N00N&lt;sub&gt;8129&lt;/sub&gt;LN, and its algebraic form is 13249×24&lt;sup&gt;8131&lt;/sup&gt;−49), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel24 Data of Athena problem base 24]

Base 25: 133639 known primes (including many unproven probable primes) and 85 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel25 Data of Athena problem base 25] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left25 Data of unsolved families for base 25]

Base 26: 25256 known primes (including 7 unproven probable primes: 5&lt;sub&gt;19391&lt;/sub&gt;6F, 7&lt;sub&gt;20279&lt;/sub&gt;OL, LD0&lt;sub&gt;20975&lt;/sub&gt;7, 6K&lt;sub&gt;23300&lt;/sub&gt;5, J0&lt;sub&gt;44303&lt;/sub&gt;KCB, M0&lt;sub&gt;61186&lt;/sub&gt;2BB, 85M&lt;sub&gt;197060&lt;/sub&gt;B) and 3 unsolved families ({A}6F, {H}MH, {I}GL, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel26 Data of Athena problem base 26]

Base 27: 102852 known primes (including many unproven probable primes) and 44 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel27 Data of Athena problem base 27] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left27 Data of unsolved families for base 27]

Base 28: 25528 known primes (including 3 unproven probable primes: N6&lt;sub&gt;24051&lt;/sub&gt;LR, 5OA&lt;sub&gt;31238&lt;/sub&gt;F, O4O&lt;sub&gt;94535&lt;/sub&gt;9) and 1 unsolved family (O{A}F, no primes or probable primes with length ≤ 900000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel28 Data of Athena problem base 28]

Base 29: 355242 known primes (including many unproven probable primes) and 125 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel29 Data of Athena problem base 29] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left29 Data of unsolved families for base 29]

Base 30: 2619 primes (including 1 unproven probable prime: I0&lt;sub&gt;24608&lt;/sub&gt;D), the largest of which has 34206 digits (it is OT&lt;sub&gt;34205&lt;/sub&gt;, and its algebraic form is 25×30&lt;sup&gt;34205&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel30 Data of Athena problem base 30]

Base 31: 569323 known primes (including many unproven probable primes) and 77 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel31 Data of Athena problem base 31] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left31 Data of unsolved families for base 31]

Base 32: 168882 known primes (including many unproven probable primes) and 120 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel32 Data of Athena problem base 32] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left32 Data of unsolved families for base 32]

Base 33: 280012 known primes (including many unproven probable primes) and 81 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel33 Data of Athena problem base 33] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left33 Data of unsolved families for base 33]

Base 34: 184785 known primes (including many unproven probable primes) and 47 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel34 Data of Athena problem base 34] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left34 Data of unsolved families for base 34]

Base 35: 720002 known primes (including many unproven probable primes) and 60 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel35 Data of Athena problem base 35] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left35 Data of unsolved families for base 35]

Base 36: 35286 known primes (including 3 unproven probable primes: 7K&lt;sub&gt;26567&lt;/sub&gt;Z, S0&lt;sub&gt;75007&lt;/sub&gt;8H, P&lt;sub&gt;81993&lt;/sub&gt;SZ) and 4 unsolved families (B{0}EUV, HM{0}N, N{0}YYN, O{L}Z, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel36 Data of Athena problem base 36]

== The fully proof of Athena problem in decimal (base ''b'' = 10) ==

'''Bold''' for the minimal elements, ''x'' ◁ ''y'' means ''x'' is a subsequence of ''y''.

Assume ''p'' is a prime &gt; 10, and the last digit of ''p'' must lie in {1,3,7,9}.

Case 1: ''p'' ends with 1.

In this case we can write ''p'' = ''x''1. If ''x'' contains 1, 3, 4, 6, or 7, then (respectively) '''11''' ◁ ''p'', '''31''' ◁ ''p'', '''41''' ◁ ''p'', '''61''' ◁ ''p'', or '''71''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 8, or 9.

Case 1.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''1. If 5 ◁ ''y'', then '''251''' ◁ ''p''. If 8 ◁ ''y'', then '''281''' ◁ ''p''. If 9 ◁ ''y'', then 29 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then '''2221''' ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 2{0}1. But then, since the sum of the digits of ''p'' is 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 2''z''2''w''1, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''20201''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 22{0}1, and the smallest prime ''p'' ∈ 22{0}1 is '''22000001'''.

If ''w'' is empty, then ''p'' ∈ 2{0}21, and the smallest prime ''p'' ∈ 2{0}21 is '''20021'''.

Case 1.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''1. If 2 ◁ ''y'', then '''521''' ◁ ''p''. If 9 ◁ ''y'', then 59 ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 5, or 8.

If 05 ◁ ''y'', then '''5051''' ◁ ''p''. If 08 ◁ ''y'', then '''5081''' ◁ ''p''. If 50 ◁ ''y'', then '''5501''' ◁ ''p''. If 58 ◁ ''y'', then '''5581''' ◁ ''p''. If 80 ◁ ''y'', then '''5801''' ◁ ''p''. If 85 ◁ ''y'', then '''5851''' ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ {5} ∪ {8}.

If ''y'' ∈ {0}, then ''p'' ∈ 5{0}1. But then, since the sum of the digits of ''p'' is 6, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' ∈ {5}, then ''p'' ∈ 5{5}1, and the smallest prime ''p'' ∈ 5{5}1 is '''555555555551'''.

If ''y'' ∈ {8}, since if 88 ◁ ''y'', then 881 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'',8}, and thus ''p'' ∈ {51,581}, but 51 and 581 are both composite.

Case 1.3: ''p'' begins with 8.

In this case we can write p = 8''y''1. If 2 ◁ ''y'', then '''821''' ◁ ''p''. If 8 ◁ ''y'', then '''881''' ◁ ''p''. If 9 ◁ ''y'', then 89 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 5.

If 50 ◁ ''y'', then '''8501''' ◁ ''p''. Hence we may assume y ∈ {0}{5}.

If 005 ◁ ''y'', then '''80051''' ◁ p. Hence we may assume y ∈ {0} ∪ {5} ∪ 0{5}.

If y ∈ {0}, then ''p'' ∈ 8{0}1. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ {5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'', 5, 55, 555, 5555, 55555, 555555, 5555555, 55555555, 555555555, 5555555555}, and thus ''p'' ∈ {81, 851, 8551, 85551, 855551, 8555551, 85555551, 855555551, 8555555551, 85555555551, 855555555551}, but all of these numbers are composite.

If y ∈ 0{5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {0, 05, 055, 0555, 05555, 055555, 0555555, 05555555, 055555555, 0555555555, 05555555555}, and thus ''p'' ∈ {801, 8051, 80551, 805551, 8055551, 80555551, 805555551, 8055555551, 80555555551, 805555555551, 8055555555551}, and of these numbers only 80555551 and 8055555551 are primes, but 80555551 ◁ 8055555551, thus only '''80555551''' is a minimal element.

Case 1.4: ''p'' begins with 9.

In this case we can write p = 9''y''1. If 9 ◁ ''y'', then '''991''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 2, 5, or 8.

If 00 ◁ ''y'', then '''9001''' ◁ ''p''. If 22 ◁ ''y'', then '''9221''' ◁ ''p''. If 55 ◁ ''y'', then '''9551''' ◁ ''p''. If 88 ◁ ''y'', then 881 ◁ ''p''. Hence we may assume ''y'' contains at most one 0, at most one 2, at most one 5, and at most one 8.

If ''y'' only contains at most one 0 and does not contain any of {2,5,8}, then ''y'' ∈ {''𝜆'',0}, and thus ''p'' ∈ {91,901}, but 91 and 901 are both composite. If ''y'' only contains at most one 0 and only one of {2,5,8}, then the sum of the digits of ''p'' is divisible by 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume ''y'' contains at least two of {2,5,8}.

If 25 ◁ ''y'', then 251 ◁ ''p''. If 28 ◁ ''y'', then 281 ◁ ''p''. If 52 ◁ ''y'', then 521 ◁ ''p''. If 82 ◁ ''y'', then 821 ◁ ''p''. Hence we may assume ''y'' contains no 2's (since if ''y'' contains 2, then ''y'' cannot contain either 5's or 8's, which is a contradiction).

If 85 ◁ ''y'', then '''9851''' ◁ ''p''. Hence we may assume ''y'' ∈ {58,580,508,058}, and thus ''p'' ∈ {9581,95801,95081,90581}, and of these numbers only 95801 is prime, but 95801 is not a minimal element since 5801 ◁ 95801.

Case 2: ''p'' ends with 3.

In this case we can write p = ''x''3. If ''x'' contains 1, 2, 4, 5, 7, or 8, then (respectively) '''13''' ◁ ''p'', '''23''' ◁ ''p'', '''43''' ◁ ''p'', '''53''' ◁ ''p'', '''73''' ◁ ''p'', or '''83''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 3: ''p'' ends with 7.

In this case we can write ''p'' = ''x''7. If ''x'' contains 1, 3, 4, 6, or 9, then (respectively) '''17''' ◁ ''p'', '''37''' ◁ ''p'', '''47''' ◁ ''p'', '''67''' ◁ ''p'', or '''97''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 7, or 8.

Case 3.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''7. If 2 ◁ ''y'', then '''227''' ◁ ''p''. If 5 ◁ ''y'', then '''257''' ◁ ''p''. If 7 ◁ ''y'', then '''277''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 8.

If 08 ◁ ''y'', then '''2087''' ◁ ''p''. If 88 ◁ ''y'', then 887 ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ 8{0}.

If ''y'' ∈ {0}, then ''p'' ∈ 2{0}7. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ 8{0}, then ''p'' ∈ 28{0}7. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 40&lt;sub&gt;''n''&lt;/sub&gt;1 = 280&lt;sub&gt;''n''&lt;/sub&gt;7.

Case 3.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''7. If 5 ◁ ''y'', then '''557''' ◁ ''p''. If 7 ◁ ''y'', then '''577''' ◁ ''p''. If 8 ◁ ''y'', then '''587''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then 227 ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 5{0}7. But then, since the sum of the digits of ''p'' is 12, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 5''z''2''w''7, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''50207''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 52{0}7, and the smallest prime ''p'' ∈ 52{0}7 is '''5200007'''.

If ''w'' is empty, then ''p'' ∈ 5{0}27, and the smallest prime ''p'' ∈ 5{0}27 is '''5000000000000000000000000000027'''.

Case 3.3: ''p'' begins with 7.

In this case we can write ''p'' = 7''y''7. If 2 ◁ ''y'', then '''727''' ◁ ''p''. If 5 ◁ ''y'', then '''757''' ◁ ''p''. If 8 ◁ ''y'', then '''787''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 7, and thus all digits of ''p'' are 0 or 7. But then, since the digits of ''p'' all have a common factor 7, ''p'' is divisible by 7, so ''p'' cannot be prime.

Case 3.4: ''p'' begins with 8.

In this case we can write ''p'' = 8''y''7. If 2 ◁ ''y'', then '''827''' ◁ ''p''. If 5 ◁ ''y'', then '''857''' ◁ ''p''. If 7 ◁ ''y'', then '''877''' ◁ ''p''. If 8 ◁ ''y'', then '''887''' ◁ ''p''. Hence we may assume ''y'' ∈ {0}, and thus ''p'' ∈ 8{0}7. But then, since the sum of the digits of ''p'' is 15, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 4: ''p'' ends with 9.

In this case we can write ''p'' = ''x''9. If ''x'' contains 1, 2, 5, 7, or 8, then (respectively) '''19''' ◁ ''p'', '''29''' ◁ ''p'', '''59''' ◁ ''p'', '''79''' ◁ ''p'', or '''89''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 4, 6, or 9.

If 44 ◁ ''x'', then '''449''' ◁ ''p''. Hence we may assume ''x'' contains zero or one 4's.

If x contains no 4's, then all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume that ''x'' contains exactly one 4.

Case 4.1: ''p'' begins with 3.

In this case we can write ''p'' = 3''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. We must have '''349''' ◁ ''p''.

Case 4.2: ''p'' begins with 4.

In this case we can write ''p'' = 4''y''9, where all digits of ''y'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''409''' ◁ ''p''. If 3 ◁ ''y'', then 43 ◁ ''p''. If 9 ◁ ''y'', then '''499''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}, and thus ''p'' ∈ 4{6}9. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 6&lt;sub&gt;''n''&lt;/sub&gt;7 = 46&lt;sub&gt;''n''&lt;/sub&gt;9.

Case 4.3: ''p'' begins with 6.

In this case we can write p = 6''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 6 ◁ ''z'', then '''6469''' ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' is empty.

If 3 ◁ ''y'', then 349 ◁ ''p''. If 9 ◁ ''y'', then '''6949''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 6.

If 06 ◁ ''y'', then '''60649''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}{0}.

If 666 ◁ ''y'', then '''666649''' ◁ ''p''. If 00000 ◁ ''y'', then '''60000049''' ◁ ''p''. Hence we may assume ''y'' ∈ {''𝜆'', 0, 00, 000, 0000, 6, 60, 600, 6000, 60000, 66, 660, 6600, 66000, 660000}, and thus ''p'' ∈ {649, 6049, 60049, 600049, 6000049, 6649, 66049, 660049, 6600049, 66000049, 66649, 666049, 6660049, 66600049, 666000049}, and of these numbers only '''66000049''' and '''66600049''' are primes.

Case 4.4: ''p'' begins with 9.

In this case we can write p = 9''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''9049''' ◁ ''p''. If 3 ◁ ''y'', then 349 ◁ ''p''. If 6 ◁ ''y'', then '''9649''' ◁ ''p''. If 9 ◁ ''y'', then '''9949''' ◁ ''p''. Hence we may assume ''y'' is empty.

If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' ∈ {6}, and thus ''p'' ∈ 94{6}9, and the smallest prime ''p'' ∈ 94{6}9 is 946669.

[[Category:Number theory]]</text>
      <sha1>mwg1oxzockplpgxuayvfc6z26dkko01</sha1>
    </revision>
    <revision>
      <id>2818462</id>
      <parentid>2818461</parentid>
      <timestamp>2026-07-17T17:06:44Z</timestamp>
      <contributor>
        <username>Athene241</username>
        <id>3100061</id>
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      <comment>/* Solve the problem */</comment>
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      <text bytes="52513" sha1="7zs8u4ipxw59y46o2sc3wocl1mq11fi" xml:space="preserve">{{mathematics}}
'''Athena problem''' is an [[:w:List of unsolved problems in mathematics|unsolved problem]] in [[:w:Number theory|number theory]] and [[:w:Formal language theory|formal language theory]] and [[:w:Order theory|order theory]], this problem is named after the ancient Greek goddess [[:w:Athena|Athena]] (which is associated with [[:w:Wisdom|wisdom]]). Athena problem is: Give a [[:w:Natural number|natural number]] ''b'' &gt; 1, find the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the set of the &quot;[[:w:Prime number|prime number]] [[:w:Greater than|&gt;]] ''b''&quot; [[:w:Numerical digit|digit]] [[:w:String (computer science)|string]]s in the [[:w:Positional numeral system|positional numeral system]] with [[:w:Radix|base]] ''b'' for the [[:w:Subsequence|subsequence]] [[:w:Partially ordered set|ordering]]. (A string ''x'' is a subsequence of another string ''y'', if ''x'' can be obtained from ''y'' by deleting zero or more of the [[:w:Character (computing)|character]]s in ''y''. For example, 514 is a subsequence of 352148, &quot;string&quot; is a subsequence of &quot;meistersinger&quot;. In contrast, 758 is not a subsequence of 378259, &quot;abc&quot; is not a subsequence of &quot;cbacacba&quot;, since the characters must be in the same order) (Unlike [[:w:Substring|substring]], subsequence is not required to occupy consecutive positions within the original sequences, e.g. the [[:w:Longest common subsequence|longest common subsequence problem]] is different from the [[:w:Longest common substring|longest common substring problem]])

Using [[:w:Formal language theory|formal language theory]] terminology, Athena problem is finding the [[:w:Set (mathematics)|set]] of the [[:w:Minimal element|minimal element]]s of the [[:w:Formal language|language]] of base-''b'' [[:w:Representation (mathematics)|representation]]s of the [[:w:Prime number|prime number]]s [[:w:Greater than|&gt;]] ''b'' (which is a set of [[:w:String (computer science)|string]]s of [[:w:Symbol|symbol]]s over the [[:w:Alphabet (formal languages)|alphabet]] ''Σ''&lt;sub&gt;''b''&lt;/sub&gt; := {0, 1, ..., ''b''−1}), under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), for a given natural number ''b'' &gt; 1. (You can draw this partial ordering as a [[:w:Hasse diagram|Hasse diagram]] to find all [[:w:Minimal element|minimal element]]s)

By [[:w:Higman's lemma|Higman's lemma]], there are no [[:w:Infinite set|infinite]] [[:w:Antichain|antichain]]s for the subsequence ordering (i.e. the subsequence ordering is always a [[:w:Well-quasi-ordering|well quasi order]]) (i.e. under the subsequence ordering (i.e. the [[:w:Binary relation|binary relation]] &quot;is a subsequence of&quot;, which is a [[:w:Partially ordered set|partial ordering]]), every set of pairwise incomparable (i.e. not [[:w:Comparability|comparable]]) strings is finite), thus there must be only finitely many such minimal elements. In other words, the set of such minimal elements must be a [[:w:Finite set|finite set]], e.g. in [[:w:Decimal|decimal]] (base ''b'' = 10), this set has exactly 77 [[:w:Element of a set|element]]s: {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}.

For bases 2 ≤ ''b'' ≤ 36, Athena problem is fully solved in bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24, and also solved in bases ''b'' = 11, 13, 16, 22, 30 if [[:w:Probable prime|probable prime]]s are allowed. For the unsolved bases ''b'' = 17, 19, 21, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, Athena problem is solved (if probable primes are allowed) except 771 [[:w:Indexed family|families]] of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be [[:w:Empty string|empty]]) of digits in base ''b'', ''y'' is a digit in base ''b'') = sequence {''xz'', ''xyz'', ''xyyz'', ''xyyyz'', ''xyyyyz'', ''xyyyyyz'', ...} (i.e. &quot;''xy''&lt;sup&gt;+&lt;/sup&gt;''z''&quot; in [[:w:Regular expression|regular expression]]), all of these 771 families contain no primes &gt; ''b'' or probable primes &gt; ''b'' with length ≤ 100000.

== Solve the problem ==

To solve the Athena problem for a given base ''b'', we must [[:w:Computing|compute]] the elements up to families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), and find the smallest prime &gt; ''b'' in all such families.

We call families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') &quot;linear&quot; families, and we reduce these families by removing all trailing digits ''y'' from ''x'', and removing all leading digits ''y'' from ''z'', to make the families be easier, e.g. family 12333{3}33345 in base ''b'' is reduced to family 12{3}45 in base ''b'', since they are in fact the same family. Our [[:w:Algorithm|algorithm]] then proceeds as follows:

* 1. ''M'' := {minimal primes in base ''b'' of length 2 or 3}, ''L'' := union of all ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'') such that ''x'' ≠ 0 and ''gcd''(''z'', ''b'') = 1 and ''Y'' is the set of digits ''y'' in base ''b'' such that ''xyz'' has no subsequence in ''M''.
* 2. While ''L'' contains nonlinear families (families which are not linear families): Explore each family of ''L'', and update ''L''. Examine each family of ''L'' by:
* 2.1. Let ''w'' be the shortest string in the family. If ''w'' has a subsequence in ''M'', then remove the family from ''L''. If ''w'' represents a prime, then add ''w'' to ''M'' and remove the family from ''L''.
* 2.2. If possible, simplify the family.
* 2.3. Using the techniques below (covering congruence, algebraic factorization, or combine of them), check if the family can be proven to only contain composites (only count the numbers &gt; ''b''), and if so then remove the family from ''L''.
* 3. Update ''L'', after each split examine the new families as in step 2.

e.g. in decimal (base ''b'' = 10):

''M'' := {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991}

''L'' := {2{0,2}1, 2{0,8}7, 3{0,3,6,9}3, 3{0,3,6,9}9, 4{6}9, 5{0,5,8}1, 5{0,2}7, 6{0,3,6,9}3, 6{0,3,4,6,9}9, 7{0,7}7, 8{0,5}1, 8{0}7, 9{0,2,5,8}1, 9{0,3,6,9}3, 9{0,3,4,6,9}9}

and since 2221 is prime, it follows that the family 2{0,2}1 splits into the families 2{0}1 and 2{0}2{0}1

and since the family 2{0}1 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

and since 20201 is prime, it follows that the family 2{0}2{0}1 splits into the families 2{0}21 and 22{0}1

221 and 2021 are composites, but 20021 is prime, thus add 20021 to ''L''

none of 221, 2201, 22001, 220001, 2200001 are primes, but 22000001 is prime, thus add 22000001 to ''L''

and since the family 3{0,3,6,9}3 can be proven to contain no primes &gt; base (since all numbers in this family are divisible by 3), it can be removed

etc.

Since the number of possible (first digit,last digit) (also called (initial digit,final digit)) combos ([[:w:Ordered pair|ordered pair]]s) of a prime &gt; ''b'' in base ''b'' is (''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(''b'') (all digits except 0 can be the first digit of a prime &gt; ''b'' in base ''b'' (thus ''b''−1 possible digits), but only the digits coprime to ''b'' can be the last digit of a prime &gt; ''b'' in base ''b'' (thus ''eulerphi''(''b'') possible digits), and by the [[:w:Rule of product|rule of product]], there are (''b''−1)×''eulerphi''(''b'') choices of the (first digit,last digit) combo, also, both &quot;numbers of primes in the set of the Athena problem in base ''b''&quot; and &quot;length of the largest prime in the set of the Athena problem in base ''b''&quot; are [[:w:Asymptotic analysis|roughly]] ''[[:w:E (mathematical_constant)|e]]''&lt;sup&gt;''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)&lt;/sup&gt;.

Shrinking the family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'')

* If ''y'' ∈ ''Y'' and the string ''xyyz'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''}''z'' ∪ ''x''{''Y'' \ ''y''}''y''{''Y'' \ ''y''}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and the string ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' represents a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or has a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.
* If ''y''&lt;sub&gt;1&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;2&lt;/sub&gt; ∈ ''Y'' and ''y''&lt;sub&gt;1&lt;/sub&gt; ≠ ''y''&lt;sub&gt;2&lt;/sub&gt; and both the strings ''xy''&lt;sub&gt;1&lt;/sub&gt;''y''&lt;sub&gt;2&lt;/sub&gt;''z'' and ''xy''&lt;sub&gt;2&lt;/sub&gt;''y''&lt;sub&gt;1&lt;/sub&gt;''z'' represent a prime &gt; ''b'' in base ''b'' (in this case, add this prime to the list) or have a subsequence which represents a prime &gt; ''b'' in base ''b'', then ''x''{''Y''}''z'' can be replaced with ''x''{''Y'' \ ''y''&lt;sub&gt;1&lt;/sub&gt;}''z'' ∪ ''x''{''Y'' \ ''y''&lt;sub&gt;2&lt;/sub&gt;}''z''.

e.g. in decimal (base ''b'' = 10):

* 2221 is a prime &gt; 10, thus the family 2{0,2}1 splits into the two families 2{0}1 and 2{0}2{0}1.
* 227 is a prime &gt; 10, and it is a subsequence of 5227, thus the family 5{0,2}7 splits into the two families 5{0}7 and 5{0}2{0}7.
* 449 is a prime &gt; 10, and it is a subsequence of 6449, thus the family 6{0,3,4,6,9}9 splits into the two families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9.
* Both 5051 and 5501 are primes &gt; 10, thus the family 5{0,5}1 splits into the two families 5{0}1 and 5{5}1 = {5}1.
* 8501 is a prime &gt; 10, thus the family 8{0,5}1 splits into the family 8{0}{5}1.
* 887 is a prime &gt; 10, and it is a subsequence of 2887, also 2087 is a prime &gt; 10, thus the family 2{0,8}7 splits into the two families 2{0}7 and 28{0}7.
* 349 and 449 are primes &gt; 10, and they are subsequences of 9349 and 9449, respectively, also 9049, 9649, 9949 are primes &gt; 10, thus the family 9{0,3,4,6,9}9 splits into the two families 9{0,3,6,9}9 and 94{0,3,6,9}9.
* 251, 281, 521, 821, 881 are primes &gt; 10, and they are subsequences of 9251, 9281, 9521, 9821, 9881, respectively, also 9001, 9221, 9551, 9851 are primes &gt; 10, thus the family 9{0,2,5,8}1 splits into the numbers {91, 901, 921, 951, 981, 9021, 9051, 9081, 9201, 9501, 9581, 9801, 90581, 95081, 95801}.

If the methods we have discussed cannot be used to rule out or shrink ''x''{''Y''}''z'' where ''Y'' = {''y''&lt;sub&gt;1&lt;/sub&gt;, ''y''&lt;sub&gt;2&lt;/sub&gt;, ..., ''y''&lt;sub&gt;''n''&lt;/sub&gt;}, then we can replace ''x''{''Y''}''z'' by ''xy''&lt;sub&gt;1&lt;/sub&gt;{''Y''}''z'' ∪ ''xy''&lt;sub&gt;2&lt;/sub&gt;{''Y''}''z'' ∪ ... ∪ ''xy''&lt;sub&gt;''n''&lt;/sub&gt;{''Y''}''z'' and re-run the methods on this new [[:w:Formal language|language]].

If all remain families are linear families (i.e. of the form ''x''{''y''}''z'', where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b''), then we search the smallest (probable) primes in these families and add these primes to the list.

e.g. in decimal (base ''b'' = 10):

* The smallest prime in the family 5{0}27 is 5000000000000000000000000000027.
* The smallest prime in the family {5}1 is 555555555551.
* The smallest prime in the family 8{5}1 is 8555555555555555555551, but 8555555555555555555551 is not a minimal element since 555555555551 is a subsequence of 8555555555555555555551.

There is no guarantee that the techniques discussed will ever terminate, but in practice they often do. They are able to determine the set of the minimal elements in base ''b'' for 2 ≤ ''b'' ≤ 16 and ''b'' = 18, 20, 22, 24, 30. The bases ''b'' = 17, 19, 21, 23, 25 ≤ ''b'' ≤ 29, 31 ≤ ''b'' ≤ 36 are solved with the exception of 771 families of the form ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'').

The following is a &quot;[[:w:Semi-algorithm|semi-algorithm]]&quot; that is guaranteed to solve the Athena problem for a given base ''b'', but it is not so easy to implement:

# ''M'' = ''[[:w:Empty string|∅]]''
# while (''L'' ≠ ''∅'') do
# choose ''x'', a shortest string in ''L''
# ''M'' := ''M'' ∪ {''x''}
# ''L'' := ''L'' − ''sup''({''x''})

In practice, for arbitrary ''L'', we cannot feasibly carry out step 5. Instead, we work with ''L''&amp;#39;, some regular overapproximation to ''L'', until we can show ''L''&amp;#39; = ''∅'' (which implies ''L'' = ''∅''). In practice, ''L''&amp;#39; is usually chosen to be a finite [[:w:Union (set theory)|union]] of sets of the form ''L''&lt;sub&gt;1&lt;/sub&gt;{''L''&lt;sub&gt;2&lt;/sub&gt;}''L''&lt;sub&gt;3&lt;/sub&gt;, where each of ''L''&lt;sub&gt;1&lt;/sub&gt;, ''L''&lt;sub&gt;2&lt;/sub&gt;, ''L''&lt;sub&gt;3&lt;/sub&gt; is finite. In the case we consider in this project, we then have to determine whether such a family contains a prime or not.

Thus, the [[:w:Time complexity|time complexity]] of the Athena problem in base ''b'' may be ''[[:w:Big O notation|O]]''(''[[:w:E (mathematical_constant)|e]]''&lt;sup&gt;''[[:w:Euler's constant|γ]]''×(''b''−1)×''[[:w:Euler's totient function|eulerphi]]''(*b*)&lt;/sup&gt;), and the [[:w:CPU time|CPU time]] of the Athena problem in base ''b'' may be longer than [[:w:Age of the universe|the age of the universe]] for bases ''b'' = 19, 23, 25, 27, 29, 31, 32, 33, 34, 35, also, Athena problem in bases ''b'' around 500 may be [[:w:NP-complete|NP-complete]] or [[:w:NP-hard|NP-hard]], or an [[:w:Undecidable problem|undecidable problem]], or an example of [[:w:Gödel's incompleteness theorems|Gödel's incompleteness theorems]] (like the [[:w:Continuum hypothesis|continuum hypothesis]] and the [[:w:Halting problem|halting problem]]).

To solve the Athena problem, we need to determine whether a given family contains a prime. In practice, if family ''x''{''Y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''Y'' is a set of digits in base ''b'') could not be ruled out as only containing composites and ''Y'' contains two or more digits, then a relatively small prime &gt; ''b'' could always be found in this family. Intuitively, this is because there are a large number of small strings in such a family, and at least one is likely to be prime (e.g. there are 2&lt;sup&gt;''n''−2&lt;/sup&gt; strings of length ''n'' in the family 1{3,7}9, and there are over a thousand strings of length 12 in the family 1{3,7}9, thus it is very impossible that these numbers are all composite). In the case ''Y'' contains only one digit, this family is of the form ''x''{''y''}''z'', and there is only a single string of each length &gt; (the length of ''x'' + the length of ''z''), and it is not known if the following [[:w:Decision problem|decision problem]] is recursively solvable (just like [[:w:Sierpiński number|Sierpiński problem]] and [[:w:Riesel number|Riesel problem]], Sierpiński problem and Riesel problem can be generalized to other bases ''b'', in fact, Athena problem base ''b'' covers the Sierpiński problem base ''b'' and the Riesel problem base ''b'' with ''k'' &lt; ''b'', i.e. finding the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (or prove such prime does not exist) with ''k'' &lt; ''b'' (specially, for bases ''b'' such that the conjectured smallest Sierpiński number or the conjectured smallest Riesel number is &lt; ''b'', Athena problem base ''b'' covers the Sierpiński problem base ''b'' or the Riesel problem base ''b'', respectively), since the smallest prime of the form ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 and ''k''×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (if exists) must be a minimal element in base ''b'', also, Athena problem base ''b'' covers finding the smallest prime of these forms in base ''b'' (or proving that such prime does not exist): (''b''&lt;sup&gt;''n''&lt;/sup&gt;−1)/(''b''−1) (for this form, ''n'' must be prime, and we want ''n'' ≥ 2), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 1), (''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/2 (for odd ''b'') (for this form, ''n'' must be power of 2, and we want ''n'' ≥ 2), (''sqrt''(''b'')×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''sqrt''(''b'')+1) (for square ''b'') (for this form, 2×''n''+1 must be prime, and we want ''n'' ≥ 2), ((''b''−2)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1)/(''b''−1) (''n'' ≥ 2), 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (''n'' ≥ 1), 2×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (''n'' ≥ 1), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+2 (''n'' ≥ 1), ''b''&lt;sup&gt;''n''&lt;/sup&gt;−2 (''n'' ≥ 2), (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;+1 (''n'' ≥ 1), (''b''−1)×''b''&lt;sup&gt;''n''&lt;/sup&gt;−1 (''n'' ≥ 1), ''b''&lt;sup&gt;''n''&lt;/sup&gt;+(''b''−1) (''n'' ≥ 1), ''b''&lt;sup&gt;''n''&lt;/sup&gt;−(''b''−1) (''n'' ≥ 2)):

Problem: Given strings ''x'', ''z'' (may be empty), a digit ''y'', and a base ''b'' (''x'' does not [[:w:Leading zero|start with the digit 0]], ''z'' ends with a digit which [[:w:Coprime integers|coprime]] to ''b'', ''y'' is not 0 if ''x'' is empty, ''y'' is coprime to ''b'' if ''z'' is empty), does there exist a prime number whose base-''b'' expansion is of the form ''xy''&lt;sub&gt;''n''&lt;/sub&gt;''z'' for some ''n'' ≥ 0?

Some families can be ruled out to contain no prime &gt; ''b'' by [[:w:Covering set|covering congruence]], [[:w:Factorization of polynomials|algebraic factorization]] (e.g. [[:w:Difference of two squares|difference of two squares]], [[:w:Sum of two cubes|sum of two cubes]], [[:w:Sophie Germain's identity|Sophie Germain's identity of ''x''&lt;sup&gt;4&lt;/sup&gt;+4×''y''&lt;sup&gt;4&lt;/sup&gt;]]), or combine of them, e.g.

* The base 9 family 2{7}: Always divisible by 2 or 5
* The base 16 family {8}F: Always divisible by 3, 7, or 13
* The base 21 family {7}D: Always divisible by 2, 13, or 17
* The base 23 family {D}GA: Always divisible by 2, 5, 7, 37, or 79
* The base 9 family 3{8}: Can be written as 4×9&lt;sup&gt;''n''&lt;/sup&gt;−1 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−1) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 8 family 1{0}1: Can be written as 8&lt;sup&gt;''n''&lt;/sup&gt;+1 and can be factored as (2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (4&lt;sup&gt;''n''&lt;/sup&gt;−2&lt;sup&gt;''n''&lt;/sup&gt;+1)
* The base 16 family {4}1: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;−49)/15 and can be factored as (2×3&lt;sup&gt;''n''&lt;/sup&gt;−7) × (2×3&lt;sup&gt;''n''&lt;/sup&gt;+7) / 15
* The base 16 family {C}D: Can be written as (4×16&lt;sup&gt;''n''&lt;/sup&gt;+1)/5 and can be factored as (2×4&lt;sup&gt;''n''&lt;/sup&gt;−2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) × (2×4&lt;sup&gt;''n''&lt;/sup&gt;+2×2&lt;sup&gt;''n''&lt;/sup&gt;+1) / 5
* The base 14 family 8{D}: Can be written as 9×14&lt;sup&gt;''n''&lt;/sup&gt;−1, it is divisible by 5 if ''n'' is odd and can be factored as (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (3×14&lt;sup&gt;''n''/2&lt;/sup&gt;+1) if ''n'' is even
* The base 12 family {B}9B: Can be written as 12&lt;sup&gt;''n''&lt;/sup&gt;−25, it is divisible by 13 if ''n'' is odd and can be factored as (12&lt;sup&gt;''n''/2&lt;/sup&gt;−5) × (12&lt;sup&gt;''n''/2&lt;/sup&gt;+5) if ''n'' is even
* The base 17 family 1{9}: Can be written as (25×17&lt;sup&gt;''n''&lt;/sup&gt;−9)/16, it is divisible by 2 if ''n'' is odd and can be factored as (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;−3) × (5×17&lt;sup&gt;''n''/2&lt;/sup&gt;+3) / 16 if ''n'' is even
* The base 19 family 1{6}: Can be written as (4×19&lt;sup&gt;''n''&lt;/sup&gt;−1)/3, it is divisible by 5 if ''n'' is odd and can be factored as (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;−1) × (2×19&lt;sup&gt;''n''/2&lt;/sup&gt;+1) / 3 if ''n'' is even

By the [[:w:Prime number theorem|prime number theorem]], the [[:w:Probability|chance]] that a [[:w:Random number|random]] ''n''-digit base ''b'' number is prime is [[:w:Asymptotic analysis|approximately]] 1/''n'' (more accurately, the chance is approximately 1/(''n''×''ln''(''b'')), where ''ln'' is the [[:w:Natural logarithm|natural logarithm]]). If one conjectures the numbers ''x''{''y''}''z'' behave similarly (i.e. the numbers ''x''{''y''}''z'' is a [[:w:Pseudorandomness|pseudorandom sequence]]) you would expect [[:w:Harmonic_series (mathematics)|1/1 + 1/2 + 1/3 + 1/4 + ... = ∞]] primes of the form ''x''{''y''}''z'' (of course, this does not always happen, since some ''x''{''y''}''z'' families can be ruled out to contain no prime &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them), but it is at least a reasonable conjecture in the absence of evidence to the contrary. Hence, the [[:w:Heuristic argument|heuristic argument]] suggests there are always infinitely many primes in family ''x''{''y''}''z'' (where ''x'' and ''z'' are strings (may be empty) of digits in base ''b'', ''y'' is a digit in base ''b'') if it cannot be ruled out to contain no prime or only contain finitely many primes, by covering congruence, algebraic factorization, or combine of them. However, some families ''x''{''y''}''z'' could not be proven to contain no primes &gt; ''b'' (by covering congruence, algebraic factorization, or combine of them) but no primes &gt; ''b'' could be found in the family, even after searching through numbers with over 100000 digits. In such a case, the only way to proceed is to [[:w:Primality test|test the primality]] of larger and larger numbers of such form and hope a prime is eventually discovered. e.g. the smallest (probable) prime in the family A{3}A in base ''b'' = 13 is A3&lt;sub&gt;592197&lt;/sub&gt;A, its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4, when written in decimal contains 659677 digits (it is only probable prime, i.e. not definitely prime).

== Data ==

These are the results of the Athena problem in bases 2 ≤ ''b'' ≤ 36 (we stop at base 36 since this base is the maximum base for which it is possible to write the numbers with the [[:w:Symbol|symbol]]s 0, 1, 2, ..., 9 and A, B, C, ..., Z (i.e. the 10 [[:w:Arabic numerals|Arabic numerals]] and the 26 [[:w:Latin script|Latin letters]]): (some large primes are only probable primes, i.e. not definitely primes, since they are too large to be [[:w:Elliptic curve primality|ECPP proved]] and [[:w:Pocklington primality test#Extensions and variants|neither ''N''−1 nor ''N''+1 can be ≥ 1/3 factored]], all of them pass the [[:w:Baillie–PSW primality test|Baillie–PSW primality test]] and the [[:w:Strong pseudoprime|strong primality test]] (i.e. the [[:w:Miller–Rabin primality test|Miller–Rabin primality test]]) with all prime bases ''p'' ≤ 61, however, all primes &lt; 10&lt;sup&gt;25000&lt;/sup&gt; for bases ''b'' = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 36 are definitely primes, most of them &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Elliptic curve primality|ECPP proving]], others &gt; 10&lt;sup&gt;299&lt;/sup&gt; are proven primes with [[:w:Pocklington primality test#Extensions and variants|''N''−1 or ''N''+1 proving]])

All numbers are written in base ''b'', [[:w:Senary#Base 36 as senary compression|using A to Z to represent digit values 10 to 35]], &quot;{}&quot; means repeating, e.g. family 12{3}45 means the sequence {1245, 12345, 123345, 1233345, 12333345, 123333345, ...} (where the members are expressed as base ''b'' strings), subscripts are used to indicate repetitions of digits, e.g. 123&lt;sub&gt;4&lt;/sub&gt;567 means 123333567 (all subscripts are written in decimal).

Base 2: 1 prime (the largest of which has 2 digits (it is 11, and its value is 3 in decimal)): {11}

Base 3: 3 primes (the largest of which has 3 digits (it is 111, and its value is 13 in decimal)): {12, 21, 111}

Base 4: 5 primes (the largest of which has 3 digits (it is 221, and its value is 41 in decimal)): {11, 13, 23, 31, 221}

Base 5: 22 primes (the largest of which has 96 digits (it is 10&lt;sub&gt;93&lt;/sub&gt;13, and its algebraic form is 5&lt;sup&gt;95&lt;/sup&gt;+8)): {12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013}

Base 6: 11 primes (the largest of which has 5 digits (it is 40041, and its value is 5209 in decimal)): {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041}

Base 7: 71 primes (the largest of which has 17 digits (it is 3&lt;sub&gt;16&lt;/sub&gt;1, and its algebraic form is (7&lt;sup&gt;17&lt;/sup&gt;−5)/2)): {14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331}

Base 8: 75 primes (the largest of which has 221 digits (it is 4&lt;sub&gt;220&lt;/sub&gt;7, and its algebraic form is (4×8&lt;sup&gt;221&lt;/sup&gt;+17)/7)): {13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447}

Base 9: 151 primes (the largest of which has 1161 digits (it is 30&lt;sub&gt;1158&lt;/sub&gt;11, and its algebraic form is 3×9&lt;sup&gt;1160&lt;/sup&gt;+10)): {12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011}

Base 10: 77 primes (the largest of which has 31 digits (it is 50&lt;sub&gt;28&lt;/sub&gt;27, and its algebraic form is 5×10&lt;sup&gt;30&lt;/sup&gt;+27)): {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}

Base 11: 1068 primes (including 1 unproven probable prime: 57&lt;sub&gt;62668&lt;/sub&gt;), the largest of which has 62669 digits (it is 57&lt;sub&gt;62668&lt;/sub&gt;, and its algebraic form is (57×11&lt;sup&gt;62668&lt;/sup&gt;−7)/10), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel11 Data of Athena problem base 11]

Base 12: 106 primes (the largest of which has 42 digits (it is 40&lt;sub&gt;39&lt;/sub&gt;77, and its algebraic form is 4×12&lt;sup&gt;41&lt;/sup&gt;+91)): {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, BBBBBB99B, B0000000000000000000000000009B, 400000000000000000000000000000000000000077}

Base 13: 3197 primes (including 4 unproven probable primes: C5&lt;sub&gt;23755&lt;/sub&gt;C, 80&lt;sub&gt;32017&lt;/sub&gt;111, 95&lt;sub&gt;197420&lt;/sub&gt;, A3&lt;sub&gt;592197&lt;/sub&gt;A), the largest of which has 592199 digits (it is A3&lt;sub&gt;592197&lt;/sub&gt;A, and its algebraic form is (41×13&lt;sup&gt;592198&lt;/sup&gt;+27)/4), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel13 Data of Athena problem base 13]

Base 14: 650 primes, the largest of which has 19699 digits (it is 4D&lt;sub&gt;19698&lt;/sub&gt;, and its algebraic form is 5×14&lt;sup&gt;19698&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel14 Data of Athena problem base 14]

Base 15: 1284 primes, the largest of which has 157 digits (it is 7&lt;sub&gt;155&lt;/sub&gt;97, and its algebraic form is (15&lt;sup&gt;157&lt;/sup&gt;+59)/2), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel15 Data of Athena problem base 15]

Base 16: 2347 primes (including 3 unproven probable primes: DB&lt;sub&gt;32234&lt;/sub&gt;, 4&lt;sub&gt;72785&lt;/sub&gt;DD, 3&lt;sub&gt;116137&lt;/sub&gt;AF), the largest of which has 116139 digits (it is 3&lt;sub&gt;116137&lt;/sub&gt;AF, and its algebraic form is (16&lt;sup&gt;116139&lt;/sup&gt;+619)/5), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel16 Data of Athena problem base 16]

Base 17: 10415 known primes (including many unproven probable primes) and 12 unsolved families (1{7}, 1F{0}7, 4{7}A, 70F{0}D, 8{B}9, 9{5}9, A{D}F, B{0}B3, {B}E9, {B}EE, F1{9}, FD0{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel17 Data of Athena problem base 17] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left17 Data of unsolved families for base 17]

Base 18: 549 primes, the largest of which has 6271 digits (it is C0&lt;sub&gt;6268&lt;/sub&gt;C5, and its algebraic form is 12×18&lt;sup&gt;6270&lt;/sup&gt;+221), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel18 Data of Athena problem base 18]

Base 19: 31417 known primes (including many unproven probable primes) and 17 unsolved families (4B5{0}H, {5}3, 5{H}05, 5{H}0H, 5{H}5, 66{B}, 71{0}177, 7AF{0}H, 97{0}3, C{H}C, EE1{6}, F{7}5, F{B}G, F{D}F, H0F{0}7A, HB{0}5B5, II{D}, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel19 Data of Athena problem base 19] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left19 Data of unsolved families for base 19]

Base 20: 3314 primes, the largest of which has 6271 digits (it is G0&lt;sub&gt;6269&lt;/sub&gt;D, and its algebraic form is 16×20&lt;sup&gt;6270&lt;/sup&gt;+13), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel20 Data of Athena problem base 20]

Base 21: 13386 known primes (including many unproven probable primes) and 8 unsolved families (5{0}DJ, {9}D, B3{0}EB, B{H}6H, C{F}0K, {F}35, G{0}FK, H{0}7771, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel21 Data of Athena problem base 21] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left21 Data of unsolved families for base 21]

Base 22: 8003 primes (including 1 unproven probable prime: BK&lt;sub&gt;22001&lt;/sub&gt;5), the largest of which has 22003 digits (it is BK&lt;sub&gt;22001&lt;/sub&gt;5, and its algebraic form is (251×22&lt;sup&gt;22002&lt;/sup&gt;−335)/21), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel22 Data of Athena problem base 22]

Base 23: 65178 known primes (including many unproven probable primes) and 87 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel23 Data of Athena problem base 23] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left23 Data of unsolved families for base 23]

Base 24: 3409 primes, the largest of which has 8134 digits (it is N00N&lt;sub&gt;8129&lt;/sub&gt;LN, and its algebraic form is 13249×24&lt;sup&gt;8131&lt;/sup&gt;−49), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel24 Data of Athena problem base 24]

Base 25: 133639 known primes (including many unproven probable primes) and 85 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel25 Data of Athena problem base 25] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left25 Data of unsolved families for base 25]

Base 26: 25256 known primes (including 7 unproven probable primes: 5&lt;sub&gt;19391&lt;/sub&gt;6F, 7&lt;sub&gt;20279&lt;/sub&gt;OL, LD0&lt;sub&gt;20975&lt;/sub&gt;7, 6K&lt;sub&gt;23300&lt;/sub&gt;5, J0&lt;sub&gt;44303&lt;/sub&gt;KCB, M0&lt;sub&gt;61186&lt;/sub&gt;2BB, 85M&lt;sub&gt;197060&lt;/sub&gt;B) and 3 unsolved families ({A}6F, {H}MH, {I}GL, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel26 Data of Athena problem base 26]

Base 27: 102852 known primes (including many unproven probable primes) and 44 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel27 Data of Athena problem base 27] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left27 Data of unsolved families for base 27]

Base 28: 25528 known primes (including 3 unproven probable primes: N6&lt;sub&gt;24051&lt;/sub&gt;LR, 5OA&lt;sub&gt;31238&lt;/sub&gt;F, O4O&lt;sub&gt;94535&lt;/sub&gt;9) and 1 unsolved family (O{A}F, no primes or probable primes with length ≤ 900000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel28 Data of Athena problem base 28]

Base 29: 355242 known primes (including many unproven probable primes) and 125 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel29 Data of Athena problem base 29] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left29 Data of unsolved families for base 29]

Base 30: 2619 primes (including 1 unproven probable prime: I0&lt;sub&gt;24608&lt;/sub&gt;D), the largest of which has 34206 digits (it is OT&lt;sub&gt;34205&lt;/sub&gt;, and its algebraic form is 25×30&lt;sup&gt;34205&lt;/sup&gt;−1), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel30 Data of Athena problem base 30]

Base 31: 569323 known primes (including many unproven probable primes) and 77 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel31 Data of Athena problem base 31] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left31 Data of unsolved families for base 31]

Base 32: 168882 known primes (including many unproven probable primes) and 120 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel32 Data of Athena problem base 32] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left32 Data of unsolved families for base 32]

Base 33: 280012 known primes (including many unproven probable primes) and 81 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel33 Data of Athena problem base 33] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left33 Data of unsolved families for base 33]

Base 34: 184785 known primes (including many unproven probable primes) and 47 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel34 Data of Athena problem base 34] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left34 Data of unsolved families for base 34]

Base 35: 720002 known primes (including many unproven probable primes) and 60 unsolved families (no primes or probable primes with length ≤ 100000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel35 Data of Athena problem base 35] and [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/left35 Data of unsolved families for base 35]

Base 36: 35286 known primes (including 3 unproven probable primes: 7K&lt;sub&gt;26567&lt;/sub&gt;Z, S0&lt;sub&gt;75007&lt;/sub&gt;8H, P&lt;sub&gt;81993&lt;/sub&gt;SZ) and 4 unsolved families (B{0}EUV, HM{0}N, N{0}YYN, O{L}Z, no primes or probable primes with length ≤ 200000, nor can be proven to only contain composites), see [https://raw.githubusercontent.com/xayahrainie4793/minimal-elements-of-the-prime-numbers/main/kernel36 Data of Athena problem base 36]

== The fully proof of Athena problem in decimal (base ''b'' = 10) ==

'''Bold''' for the minimal elements, ''x'' ◁ ''y'' means ''x'' is a subsequence of ''y''.

Assume ''p'' is a prime &gt; 10, and the last digit of ''p'' must lie in {1,3,7,9}.

Case 1: ''p'' ends with 1.

In this case we can write ''p'' = ''x''1. If ''x'' contains 1, 3, 4, 6, or 7, then (respectively) '''11''' ◁ ''p'', '''31''' ◁ ''p'', '''41''' ◁ ''p'', '''61''' ◁ ''p'', or '''71''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 8, or 9.

Case 1.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''1. If 5 ◁ ''y'', then '''251''' ◁ ''p''. If 8 ◁ ''y'', then '''281''' ◁ ''p''. If 9 ◁ ''y'', then 29 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then '''2221''' ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 2{0}1. But then, since the sum of the digits of ''p'' is 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 2''z''2''w''1, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''20201''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 22{0}1, and the smallest prime ''p'' ∈ 22{0}1 is '''22000001'''.

If ''w'' is empty, then ''p'' ∈ 2{0}21, and the smallest prime ''p'' ∈ 2{0}21 is '''20021'''.

Case 1.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''1. If 2 ◁ ''y'', then '''521''' ◁ ''p''. If 9 ◁ ''y'', then 59 ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 5, or 8.

If 05 ◁ ''y'', then '''5051''' ◁ ''p''. If 08 ◁ ''y'', then '''5081''' ◁ ''p''. If 50 ◁ ''y'', then '''5501''' ◁ ''p''. If 58 ◁ ''y'', then '''5581''' ◁ ''p''. If 80 ◁ ''y'', then '''5801''' ◁ ''p''. If 85 ◁ ''y'', then '''5851''' ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ {5} ∪ {8}.

If ''y'' ∈ {0}, then ''p'' ∈ 5{0}1. But then, since the sum of the digits of ''p'' is 6, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' ∈ {5}, then ''p'' ∈ 5{5}1, and the smallest prime ''p'' ∈ 5{5}1 is '''555555555551'''.

If ''y'' ∈ {8}, since if 88 ◁ ''y'', then 881 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'',8}, and thus ''p'' ∈ {51,581}, but 51 and 581 are both composite.

Case 1.3: ''p'' begins with 8.

In this case we can write p = 8''y''1. If 2 ◁ ''y'', then '''821''' ◁ ''p''. If 8 ◁ ''y'', then '''881''' ◁ ''p''. If 9 ◁ ''y'', then 89 ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 5.

If 50 ◁ ''y'', then '''8501''' ◁ ''p''. Hence we may assume y ∈ {0}{5}.

If 005 ◁ ''y'', then '''80051''' ◁ p. Hence we may assume y ∈ {0} ∪ {5} ∪ 0{5}.

If y ∈ {0}, then ''p'' ∈ 8{0}1. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ {5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {''𝜆'', 5, 55, 555, 5555, 55555, 555555, 5555555, 55555555, 555555555, 5555555555}, and thus ''p'' ∈ {81, 851, 8551, 85551, 855551, 8555551, 85555551, 855555551, 8555555551, 85555555551, 855555555551}, but all of these numbers are composite.

If y ∈ 0{5}, since if 55555555555 ◁ ''y'', then 555555555551 ◁ ''p'', hence we may assume ''y'' ∈ {0, 05, 055, 0555, 05555, 055555, 0555555, 05555555, 055555555, 0555555555, 05555555555}, and thus ''p'' ∈ {801, 8051, 80551, 805551, 8055551, 80555551, 805555551, 8055555551, 80555555551, 805555555551, 8055555555551}, and of these numbers only 80555551 and 8055555551 are primes, but 80555551 ◁ 8055555551, thus only '''80555551''' is a minimal element.

Case 1.4: ''p'' begins with 9.

In this case we can write p = 9''y''1. If 9 ◁ ''y'', then '''991''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0, 2, 5, or 8.

If 00 ◁ ''y'', then '''9001''' ◁ ''p''. If 22 ◁ ''y'', then '''9221''' ◁ ''p''. If 55 ◁ ''y'', then '''9551''' ◁ ''p''. If 88 ◁ ''y'', then 881 ◁ ''p''. Hence we may assume ''y'' contains at most one 0, at most one 2, at most one 5, and at most one 8.

If ''y'' only contains at most one 0 and does not contain any of {2,5,8}, then ''y'' ∈ {''𝜆'',0}, and thus ''p'' ∈ {91,901}, but 91 and 901 are both composite. If ''y'' only contains at most one 0 and only one of {2,5,8}, then the sum of the digits of ''p'' is divisible by 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume ''y'' contains at least two of {2,5,8}.

If 25 ◁ ''y'', then 251 ◁ ''p''. If 28 ◁ ''y'', then 281 ◁ ''p''. If 52 ◁ ''y'', then 521 ◁ ''p''. If 82 ◁ ''y'', then 821 ◁ ''p''. Hence we may assume ''y'' contains no 2's (since if ''y'' contains 2, then ''y'' cannot contain either 5's or 8's, which is a contradiction).

If 85 ◁ ''y'', then '''9851''' ◁ ''p''. Hence we may assume ''y'' ∈ {58,580,508,058}, and thus ''p'' ∈ {9581,95801,95081,90581}, and of these numbers only 95801 is prime, but 95801 is not a minimal element since 5801 ◁ 95801.

Case 2: ''p'' ends with 3.

In this case we can write p = ''x''3. If ''x'' contains 1, 2, 4, 5, 7, or 8, then (respectively) '''13''' ◁ ''p'', '''23''' ◁ ''p'', '''43''' ◁ ''p'', '''53''' ◁ ''p'', '''73''' ◁ ''p'', or '''83''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 3: ''p'' ends with 7.

In this case we can write ''p'' = ''x''7. If ''x'' contains 1, 3, 4, 6, or 9, then (respectively) '''17''' ◁ ''p'', '''37''' ◁ ''p'', '''47''' ◁ ''p'', '''67''' ◁ ''p'', or '''97''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 2, 5, 7, or 8.

Case 3.1: ''p'' begins with 2.

In this case we can write ''p'' = 2''y''7. If 2 ◁ ''y'', then '''227''' ◁ ''p''. If 5 ◁ ''y'', then '''257''' ◁ ''p''. If 7 ◁ ''y'', then '''277''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 8.

If 08 ◁ ''y'', then '''2087''' ◁ ''p''. If 88 ◁ ''y'', then 887 ◁ ''p''. Hence we may assume ''y'' ∈ {0} ∪ 8{0}.

If ''y'' ∈ {0}, then ''p'' ∈ 2{0}7. But then, since the sum of the digits of ''p'' is 9, ''p'' is divisible by 3, so ''p'' cannot be prime.

If y ∈ 8{0}, then ''p'' ∈ 28{0}7. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 40&lt;sub&gt;''n''&lt;/sub&gt;1 = 280&lt;sub&gt;''n''&lt;/sub&gt;7.

Case 3.2: ''p'' begins with 5.

In this case we can write ''p'' = 5''y''7. If 5 ◁ ''y'', then '''557''' ◁ ''p''. If 7 ◁ ''y'', then '''577''' ◁ ''p''. If 8 ◁ ''y'', then '''587''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 2.

If 22 ◁ ''y'', then 227 ◁ ''p''. Hence we may assume ''y'' contains zero or one 2's.

If ''y'' contains no 2's, then ''p'' ∈ 5{0}7. But then, since the sum of the digits of ''p'' is 12, ''p'' is divisible by 3, so ''p'' cannot be prime.

If ''y'' contains exactly one 2, then we can write ''p'' = 5''z''2''w''7, where ''z'',''w'' ∈ {0}. If 0 ◁ ''z'' and 0 ◁ ''w'', then '''50207''' ◁ ''p''. Hence we may assume either ''z'' or ''w'' is empty.

If ''z'' is empty, then ''p'' ∈ 52{0}7, and the smallest prime ''p'' ∈ 52{0}7 is '''5200007'''.

If ''w'' is empty, then ''p'' ∈ 5{0}27, and the smallest prime ''p'' ∈ 5{0}27 is '''5000000000000000000000000000027'''.

Case 3.3: ''p'' begins with 7.

In this case we can write ''p'' = 7''y''7. If 2 ◁ ''y'', then '''727''' ◁ ''p''. If 5 ◁ ''y'', then '''757''' ◁ ''p''. If 8 ◁ ''y'', then '''787''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 7, and thus all digits of ''p'' are 0 or 7. But then, since the digits of ''p'' all have a common factor 7, ''p'' is divisible by 7, so ''p'' cannot be prime.

Case 3.4: ''p'' begins with 8.

In this case we can write ''p'' = 8''y''7. If 2 ◁ ''y'', then '''827''' ◁ ''p''. If 5 ◁ ''y'', then '''857''' ◁ ''p''. If 7 ◁ ''y'', then '''877''' ◁ ''p''. If 8 ◁ ''y'', then '''887''' ◁ ''p''. Hence we may assume ''y'' ∈ {0}, and thus ''p'' ∈ 8{0}7. But then, since the sum of the digits of ''p'' is 15, ''p'' is divisible by 3, so ''p'' cannot be prime.

Case 4: ''p'' ends with 9.

In this case we can write ''p'' = ''x''9. If ''x'' contains 1, 2, 5, 7, or 8, then (respectively) '''19''' ◁ ''p'', '''29''' ◁ ''p'', '''59''' ◁ ''p'', '''79''' ◁ ''p'', or '''89''' ◁ ''p''. Hence we may assume all digits of ''x'' are 0, 3, 4, 6, or 9.

If 44 ◁ ''x'', then '''449''' ◁ ''p''. Hence we may assume ''x'' contains zero or one 4's.

If x contains no 4's, then all digits of ''x'' are 0, 3, 6, or 9, and thus all digits of ''p'' are 0, 3, 6, or 9. But then, since the digits of ''p'' all have a common factor 3, ''p'' is divisible by 3, so ''p'' cannot be prime. Hence we may assume that ''x'' contains exactly one 4.

Case 4.1: ''p'' begins with 3.

In this case we can write ''p'' = 3''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. We must have '''349''' ◁ ''p''.

Case 4.2: ''p'' begins with 4.

In this case we can write ''p'' = 4''y''9, where all digits of ''y'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''409''' ◁ ''p''. If 3 ◁ ''y'', then 43 ◁ ''p''. If 9 ◁ ''y'', then '''499''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}, and thus ''p'' ∈ 4{6}9. But then ''p'' is divisible by 7, since for ''n'' ≥ 0 we have 7 × 6&lt;sub&gt;''n''&lt;/sub&gt;7 = 46&lt;sub&gt;''n''&lt;/sub&gt;9.

Case 4.3: ''p'' begins with 6.

In this case we can write p = 6''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 6 ◁ ''z'', then '''6469''' ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' is empty.

If 3 ◁ ''y'', then 349 ◁ ''p''. If 9 ◁ ''y'', then '''6949''' ◁ ''p''. Hence we may assume all digits of ''y'' are 0 or 6.

If 06 ◁ ''y'', then '''60649''' ◁ ''p''. Hence we may assume ''y'' ∈ {6}{0}.

If 666 ◁ ''y'', then '''666649''' ◁ ''p''. If 00000 ◁ ''y'', then '''60000049''' ◁ ''p''. Hence we may assume ''y'' ∈ {''𝜆'', 0, 00, 000, 0000, 6, 60, 600, 6000, 60000, 66, 660, 6600, 66000, 660000}, and thus ''p'' ∈ {649, 6049, 60049, 600049, 6000049, 6649, 66049, 660049, 6600049, 66000049, 66649, 666049, 6660049, 66600049, 666000049}, and of these numbers only '''66000049''' and '''66600049''' are primes.

Case 4.4: ''p'' begins with 9.

In this case we can write p = 9''y''4''z''9, where all digits of ''y'', ''z'' are 0, 3, 6, or 9. If 0 ◁ ''y'', then '''9049''' ◁ ''p''. If 3 ◁ ''y'', then 349 ◁ ''p''. If 6 ◁ ''y'', then '''9649''' ◁ ''p''. If 9 ◁ ''y'', then '''9949''' ◁ ''p''. Hence we may assume ''y'' is empty.

If 0 ◁ ''z'', then 409 ◁ ''p''. If 3 ◁ ''z'', then 43 ◁ ''p''. If 9 ◁ ''z'', then 499 ◁ ''p''. Hence we may assume ''z'' ∈ {6}, and thus ''p'' ∈ 94{6}9, and the smallest prime ''p'' ∈ 94{6}9 is 946669.

[[Category:Number theory]]</text>
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    <title>Integrated Building Performance Simulation</title>
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== '''About''' ==
Course offered by the Graduate Programs in Mechanical Engineering (PPGEM) and Smart and Sustainable Cities (PPGCIS) at Pontifícia Universidade Católica do Paraná (PUCPR).

'''Areas of Interest''': Engineering, Data Science, Computer Science, Architecture and Urban Planning, and other fields.

'''Course instructors''': Nathan Mendes, Walter Mazuroski, Luciano Ayres de Mello, and Marcos Batistella Lopes.

== '''Course Description''' ==
This course explores the physical principles governing heat, air, moisture, and energy processes in buildings, emphasizing their role in building performance, energy efficiency, indoor air quality (IAQ), renewable energy integration, and sustainability. The course introduces integrated simulation approaches for analyzing the interactions among building components, HVAC (Heating, Ventilation and Air Conditioning) systems, occupants, and the surrounding environment. Special emphasis is placed on co-simulation and interoperable modeling using the Functional Mock-up Interface (FMI) standard, enabling the integration of multiple simulation tools and domains. Applications extend from individual buildings to districts and urban energy systems, providing the foundations for advanced research in sustainable buildings, energy transition, and future Digital Twin applications. The course combines theoretical foundations, computational modeling, hands-on simulation, and current research challenges in building performance and sustainable cities.

== '''Place and Time''' ==
'''Location''': To be determined (hybrid classes).
'''When''': 2026 2nd semester

&lt;u&gt;Wednesday&lt;/u&gt;, from 3pm to 5pm (BRT), on the following dates:

* August: 05, 12, 19, and 26
* September: 02, 09, and 16
* October: 07, 14, 21, and 28
* November: 04 and 11

&lt;u&gt;Saturday&lt;/u&gt;, from 9am to 12pm (BRT), on the following dates:

* September: 26
* October: 02

== '''Workload''' ==
45 hours (3 credits)

== '''Objectives''' ==

* Understand the physical principles governing integrated building performance;
* Develop and apply simulation and co-simulation models using interoperable computational tools;
* Assess building performance in terms of energy efficiency, indoor environmental quality, and moisture-related phenomena;
* Apply computational modeling to support the design, operation, and optimizazion.

== '''Tentative Topics of Study''' ==
Topics may be adjusted according to the interests of the students, ongoing research projects, and the expertise of participants.

'''Fundamentals (Nathan Mendes)'''

* Building physics fundamentals, including heat, air and moisture transfer processes
* Mass and energy balances in buildings and urban systems

'''Energy and Thermal Performance and Simulation (Nathan Mendes)'''

* Building envelope performance
* HVAC systems modeling and performance
* Thermal comfort
* Moisture-related risks
* Building energy performance evaluation

'''Co-Simulation and Interoperability (Walter Mazuroski)'''

* Co-simulation concepts and frameworks
* Functional Mock-up Interface (FMI) and Functional Mock-up Units (FMUs)
* Data exchange and interoperability
* Applications: Integration of building, HVAC, urban climate, and energy system models

'''IAQ and Co-simulation for assessment of contaminants (Marcos Batistella Lopes)'''

* Fundamentals of Indoor Air Quality
* Smart ventilation
* Contam
* Indoor air quality assessment
* Domus-Contam Co-Simulation

'''Emerging Research topics, Applications and Case Studies (All lecturers)'''

* Co-Simulation BPS-CFD
* High-performance buildings
* Net-zero and positive-energy buildings
* Urban Heat Island mitigation strategies
* Energy efficiency
* Energy transition in the city scale
* Anthropogenic heat emissions
* Positive Energy Districts
* Urban energy transition scenarios
* Climate change adaptation strategies
* Digital Twins for buildings, districts, and cities

== '''Simulation Tools''' ==

* '''DOMUS/EnergyPlus''' - building energy simulation
* '''CONTAM''' – airflow and indoor air quality analysis
* '''Python / MATLAB''' – data processing, model integration, parametric studies, and optimization
* Additional simulation, co-simulation platforms according to project requirements

== '''Evaluation''' ==

* 30% Computer assignments, simulations, projects, and exercises
* 15% Seminar presentation
* 35% Scientific article
* 20% Examination

Additional evaluation methods may be adopted depending on class size and course dynamics.

== '''References''' ==

# Hagentoft, C.-E. Introduction to Building Physics. Studentlitteratur, Lund, 2001.
# Hens, H. Building Physics: Heat, Air and Moisture – Fundamentals and Engineering Methods with Examples and Exercises. Ernst &amp; Sohn, 3rd Edition, 2017.
# Mendes, N.; Chhay, M.; Berger, J.; Dutykh, D. Numerical Methods for Diffusion Phenomena in Building Physics: A Practical Introduction. Springer, 2019.
# ASHRAE Handbook Fundamentals. Latest Edition.
# Selected journal papers on Building Performance Simulation, Heat and Moisture Transfer, Urban Energy Systems, Positive Energy Districts, Digital Twins, and Sustainable Cities.
# Ongoing international collaborative research projects.</text>
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}}

== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
      <sha1>rmysc1dta2lirlir4tjyvat9w3aj0xj</sha1>
    </revision>
  </page>
  <page>
    <title>File:Python.Work2.Library.1A.20260715.pdf</title>
    <ns>6</ns>
    <id>330629</id>
    <revision>
      <id>2818465</id>
      <timestamp>2026-07-17T17:28:47Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>{{Information
|Description=Work2.1A: Libraries (20260715 - 20260714)
|Source={{own|Young1lim}}
|Date=2026-07-17
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}</comment>
      <origin>2818465</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="264" sha1="5gvzro5jl5sbhgjmc3pboabli75umta" xml:space="preserve">== Summary ==
{{Information
|Description=Work2.1A: Libraries (20260715 - 20260714)
|Source={{own|Young1lim}}
|Date=2026-07-17
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}

== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
      <sha1>5gvzro5jl5sbhgjmc3pboabli75umta</sha1>
    </revision>
  </page>
  <page>
    <title>User:Viththiyatharan Ananthajith/sandbox</title>
    <ns>2</ns>
    <id>330630</id>
    <revision>
      <id>2818468</id>
      <timestamp>2026-07-18T01:49:18Z</timestamp>
      <contributor>
        <username>Viththiyatharan Ananthajith</username>
        <id>3101703</id>
      </contributor>
      <comment>I write this article about JMC INNOVATORS LEARNING PLATFORM.</comment>
      <origin>2818468</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="38" sha1="8wwfhiu0gupk1alg5sgmv56o08o2yj3" xml:space="preserve">'''JMC INNOVATORS LEARNING PLATFORM'''</text>
      <sha1>8wwfhiu0gupk1alg5sgmv56o08o2yj3</sha1>
    </revision>
  </page>
  <page>
    <title>Babylonian Tonal System</title>
    <ns>0</ns>
    <id>330631</id>
    <revision>
      <id>2818469</id>
      <timestamp>2026-07-18T06:08:13Z</timestamp>
      <contributor>
        <username>Grimes2</username>
        <id>2895877</id>
      </contributor>
      <comment>Babylonian Tonal System</comment>
      <origin>2818469</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9832" sha1="pgp9x83bn6d97w3swwom7qof1b4cgsi" xml:space="preserve">==Basic tetrachords==
1½1, ½11, 11½
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; }
\relative c' { 
  d1 e -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup f\stopGroup g}
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; }
\relative c' { 
  e1 -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup f\stopGroup g a}
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; }
\relative c' { 
  c1 d e -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup f\stopGroup}
&lt;/score&gt;
and the tritone&lt;br&gt;
111
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; }
\relative c' { 
  f1 g a b}
&lt;/score&gt;

==7 heptachord modes==
The 7 modal patterns for the Babylonian heptachords are: 

{| class=&quot;wikitable sortable&quot; style=&quot;text-align: center;
|+ Babylonian tonal system
! Mode !! Pattern !! Center note || Semitones || Tritones
|-
| ''kitmum'' || 1½11½1 || D || 10 || 0
|-
| ''pītum'' || ½11½11 || E || 10 || 0
|-
| ''qablītum'' || 11½11½ || C || 10 || 0
|-
| ''išartum'' || 1½111½ || G || 10 || 1
|-
| ''embūbum'' || ½111½1 || A || 10 || 1
|-
| ''nīd qablim'' || 111½11 || B || 11 || 2
|-
| ''nīš tuḫrim'' || 11½111 || F || 11 || 2
|}

&lt;score&gt; \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } \relative c' { 
  a1^&quot;kitmum&quot; b -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup f\stopGroup g }
&lt;/score&gt;
&lt;score&gt; \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } \relative c' { 
  g1^&quot;pītum&quot; a b -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup f\stopGroup }
&lt;/score&gt;
&lt;score&gt; \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  b1^&quot;qablītum&quot; -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup f\stopGroup g a}
&lt;/score&gt;
&lt;score&gt; \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c { 
  e1^&quot;išartum&quot; -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup f\stopGroup g a b-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup c\stopGroup d}
&lt;/score&gt;
&lt;score&gt; \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  d1^&quot;embūbum&quot; e -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup f\stopGroup g a b-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup c\stopGroup}
&lt;/score&gt;
&lt;score&gt; \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  f,1^&quot;nīd qablim&quot; g a b -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup c\stopGroup d e}
&lt;/score&gt;
&lt;score&gt; \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  c1^&quot;nīš tuḫrim&quot; d e -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup f\stopGroup g a b}
&lt;/score&gt;

==2 consecutive heptachords==
&lt;score&gt; \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } \relative c' { 
  a1^&quot;kitmum&quot; b -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup f\stopGroup g a-\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup bes\stopGroup c d-\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup es\stopGroup f}
&lt;/score&gt;

==7 consecutive heptachords==
7 consecutive heptachords build the 7 Babylonian modes:
[[File:Kitmum.svg|thumb|left|1000px]]
[[File:Pitum.svg|thumb|left|1000px]]
[[File:Qablitum.svg|thumb|left|1000px]]
[[File:Isartum.svg|thumb|left|1000px]]
[[File:Embubum.svg|thumb|left|1000px]]
[[File:Nid quablim.svg|thumb|left|1000px]]
[[File:Nis tuhrim.svg|thumb|left|1000px]]
{{clear}}

===6 cyclic consecutive heptachords===
[[File:Circle of fourths.svg|thumb|left|Circle of fourths, ''kitmum'', 36 notes, 60 semitones = Anu 𒀭]]
{{clear}}

==Tuning==
A tuning procedure ‘loosening’ (TU.LU) in Music of Mesopotamia for a 7-stringed instrument based on a transposition to D/D♭:
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; }
\relative c' { 
  d1^&quot;išartum&quot; e -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup f\stopGroup g a b-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup c\stopGroup }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; }
\relative c' { 
  d1^&quot;kitmum&quot; e -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup f\stopGroup g a-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup bes\stopGroup c }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  d1^&quot;embūbum&quot; -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup es\stopGroup f g a-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup bes\stopGroup c }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  d1^&quot;pītum&quot; -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup es\stopGroup f g -\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup as\stopGroup bes c }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  des1^&quot;nīd qablim&quot; es f g-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup as\stopGroup bes c }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  des1^&quot;nīš tuḫrim&quot; es f -\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup ges\stopGroup as bes c }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  des1^&quot;qablītum&quot; es f -\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup ges\stopGroup as bes-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup ces\stopGroup }
&lt;/score&gt;
‘tightening’ (GÍD.I)
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; }
\relative c' { 
  d1^&quot;išartum&quot; e -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup f\stopGroup g a b-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup c\stopGroup }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  d1^&quot;qablītum&quot; e fis -\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup g\stopGroup a b-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup c\stopGroup }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  d1^&quot;niš tuĥrim&quot; e fis -\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup g\stopGroup a b cis }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  d1^&quot;nīd qablim&quot; e fis gis-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup a\stopGroup b cis }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  dis1^&quot;pītum&quot; -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup e\stopGroup fis gis -\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup a\stopGroup b cis }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  dis1^&quot;embūbum&quot; -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup e\stopGroup fis gis ais-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup b\stopGroup cis }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; }
\relative c' { 
  dis1^&quot;kitmum&quot; eis -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup fis\stopGroup gis ais-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup b\stopGroup cis }
&lt;/score&gt;

==Further reading==
* {{cite conference | first = Leon | last = Crickmore | title = A New Light on the Babylonian Tonal System | conference = Proceedings of the International Conference of Near Eastern Archaeomusicology (ICONEA 2008), The British Museum, London, December 4–6, 2008, editors: Richard Dumbrill and Irving Finkel | publisher = Iconea Publications | url = https://musicircle.net/wp-content/uploads/2018/08/Crickmore-Iconea20081.pdf | place = London | year = 2008 | pages = 11-22
}}
* {{cite web | title=Babylonian Tonal System | website=Allan Pettersson Fanpage | url=https://pettersson-fanpage.de/Babylon/Babylon.html | access-date=19 January 2026}}</text>
      <sha1>pgp9x83bn6d97w3swwom7qof1b4cgsi</sha1>
    </revision>
    <revision>
      <id>2818470</id>
      <parentid>2818469</parentid>
      <timestamp>2026-07-18T06:19:37Z</timestamp>
      <contributor>
        <username>Grimes2</username>
        <id>2895877</id>
      </contributor>
      <comment>+cat</comment>
      <origin>2818470</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9859" sha1="2yccg5f796c1fksdjqmmr59dvtgcope" xml:space="preserve">==Basic tetrachords==
1½1, ½11, 11½
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; }
\relative c' { 
  d1 e -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup f\stopGroup g}
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; }
\relative c' { 
  e1 -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup f\stopGroup g a}
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; }
\relative c' { 
  c1 d e -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup f\stopGroup}
&lt;/score&gt;
and the tritone&lt;br&gt;
111
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; }
\relative c' { 
  f1 g a b}
&lt;/score&gt;

==7 heptachord modes==
The 7 modal patterns for the Babylonian heptachords are: 

{| class=&quot;wikitable sortable&quot; style=&quot;text-align: center;
|+ Babylonian tonal system
! Mode !! Pattern !! Center note || Semitones || Tritones
|-
| ''kitmum'' || 1½11½1 || D || 10 || 0
|-
| ''pītum'' || ½11½11 || E || 10 || 0
|-
| ''qablītum'' || 11½11½ || C || 10 || 0
|-
| ''išartum'' || 1½111½ || G || 10 || 1
|-
| ''embūbum'' || ½111½1 || A || 10 || 1
|-
| ''nīd qablim'' || 111½11 || B || 11 || 2
|-
| ''nīš tuḫrim'' || 11½111 || F || 11 || 2
|}

&lt;score&gt; \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } \relative c' { 
  a1^&quot;kitmum&quot; b -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup f\stopGroup g }
&lt;/score&gt;
&lt;score&gt; \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } \relative c' { 
  g1^&quot;pītum&quot; a b -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup f\stopGroup }
&lt;/score&gt;
&lt;score&gt; \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  b1^&quot;qablītum&quot; -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup f\stopGroup g a}
&lt;/score&gt;
&lt;score&gt; \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c { 
  e1^&quot;išartum&quot; -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup f\stopGroup g a b-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup c\stopGroup d}
&lt;/score&gt;
&lt;score&gt; \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  d1^&quot;embūbum&quot; e -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup f\stopGroup g a b-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup c\stopGroup}
&lt;/score&gt;
&lt;score&gt; \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  f,1^&quot;nīd qablim&quot; g a b -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup c\stopGroup d e}
&lt;/score&gt;
&lt;score&gt; \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  c1^&quot;nīš tuḫrim&quot; d e -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup f\stopGroup g a b}
&lt;/score&gt;

==2 consecutive heptachords==
&lt;score&gt; \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } \relative c' { 
  a1^&quot;kitmum&quot; b -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup f\stopGroup g a-\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup bes\stopGroup c d-\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup es\stopGroup f}
&lt;/score&gt;

==7 consecutive heptachords==
7 consecutive heptachords build the 7 Babylonian modes:
[[File:Kitmum.svg|thumb|left|1000px]]
[[File:Pitum.svg|thumb|left|1000px]]
[[File:Qablitum.svg|thumb|left|1000px]]
[[File:Isartum.svg|thumb|left|1000px]]
[[File:Embubum.svg|thumb|left|1000px]]
[[File:Nid quablim.svg|thumb|left|1000px]]
[[File:Nis tuhrim.svg|thumb|left|1000px]]
{{clear}}

===6 cyclic consecutive heptachords===
[[File:Circle of fourths.svg|thumb|left|Circle of fourths, ''kitmum'', 36 notes, 60 semitones = Anu 𒀭]]
{{clear}}

==Tuning==
A tuning procedure ‘loosening’ (TU.LU) in Music of Mesopotamia for a 7-stringed instrument based on a transposition to D/D♭:
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; }
\relative c' { 
  d1^&quot;išartum&quot; e -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup f\stopGroup g a b-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup c\stopGroup }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; }
\relative c' { 
  d1^&quot;kitmum&quot; e -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup f\stopGroup g a-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup bes\stopGroup c }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  d1^&quot;embūbum&quot; -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup es\stopGroup f g a-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup bes\stopGroup c }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  d1^&quot;pītum&quot; -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup es\stopGroup f g -\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup as\stopGroup bes c }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  des1^&quot;nīd qablim&quot; es f g-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup as\stopGroup bes c }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  des1^&quot;nīš tuḫrim&quot; es f -\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup ges\stopGroup as bes c }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  des1^&quot;qablītum&quot; es f -\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup ges\stopGroup as bes-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup ces\stopGroup }
&lt;/score&gt;
‘tightening’ (GÍD.I)
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; }
\relative c' { 
  d1^&quot;išartum&quot; e -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup f\stopGroup g a b-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup c\stopGroup }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  d1^&quot;qablītum&quot; e fis -\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup g\stopGroup a b-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup c\stopGroup }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  d1^&quot;niš tuĥrim&quot; e fis -\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup g\stopGroup a b cis }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  d1^&quot;nīd qablim&quot; e fis gis-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup a\stopGroup b cis }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  dis1^&quot;pītum&quot; -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup e\stopGroup fis gis -\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup a\stopGroup b cis }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  dis1^&quot;embūbum&quot; -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup e\stopGroup fis gis ais-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup b\stopGroup cis }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; }
\relative c' { 
  dis1^&quot;kitmum&quot; eis -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup fis\stopGroup gis ais-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup b\stopGroup cis }
&lt;/score&gt;

==Further reading==
* {{cite conference | first = Leon | last = Crickmore | title = A New Light on the Babylonian Tonal System | conference = Proceedings of the International Conference of Near Eastern Archaeomusicology (ICONEA 2008), The British Museum, London, December 4–6, 2008, editors: Richard Dumbrill and Irving Finkel | publisher = Iconea Publications | url = https://musicircle.net/wp-content/uploads/2018/08/Crickmore-Iconea20081.pdf | place = London | year = 2008 | pages = 11-22
}}
* {{cite web | title=Babylonian Tonal System | website=Allan Pettersson Fanpage | url=https://pettersson-fanpage.de/Babylon/Babylon.html | access-date=19 January 2026}}

[[Category:Music theory]]</text>
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    <revision>
      <id>2818471</id>
      <parentid>2818470</parentid>
      <timestamp>2026-07-18T06:48:44Z</timestamp>
      <contributor>
        <username>Grimes2</username>
        <id>2895877</id>
      </contributor>
      <comment>+text</comment>
      <origin>2818471</origin>
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      <text bytes="9907" sha1="4sjiida3eaqln4qnp96v6g5t7fzn9qh" xml:space="preserve">==Basic tetrachords==
1½1, ½11, 11½
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; }
\relative c' { 
  d1 e -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup f\stopGroup g}
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; }
\relative c' { 
  e1 -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup f\stopGroup g a}
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; }
\relative c' { 
  c1 d e -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup f\stopGroup}
&lt;/score&gt;
and the tritone&lt;br&gt;
111
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; }
\relative c' { 
  f1 g a b}
&lt;/score&gt;
Two consecutive tetrachords build a heptachord.

==7 heptachord modes==
The 7 modal patterns for the Babylonian heptachords are: 

{| class=&quot;wikitable sortable&quot; style=&quot;text-align: center;
|+ Babylonian tonal system
! Mode !! Pattern !! Center note || Semitones || Tritones
|-
| ''kitmum'' || 1½11½1 || D || 10 || 0
|-
| ''pītum'' || ½11½11 || E || 10 || 0
|-
| ''qablītum'' || 11½11½ || C || 10 || 0
|-
| ''išartum'' || 1½111½ || G || 10 || 1
|-
| ''embūbum'' || ½111½1 || A || 10 || 1
|-
| ''nīd qablim'' || 111½11 || B || 11 || 2
|-
| ''nīš tuḫrim'' || 11½111 || F || 11 || 2
|}

&lt;score&gt; \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } \relative c' { 
  a1^&quot;kitmum&quot; b -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup f\stopGroup g }
&lt;/score&gt;
&lt;score&gt; \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } \relative c' { 
  g1^&quot;pītum&quot; a b -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup f\stopGroup }
&lt;/score&gt;
&lt;score&gt; \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  b1^&quot;qablītum&quot; -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup f\stopGroup g a}
&lt;/score&gt;
&lt;score&gt; \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c { 
  e1^&quot;išartum&quot; -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup f\stopGroup g a b-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup c\stopGroup d}
&lt;/score&gt;
&lt;score&gt; \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  d1^&quot;embūbum&quot; e -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup f\stopGroup g a b-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup c\stopGroup}
&lt;/score&gt;
&lt;score&gt; \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  f,1^&quot;nīd qablim&quot; g a b -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup c\stopGroup d e}
&lt;/score&gt;
&lt;score&gt; \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  c1^&quot;nīš tuḫrim&quot; d e -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup f\stopGroup g a b}
&lt;/score&gt;

==2 consecutive heptachords==
&lt;score&gt; \new Staff \with { \consists Ambitus_engraver \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } \relative c' { 
  a1^&quot;kitmum&quot; b -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup c\stopGroup d e-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup f\stopGroup g a-\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup bes\stopGroup c d-\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup es\stopGroup f}
&lt;/score&gt;

==7 consecutive heptachords==
7 consecutive heptachords build the 7 Babylonian modes:
[[File:Kitmum.svg|thumb|left|1000px]]
[[File:Pitum.svg|thumb|left|1000px]]
[[File:Qablitum.svg|thumb|left|1000px]]
[[File:Isartum.svg|thumb|left|1000px]]
[[File:Embubum.svg|thumb|left|1000px]]
[[File:Nid quablim.svg|thumb|left|1000px]]
[[File:Nis tuhrim.svg|thumb|left|1000px]]
{{clear}}

===6 cyclic consecutive heptachords===
[[File:Circle of fourths.svg|thumb|left|Circle of fourths, ''kitmum'', 36 notes, 60 semitones = Anu 𒀭]]
{{clear}}

==Tuning==
A tuning procedure ‘loosening’ (TU.LU) in Music of Mesopotamia for a 7-stringed instrument based on a transposition to D/D♭:
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; }
\relative c' { 
  d1^&quot;išartum&quot; e -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup f\stopGroup g a b-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup c\stopGroup }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; }
\relative c' { 
  d1^&quot;kitmum&quot; e -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup f\stopGroup g a-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup bes\stopGroup c }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  d1^&quot;embūbum&quot; -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup es\stopGroup f g a-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup bes\stopGroup c }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  d1^&quot;pītum&quot; -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup es\stopGroup f g -\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup as\stopGroup bes c }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  des1^&quot;nīd qablim&quot; es f g-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup as\stopGroup bes c }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  des1^&quot;nīš tuḫrim&quot; es f -\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup ges\stopGroup as bes c }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  des1^&quot;qablītum&quot; es f -\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup ges\stopGroup as bes-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup ces\stopGroup }
&lt;/score&gt;
‘tightening’ (GÍD.I)
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; }
\relative c' { 
  d1^&quot;išartum&quot; e -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup f\stopGroup g a b-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup c\stopGroup }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  d1^&quot;qablītum&quot; e fis -\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup g\stopGroup a b-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup c\stopGroup }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  d1^&quot;niš tuĥrim&quot; e fis -\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup g\stopGroup a b cis }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  d1^&quot;nīd qablim&quot; e fis gis-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup a\stopGroup b cis }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  dis1^&quot;pītum&quot; -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup e\stopGroup fis gis -\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup a\stopGroup b cis }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; } 
\relative c' { 
  dis1^&quot;embūbum&quot; -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup e\stopGroup fis gis ais-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup b\stopGroup cis }
&lt;/score&gt;
&lt;score&gt;
\new Staff \with { \consists Horizontal_bracket_engraver \remove &quot;Time_signature_engraver&quot; \remove &quot;Bar_engraver&quot; }
\relative c' { 
  dis1^&quot;kitmum&quot; eis -\tweak HorizontalBracketText.text &quot;semitone&quot;  \startGroup fis\stopGroup gis ais-\tweak HorizontalBracketText.text &quot;semitone&quot;\startGroup b\stopGroup cis }
&lt;/score&gt;

==Further reading==
* {{cite conference | first = Leon | last = Crickmore | title = A New Light on the Babylonian Tonal System | conference = Proceedings of the International Conference of Near Eastern Archaeomusicology (ICONEA 2008), The British Museum, London, December 4–6, 2008, editors: Richard Dumbrill and Irving Finkel | publisher = Iconea Publications | url = https://musicircle.net/wp-content/uploads/2018/08/Crickmore-Iconea20081.pdf | place = London | year = 2008 | pages = 11-22
}}
* {{cite web | title=Babylonian Tonal System | website=Allan Pettersson Fanpage | url=https://pettersson-fanpage.de/Babylon/Babylon.html | access-date=19 January 2026}}

[[Category:Music theory]]</text>
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