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)</text>
      <sha1>6m3u3bl4n9h8ygopzspfhua1p3ayek5</sha1>
    </revision>
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        <id>10242</id>
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      <comment>/* Resources suitable for the main namespace */ reply ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
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      <text bytes="50675" sha1="l1hkiocd3w4yo8j25zx2afz7izbxpbb" xml:space="preserve">&lt;!-- {{Out of town}} --&gt;
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{{{{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! ==
&lt;div style=&quot;border: 3px solid #FFD700; background-color: #FFFAF0; padding:0.2em 0.4em; height:auto; min-height:173px; border-radius:1em; {{box-shadow|0.1em|0.1em|0.5em|rgba(0,0,0,0.75)}}&lt;!--
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[[File:Everlasting Fireworks looped.gif|left|x173px]][[File:Happy new year 01.svg|x173px|right]]
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{{Center|{{resize|179%|'''''[[New Year|Happy New Year]]!'''''}}}}
'''Jtneill''',&lt;br /&gt;Have a prosperous, productive and enjoyable [[New Year]], and thanks for your contributions to Wikiversity.
&lt;br /&gt;[[User:Harold Foppele|Harold Foppele]] ([[User talk:Harold Foppele|discuss]] • [[Special:Contributions/Harold Foppele|contribs]]) 17:10, 2 January 2026 (UTC)&lt;br /&gt;&lt;br /&gt;
&lt;/div&gt;
&amp;nbsp;&amp;nbsp;&amp;nbsp;''{{resize|88%|Send New Year cheer by adding {{tls|Happy New Year fireworks}} to user talk pages.}}''
{{clear}}&lt;!-- From template:Happy New Year fireworks --&gt;

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

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

== DELETE request ==

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

== Archiving ==

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

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

== Email ==

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

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

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

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

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

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

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

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

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

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

== Thoughts about Wikinews closure ==

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

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

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

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

== Resources suitable for the main namespace ==

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

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

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

An example of such a project is needed [[Sweet Home 3D|here]] or [[User:Juandev/R/Compression stocking|here]], but there was an arumentation, they are underdeveloped and they should be moved to user ns. So that's why I'm asking what the evaluation criteria are, so that it doesn't end up in a way that the pages are moved away from the main ns and I end up finding out that I have to move it to my own wiki anyway. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:37, 3 July 2026 (UTC)
: Oh, sorry to hear that you are finished with Czech Wikiversity, but maybe that is good for en.wv.
: I guess we'll never really have any guarantees about anything placed on a publicly editable wiki because practices and users can change.
: I share your concerns about actual, or threats of, rather blunt approaches to educational use of AI. Of course, AI can be educational useful, and of course we are capable of finding nuanced, reasonable ways to include and use it. But as we see e.g., on Commons, there is a strong, simplistic anti-AI sentiment within the Wikimedia community.
: I don't recall much discussion about, or use of the Draft ns on en.wv. I think it was probably created very early on, to replicate Wikipedia, where a draft article makes sense before being moved to main space. I think the Draft or User space is welcome to be used for almost anything within scope, without much tension or debate. Then there is the issue around what some users consider acceptable or not for the main space on en.wv. Personally, I'm quite open. en.wv is still in early days of experimentation and trying things is needed, so I'm included to be inclusive and accepting, rather than shunting projects into Draft ns.
: I don't use Draft:, but I do use User: subpages and of course main space. I haven't had any issues with others asking me to justify main space content or proposals to move content to Draft or User.
: I'm sorry this doesn't provide any guarantees, except I guess to say I feel good about using en.wv as a working environment.
: Sincerely,&lt;br&gt; James
: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 07:25, 3 July 2026 (UTC)</text>
      <sha1>l1hkiocd3w4yo8j25zx2afz7izbxpbb</sha1>
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  <page>
    <title>Talk:Albert Einstein quote</title>
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      <text bytes="17663" sha1="m0pz6rxh5bxtcknqaz1qi6e8zwi4rjl" xml:space="preserve">[[File:Einstein 1921 by F Schmutzer - restoration.jpg|right|250px]]

&lt;blockquote&gt;'''''&quot;Imagination is more important than knowledge. For knowledge is limited, whereas imagination embraces the entire world, stimulating progress, giving birth to evolution.&quot;'''''&lt;br /&gt;-[[w:Albert Einstein|Albert Einstein]], [[q:Albert Einstein|What Life Means to Einstein]] (1924)&lt;/blockquote&gt;

== Context ==

&lt;blockquote&gt;&quot;I am enough of an artist to draw freely upon my imagination. Imagination is more important than knowledge. For knowledge is limited, whereas imagination encircles the world&quot; &lt;br /&gt;-[[w:Albert Einstein|Albert Einstein]]  As quoted in &quot;[http://www.saturdayeveningpost.com/wp-content/uploads/satevepost/what_life_means_to_einstein.pdf What Life Means to Einstein: An Interview by George Sylvester Viereck]&quot; in The Saturday Evening Post (26 October 1929) [[q:Albert_Einstein]]&lt;/blockquote&gt;

==Discussion==
You are spot on Alby. &lt;small&gt;unsigned comment added by [[Special:Contributions/65.92.111.22|65.92.111.22]] 18:36, 18 May 2008, attribution added by [[User:Abd|Abd]] 18:57, 31 July 2011 (UTC)&lt;/small&gt;

Perhaps imagination is the basis for all knowledge due to the fact that any kind of invention or theory is either created or purposed as a result of an idea formatted from imagination. Thus, proving that knowledge is just as infinite and important as imagination.  {{unsigned2| 23:08, 18 October 2008|68.198.132.165}}

:This quote inspired me to make [[commons:Image:Mathematica%C2%B2.jpg|this]] -- [[User:Amog|Amog]] |&lt;sup&gt;[[User talk:Amog|T{{font|color=red|a}}{{font|color=green|l}}{{font|color=black|k}}]]&lt;/sup&gt; 08:13, 29 April 2008 (UTC)
 
I completely Agree with this precious quote of his. Yes we are nothing without knowledge but where imagination comes into play new knowledge is born. The fact that imagination without turning into something useful is such a waste but it can't be denied that everything that has led to great discoveries are part of someone's imagination. For me imagination is way to unlock hidden doors.

I'd disagree, even the most imaginative person is nothing without knowledge, while someone with great knowledge can still make contributions without imagination.  {{unsigned2|23:31, 18 October 2008|24.208.209.177}}

You make a good point, yet I feel that only pertains to math and sciences and other forms of
&quot;one sided&quot; subjects. Not saying that they aren't a difficult concept that requires both knowledge and imagination but they are much more set in their ways. Where as in the arts, one with knowledge can create things based off of the past when one with imagination can create original work rather than replicas of the past. I feel they have some what of a Ying-Yang relation. For instance, math and music can go hand in hand in some cases, just as imagination and knowledge can.  {{unsigned2| 23:11, 21 October 2008|69.121.123.87}}

Knowledge is simply a reference to what was accomplished due to past acts of imagination. Therefor, it is inferior to imagination in that respect. It is true that someone can contribute without imagination, but none of those contributions can possibly spawn new sources of knowledge. One can teach and pass down previous knowledge without imagination, but only through imagination can new knowledge be created, making knowledge dependent upon imagination and ultimately inferior. But it is very true that knowledge can be an aid to imagination.  {{unsigned2| 23:11, 21 October 2008|69.121.123.87}}

I agree, in the sense that knowledge is dependent upon imagination but I feel that neither is superior, due to the fact that after imagination creates something new, it becomes knowledge. Thus proving that even imagination is dependent upon knowledge; mainly to allow the imagination process to be completed and to become of use.  {{unsigned2| 23:39, 21 October 2008|69.121.123.87}}

I disagree. I believe that imagination doesn't become knowledge, but instead creates knowledge. The imagination still remains after the knowledge is created, and it goes on to create even more knowledge. Imagination is the mother, and knowledge is the child.  {{unsigned2|23:45, 21 October 2008|69.119.203.219}}

I agree that imagination is more important than knowledge. Knowledge is only the root that initiates imagination. It is the imagination through which one can promote ideas and not knowlege.Knowledge is the fundamental through which scientists promoted inventions through their imagination.  {{unsigned2|03:10, 31 May 2009|Priyanka sinha}}

New Guy: My dog has knowledge but no imagination.  And like most (higher) animals, she gets along okay like that.  She knows where she lives, for instance.   If she had no knowledge, she may not survive.  Imagination is essentially of human importance, but it's probably just ego.  You really could become an automaton and still be alive, and not in serious pain.  You would no longer be what we called human.  Even if the dog analogy is only an analogy, I think it extends out.  &quot;More important to what end?&quot; may be the real question.  {{unsigned2|15:32, 5 July 2009|75.164.146.55}}

This is a slogan that the ignorant and intellectually lazy use to pretend that that they are equal to or better than those those who have actually made the effort to learn about something. (of course, that was not Einstein's intention, if he really ever said it, but it is how it gets used.) It is anti-educational and is an extremely inappropriate epigraph for something purporting to be an educational project.  {{unsigned2|04:43, 19 July 2009|75.84.85.130}}

A favorite quote of mine.  In terms of quotes, people's interpretations are rather contextual.  Some people may find the quote inspirational because they tend to be more creative.  While others may feel it is insulting to their intellectualism.  But personally, I feel the truth of this quote is its ability to convey the importance of both knowledge AND imagination.  Inserted into context, Einstein himself was a rather curious and imaginative person.  His ability to think beyond the norms is why he was able to develop his Theory of Relativity in the first place.  One also cannot deny Einstein's appreciation of creative arts such as literature and music. {{unsigned2|10:18, 23 August 2009|76.115.55.160}}

OK people many of you are missing the entire meaning of the quote altogether. Imagination is absolutely MORE important than knowledge because it is the basis of all we create and without creation man is only another animal. It is what seperates us from the lower lifeforms on our planet people. The concept is to understand that your imagination has no limits at all and your knowledge obviously does. Imagination is how we grow and how we learn more knowledge. {{unsigned2|16:08, 5 January 2010|72.200.61.226}}

I find irrelevant to measure one against the other. I myself like to think that inspiration is also important source of great accomplishments, although Thomas Edison once said &quot;Genius is one percent inspiration, ninety-nine percent perspiration&quot;. There are variations to that first statement he made such as &quot;None of my inventions came by accident. I see a worthwhile need to be met and I make trial after trial until it comes. What it boils down to is one per cent inspiration and ninety-nine per cent perspiration&quot;. We also have read about discoveries due to mishandling or accidents, which would not be evaluated by the authors if they did not have the necessary knowledge. Therefore, one can conclude that many factors contribute to accomplish great discovery or work of art: inspiration, imagination, knowlegde, skill, hard work and sometimes, luck - things also happen by accident. [[User:Bebo|Bebo]] 16:29, 31 January 2010 (UTC)

Avoiding the whole taking sides, Both are important, but look at this way, there are people who created the earth and people who make it go round, basically each serves a purpose, imagination does play a bigger role in a sense because it makes the object or thought more useful, like a basketball for instance, so you've created a ball, its round and bounces, but with creatitivy and imagination you were able to improvise and make a sport if not more i.e soccer, volleyball. Never Let ignorance control who you should favour but to respect and validate each model's contribution to society, that's the only way we can succeed as humans. {{unsigned2|11:21, 25 April 2010|74.58.6.252}}

Some of you did not understand the statement. &quot;imagination is more important than knowledge&quot; does not mean knowledge is not important is just less important, because without imagination you will not know how to make good use of the knowledge not matter how great the knowledge is. When you have both, imagination have to be greater. {{unsigned2|07:19, 26 May 2010|118.93.186.246}}

That quote was absolutely beautiful.  I almost teared up. {{unsigned2|18:38, 5 December 2010|66.206.183.154}}

A child's imagination is infinite. Adult's knowledge is limited. darcsr
...

Hello
This is a wide topic. 
Imagination often provokes us to create knowledge.
Albert reveres imagination because he has a great mind.
Whereas for most people imagination means things we can merely imagine.
Such as in an ideal world or wishful thinking and clearly impossible dreams.
Knowledge is different to imagining, it grows by deliberate experiments, development and experience. As distinct from an untried imagination.
Imaginative ideas draw on knowledge for components and veracity to be feasible. 
My theory is isolation was the origin of the actual need for human imagination.
Added 26 December 2010 signed [[wp.user:Thylagene]].

:Consider this: Imagination (fantasy) is the art to place knowledge in a form it has not been before. Which would make some form of knowledge nessesary to apply imagination. Knowledge can exist in many forms. In the universe knowledge exists as matter, advanced form of knowledge is advanced forms of matter. But the fact that there is evolution in this universe makes it a necessity that imagination is present even when no living creatures are around. Otherwise nature would not &quot;know&quot; what to create next. Evolution is the natural way imagination plays out. Which needs a place where knowledge can be naturally shuffled around through circumstances. &lt;br /&gt;
:Albert's note can be taken in many ways. I take it that for humans, because his knowledge is limited, imagination is more important to be able to create the knowledge he craves. Otherwise imagination-knowledge is more of a chicken-egg relationship. Without knowledge there cannot be an imagination but if there is no imagination then there can not be any evolution meaning that there is no creation of knowledge. Everything would still be as it was billion of years before.[[User:Martin Lenoar|Martin Lenoar]] 10:22, 16 January 2011 (UTC)
:*Imagination is required to construct insightful theories to explain observational data.  Most of the theories we can think of turn out to be incorrect, but through a process of examination and refinement, we eventually weed out the incorrect theories and arrive at the best surviving scientific model we can come up with.  We need both imagination and analysis to carry this process forward.  —[[User:Caprice|Caprice]] 13:19, 16 January 2011 (UTC)

'''Imagination and Knowledge''', both work synergistically, I would not dare try to make a distinction. Because disctintion, is an illusion, created by the filtrations and perceptions of the human brain. Illusions, mean you cannot see the truth, the truth of the universe. so I don't make distinctions. This applys to past, present, future... it applys to, time flys when you are having fun, or slowing down when doing a chore. I have Transended such things, I am in , '''some sort of limitation awareness''', where I'' try'' and find the limit, yet never really do, and yet, ''sometimes I have to impose false limits'', to come back down.....but,  I must say, I love them both, just as well, and..''I am happy''.--[[User:Gaon Yincang Abhinava|Gaon Abhinava]] 13:30, 31 July 2011 (UTC)'''W V . Instructor, Survival ''Intelligence''  (S.I)'''


'''A child can use ''Imagination,'' and Soar through the Courtyards of Castles built in the Rainbow Clouds, and an old man can use ''knowledge'' to abuse mankind. I try not to make distinction between the two, but Albert is right. This is why Imagination is more Important than knowledge, Go Albert, Go !--[[User:Gaon Yincang Abhinava|Gaon Abhinava]] 13:43, 31 July 2011 (UTC)''''''W V . Instructor, Survival ''Intelligence''  (S.I)'''

I would Like to thank , for his insightful, writings, without which, I probably would not have transcended to a higher consciousness, he says 

&quot;Think Big&quot; 
pa+st +  &lt;-*   +   pre+sent   -*  +  fut+ure/*-&gt; ''(mind filtration)= mental = perception = '''MP'''''/
so I did/ and here I am/ and now, '''&quot;Time&quot;'''.... to do some wriing, ''right Now'', '''soon,''' ok, w@ell somet/ime soo/n, m_a)ybe so.......lol... : ) --[[User:Gaon Yincang Abhinava|Gaon Abhinava]] 14:01, 31 July 2011 (UTC)'''W V . Instructor, Survival ''Intelligence''  (S.I)'''

In my own opinion, I would say 'imagination is knowledge' because 'knowledge' is acquired through imagination, meditation, processed information, research, which eventually translates into knowledge. ''Imagination is more important than Knowledge''.I would say 'imagination is knowledge' because 'knowledge' is acquired through imagination, meditation, processed information, research, which eventually translates into knowledge. {{unsigned|197.149.85.150|06:15, 9 September 2016‎}}


Knowledge without imagination is Static. If all our knowledgeable scientists work on the principle that A = B then we would seldom learn anything new. However, the scientist with knowledge and imagination will say &quot;I know that A = B, but what would happen if ?&quot; He would then investigate the possibilities and thus may discover new answers. {{unsigned|95.147.81.177}}
[[Category:Quotes]]

imagination is the construct for all development in the advances of science. science does always begin as one question but that question ceases to exist if imagination is not used. if someone were to think of an original question from their own mind then it is reasonable to say that they had to use their very own imagination to develop the question. it all begins as a mind with enough creativity to ask a question not many would ask themselves {{unsigned|???}}

* It seems that [[w:Alexander_Grothendieck|'''Grothendieck''']] agrees with '''Einstein''' {{wink}} : &lt;blockquote&gt;&quot;''What makes the quality of the researcher's inventiveness and imagination is the quality of his attention, listening to the voice of things. […] It takes a consummate flair to grasp and update new types of mathematical structures. This kind of imagination or ‘’flair’’ seems to me rare, not only among physicists (where Einstein and Schrödinger seem to have been among the rare exceptions), but even among mathematicians (and here I speak with full knowledge of the facts). To sum up, I predict that the expected renewal (if it has yet to come...) will come rather from a mathematician at heart, well informed about the great problems of physics, than from a physicist. But above all, it will take a man with ‘’philosophical openness’’ to grasp the crux of the problem. This is not a technical problem, but a fundamental problem of ‘’philosophy of nature’’'' &quot;. Alexander Grothendieck, ''LA CLEF DES SONGES''.&lt;/blockquote&gt;[[User:EclairEnZ|Claude Mariotti]] ([[User talk:EclairEnZ|discuss]] • [[Special:Contributions/EclairEnZ|contribs]]) 21:46, 7 February 2021 (UTC)

Did you notice that Einstein really has an immense amount of knowledge about physics and mathematics? He could say imagination is more important, because knowledge is not something lacking in him. For someone who has no knowledge, imagination is useless. Besides, imagination is never unlimited as some would imagine: it is very much limited by the world around us and by existing knowledge. If you checked out the famous science fictions by Isaac Asimov or other famous writers, you would find that no one was really able to imagine the current world. Their imagination was limited by what they knew. —[[User:Adah1972|Adah1972]] ([[User talk:Adah1972|discuss]] • [[Special:Contributions/Adah1972|contribs]]) 03:08, 13 June 2021 (UTC)

:its truly great that in these words alone he sparked imagination's to coagulate and form.  The words inspired each and every one of the comments to be formed with the very subject that he is describing. I do so believe anybody debating his words came to their point from their imagination. All the wisdom and knowledge any of you have gained could only allow for one thing. Imaginary understandings. So we have a contradiction that leaves room for prediction, thus stating what is needed if you are to learn at all.  [[Special:Contributions/&amp;#126;2026-38122-72|&amp;#126;2026-38122-72]] ([[User talk:&amp;#126;2026-38122-72|talk]]) 08:44, 3 July 2026 (UTC)

== The true sign of intelligence is not knowledge but imagination ==

Upsc essay [[Special:Contributions/45.248.56.102|45.248.56.102]] ([[User talk:45.248.56.102|discuss]]) 06:42, 21 June 2022 (UTC)

== Intelligence ==

sunflower perfume elizabeth arden  wisdom is just as important  as intelligence  is just as important as wisdom for instance in natural selection as calvin klein one and orange new jersey just as the almighty god and most high gods and goddesses maria and shane figeroux......

[[Special:Contributions/200.7.90.175|200.7.90.175]] ([[User talk:200.7.90.175|discuss]]) 04:35, 10 September 2023 (UTC)</text>
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My name is James Neill (''he/him''). I'm an Assistant Professor in the [https://www.canberra.edu.au/about-uc/faculties/health/study/psychology Discipline of Psychology] at the [[University of Canberra]], Australia.

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

On English Wikiversity, I'm a [[WV:Custodianship|custodian]] and [[WV:Bureaucratship|bureaucrat]]&lt;small&gt;&lt;sup&gt;[https://en.wikiversity.org/w/index.php?title=Special:ListUsers&amp;limit=1&amp;username=Jtneill (verify)]&lt;/sup&gt;&lt;/small&gt;. Since 2005, I've made:
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My [[User:Jtneill/Teaching/Philosophy|teaching philosophy]] is based on experiential learning. [[/Teaching|I teach]] a 3rd-year undergraduate [[psychology]] unit, [[motivation and emotion]], and a 4th-year Honours unit about [[research methods in psychology]].

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

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

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  <page>
    <title>Social Victorians/People/Abercorn</title>
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      <text bytes="37220" sha1="m1p75kkfcfbunn9injv09eks4m1psfr" xml:space="preserve">== Overview ==
The Dukedom of Abercorn is the last non-royal dukedom created. Queen Victoria created it in 1869. 

This page includes the Earl of Wicklow, the family of which married into the Abercorn family in 1816 when William Howard, 4th Earl of Wicklow married Lady Cecil Frances Hamilton — the daughter and only child of John Hamilton, 1st Marquess of Abercorn.&lt;ref&gt;{{Cite journal|date=2026-06-24|title=William Howard, 4th Earl of Wicklow|url=https://en.wikipedia.org/w/index.php?title=William_Howard,_4th_Earl_of_Wicklow&amp;oldid=1360966619|journal=Wikipedia|language=en}}&lt;/ref&gt; William Howard, 4th Earl of Wicklow was succeeded by his nephew, Charles Howard, 5th Earl of Wicklow (5 November 1839 – 20 June 1881).&lt;ref&gt;{{Cite journal|date=2024-08-26|title=Charles Howard, 5th Earl of Wicklow|url=https://en.wikipedia.org/w/index.php?title=Charles_Howard,_5th_Earl_of_Wicklow&amp;oldid=1242455245|journal=Wikipedia|language=en}}&lt;/ref&gt; Also Ralph Howard, 7th Earl of Wicklow married Lady Gladys Mary Hamilton (daughter of the 2nd Duke of Abercorn) in 1902.&lt;ref name=&quot;:18&quot;&gt;{{Cite journal|date=2025-08-05|title=Cecil Howard, 6th Earl of Wicklow|url=https://en.wikipedia.org/w/index.php?title=Cecil_Howard,_6th_Earl_of_Wicklow&amp;oldid=1304372795|journal=Wikipedia|language=en}}&lt;/ref&gt;

The National Library of Ireland has papers from Sarah Howard and her children, including Lady Caroline Howard. 

== Also Known As ==
*Family name: Hamilton
*the Duke of Abercorn
**James Hamilton, 1st Duke of Abercorn (10 August 1868 – 31 October 1885)&lt;ref name=&quot;:0&quot;&gt;&quot;James Hamilton, 1st Duke of Abercorn.&quot; {{Cite web|url=http://www.thepeerage.com/p10144.htm#i101433|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;
**James Hamilton, 2nd Duke of Abercorn (31 October 1885 – 3 January 1913)&lt;ref name=&quot;:12&quot;&gt;&quot;James Hamilton, 2nd Duke of Abercorn.&quot; {{Cite web|url=http://www.thepeerage.com/p10104.htm#i101033|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;
**James Albert Edward Hamilton, 3rd Duke of Abercorn (3 January 1913 – 12 September 1953)&lt;ref name=&quot;:13&quot;&gt;&quot;James Albert Edward Hamilton, 3rd Duke of Abercorn.&quot; {{Cite web|url=http://www.thepeerage.com/p10104.htm#i101031|title=Person Page|website=www.thepeerage.com|access-date=2020-10-09}}&lt;/ref&gt;
*the Duchess of Abercorn
**Louisa Russell Hamilton, Duchess of Abercorn (10 August 1868 – 31 October 1885)
**Maria Anna Curzon-Howe Hamilton (31 October 1885 – 3 January 1913)
*Dowager Duchess of Hamilton
**Louisa Russell Hamilton, Duchess of Abercorn (31 October 1885 – March 1905)
**Maria Anna Curzon-Howe Hamilton (3 January 1913 – )
*Subsidiary titles:
**Marquess of Hamilton (courtesy title for the heir apparent)
***James Albert Edward Hamilton, 3rd Duke of Abercorn (31 October 1885 – 12 September 1953)
**Viscount Strabane (courtesy title for the heir apparent of the Marquess of Hamilton)

== Acquaintances, Friends and Enemies ==
=== Friends ===
*The Royal Family, especially [[Social Victorians/People/Albert Edward, Prince of Wales | Albert Edward, Prince]] and [[Social Victorians/People/Alexandra, Princess of Wales | Alexandra, Princess]] of Wales, in the generation of the 2nd duke.

== Timeline ==
A lot of people are treated on this page, so this timeline will be somewhat chaotic to read. These events probably didn't directly affect every single person treated on this page, but discussions about them probably circulated through the families. The detail about Lady Caroline Howard and her mother, the Hon. Susan Howard, is to make these people, whose papers are in the National Library of Ireland, more concrete and known.

'''1832 October 25''', James Hamilton and Louisa Russell married at Gordon Castle, Fochabers, Morayshire, in Scotland.&lt;ref name=&quot;:0&quot; /&gt;

'''1854 May 23''', Beatrix Frances Hamilton and George Frederick D'Arcy Lambton married.&lt;ref&gt;&quot;Lady Beatrix Frances Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p1147.htm#i11470|title=Person Page|website=www.thepeerage.com|access-date=2020-10-09}}&lt;/ref&gt;

'''1855 April 10''', Harriet Georgiana Louisa Hamilton and Thomas George Anson married.&lt;ref name=&quot;:2&quot;&gt;&quot;Lady Harriett Georgiana Louisa Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p1034.htm#i10332|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;

'''1858 October 26''', Katherine Elizabeth Hamilton and William Henry Edgcumbe married.&lt;ref&gt;&quot;Lady Katherine Elizabeth Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p1135.htm#i11344|title=Person Page|website=www.thepeerage.com|access-date=2020-10-09}}&lt;/ref&gt;

'''1859 November 22''', Louisa Jane Hamilton and William Montagu Douglass Scott married.&lt;ref&gt;&quot;Lady Louisa Jane Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p10359.htm#i103583|title=Person Page|website=www.thepeerage.com|access-date=2020-10-09}}&lt;/ref&gt;

'''1868''', the title the Duke of Abercorn was created.&lt;ref&gt;{{Cite journal|date=2020-07-06|title=James Hamilton, 1st Duke of Abercorn|url=https://en.wikipedia.org/w/index.php?title=James_Hamilton,_1st_Duke_of_Abercorn&amp;oldid=966293304|journal=Wikipedia|language=en}}&lt;/ref&gt;

'''1869 January 7''', James Hamilton (2nd Duke) and Maria Anna Curzon-Howe married at St. George's Church, St. George Street, Hanover Square, in London.&lt;ref name=&quot;:3&quot;&gt;&quot;Lady Mary Anna Curzon.&quot; {{Cite web|url=http://www.thepeerage.com/p10104.htm#i101034|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;

'''1869 November 8''', there may have been a double wedding: Albertha Frances Anne Hamilton and George Charles Spencer-Churchill married&lt;ref name=&quot;:8&quot;&gt;&quot;Lady Albertha Frances Anne Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p10595.htm#i105942|title=Person Page|website=www.thepeerage.com|access-date=2020-10-09}}&lt;/ref&gt;, and Maud Evelyn Hamilton and Henry Petty-Fitzmaurice married&lt;ref name=&quot;:1&quot;&gt;&quot;Lady Maud Evelyn Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p1163.htm#i11629|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;.

'''1871 February 17, Friday''', Lady Caroline Howard attended a [[Social Victorians/Timeline/1870s#Birmingham Tennis Court Club Ball|ball hosted by the &quot;bachelors of the Tennis Court Club&quot; in Birmingham]].

'''1871 May 9, Tuesday''', Lady Caroline Howard, Lady Alice Howard and Lady Louisa Howard were [[Social Victorians/Timeline/1870s#9 May 1871, Tuesday, Queen's Drawing-Room|presented to Queen Victoria at a Drawing-room]] by their mother, the Hon. Mrs. Sarah Howard.

'''1871 August 31, Thursday''', The Freeman's Journal reported that &quot;The Hon. Mrs. Howard, Lady Caroline Howard and suite have arrived at the Morrisson Hotel.&quot;&lt;blockquote&gt;The following are amongst the latest arrivals at the Morrisson Hotel: — Mrs. Percival Maxwell and the Misses Maxwell and suite, Mr and Mrs Herbert Read and suite, Rev H R Heywood, and Master H A Heywood, Mr F H Downing, Mr M Neil, Mr and Mrs Herbert and suite, Mr Abbott, Mr D'Arcy, Mr and Mrs G Woods and suite.&lt;ref&gt;&quot;Fashion and Varieties.&quot; ''Freeman's Journal'' 31 August 1871, Thursday: 4 [of 4], Col. 1a [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000056/18710831/012/0004. Same print title, n.p.&lt;/ref&gt;&lt;/blockquote&gt;'''1871 November 28''', George Francis Hamilton and Maud Caroline Lascelles married.&lt;ref name=&quot;:6&quot;&gt;&quot;Rt. Hon. Lord Sir George Francis Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p1133.htm#i11323|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;

'''1872 January 5, Friday''', the Hon. Mrs. Howard and Lady Caroline Howard were reported to &quot;have arrived at Morrisson's Hotel in Dublin.&lt;ref&gt;&quot;Fashion and Varieties.&quot; ''Morning Mail'' (Dublin) 5 January 1872, Friday: 3 [of 4, digital], Col. 2c [of 10 on digital image]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0006103/18720105/067/0003. The digital image has the last 2 columns of the prior page on this page, so the citation should be to p. 2 [of 4], Col. 8c [of 8].&lt;/ref&gt; Also at the Morrisson's Hotel at this time was Sir Roland Blennerhassett, Bart., M.P.&lt;ref&gt;&quot;Fashion and Varieties.&quot; ''Dublin Evening'' Mail 5 January 1872, Friday:  3 [of 4], Col. 8b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000433/18720105/028/0003. Same print and digital title, print n.p.&lt;/ref&gt;

'''1872 March 2, Saturday''', the ''Weekly Freeman and Irish Agriculturalist'' reported that &quot;Lady Caroline Howard, Lady Louisa Howard, and the Hon Mrs Howard and suite, Shelton Abbey, have arrived at Morrisson's Hotel.&quot; Two 1-sentence paragraphs later, the paper reported that the same group had &quot;left Morrisson's Hotel for Shelton Abbey.&quot;&lt;ref&gt;&quot;Fashion and Varieties.&quot; ''Weekly Freeman's Journal'' 2 March 1872, Saturday: 7 [of 8], Col. 1a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001446/18720302/062/0007. Print title: ''Weekly Freeman and Irish Agriculturalist'', same p.&lt;/ref&gt; Shelton Abbey was the [[Social Victorians/People/Abercorn#Residences|ancestral seat and at this time the country residence]] of the Earls of Wicklow, Arklow, Co. Wicklow.

'''1873 January 14, Saturday''', &quot;Lord Dunally and suite, Hon. Mrs. Howard, Lady Alice Howard and suite, Lady Louise Howard and suite, and Lady Caroline Howard, have arrived at Morrisson's Hotel.&quot;&lt;ref&gt;&quot;Fashionable Intelligence.&quot; ''Dublin Evening Post'' 14 January 1873, Tuesday: 3 [of 4], Col. 5a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000435/18730114/049/0003. Same print and digital title, print p. is n.p.&lt;/ref&gt;

'''1874 December 15, Tuesday''', the Right Hon. Sir Michael and Lady Lucy Hicks-Beach hosted a dinner in the Chief Secretary's Lodge, suggesting that this social event might have had a political purpose. Mr. LeFanu cannot be the Irish writer Sheridan Le Fanu, who died 7 February 1873.&lt;ref&gt;{{Cite journal|date=2026-06-28|title=Sheridan Le Fanu|url=https://en.wikipedia.org/w/index.php?title=Sheridan_Le_Fanu&amp;oldid=1361491348|journal=Wikipedia|language=en}}&lt;/ref&gt; Perhaps this LeFanu is a relation, a son or brother?&lt;blockquote&gt;THE CHIEF SECRETARY’S LODGE.&lt;p&gt;

The Right Hon. Sir Michael and Lady Lucy Hicks-Beach entertained the following at dinner on Tuesday evening at the Chief Secretary’s Lodge: — Sir Dominic Corrlgan, Sir Arthur and Lady Olive Guinness, Lady Mary Fortescue, the Hon. Mrs. Howard and Lady Caroline Howard, Mr. and Mrs. Percy Bernard, Colonel Henry, R.A., and Mrs. Henry; Mr. Donnelly. C.B., and Mrs. Donnelly; Mr., Mrs., and Miss lsaac; Mr. LeFanu, Colonel Forster, Colonel Hillier, and Mr. Caulfield.&lt;ref&gt;&quot;Fashionable Intelligence.&quot; ''Cork Constitution'' 17 December 1874, Thursday: 4 [of 4; n.p. in print], Col. 1a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001648/18741217/099/0004. Print title: ''The Cork Constitution''.&lt;/ref&gt;&lt;/blockquote&gt;'''1876 March 23''', Cecil Howard, 6th Earl of Wicklow and Francesca Maria Chamberlayne married.&lt;ref name=&quot;:18&quot; /&gt;

'''1877 July 25, Wednesday''', Miss Tottenham, Lady Caroline Howard, Miss Colley are reported to have arrived at Merton Lodge in Torquay.&lt;ref&gt;&quot;The Torquay Directory.&quot; ''Torquay Directory and South Devon Journal'' 25 July 1877, Wednesday: 4 [of 8], Col. 7a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001246/18770725/085/0004. Same print and digital title and p.&lt;/ref&gt;

'''1877 July 28, Saturday''', Lady Caroline Howard is listed as one of the guests at Merton Lodge in Lincombe Hill Road Middle, Torquay. Other guests listed are Miss Kelly, Mrs. Frank Webber, Miss Tottenham and Miss Colley.&lt;ref&gt;&quot;49. Lincombe Hill Road. Middle.&quot; &quot;Torquay Directory.&quot; ''Torquay Times and South Devon Advertiser'' 28 July 1877, Saturday: 2 [of 8, both print and digital], Col. 3c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001420/18770728/039/0002.&lt;/ref&gt;

'''1877 December 22, Saturday''', Sarah Howard, Lady Caroline Howard and Captain the Hon. Cecil Ralph Howard were visitors in Dagmar Terrace in Portsmouth. The following are all the people listed as visitors at Dagmar Terrace, with the odd numbering:&lt;blockquote&gt;D&lt;small&gt;AGMAR&lt;/small&gt; T&lt;small&gt;ER&lt;/small&gt;&lt;small&gt;RACE&lt;/small&gt;.

# Captain the Hon. Cecil Ralph Howard, late 60th Rifles, &amp; the Hon Mrs Howard Lady Caroline Howard
# Captain &amp; Mrs. Henderson
## [a] The Hon. Richard and Mrs. Bineham 
# [a] Captain and Mrs. Fearson and family
# Mr.and Mrs. Hall Mrs. and the Misses Buchannans
# The Rev Palms &amp; fam
# [a] Colonel Johnston [a] Mrs. Oldfield [a] Miss Flowers
# Captain Parkinson and family&lt;ref&gt;&quot;Visitors' List.&quot; ''Portsmouth Times and Naval Gazette'' 22 December 1877, Saturday: 3 [of 10, digital and print], Col. 5 [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001365/18771222/027/0003. Print title: ''Portsmouth Times and Naval Gazette, County Journal''.&lt;/ref&gt;
&lt;/blockquote&gt;'''1878 January 19, Saturday''', The ''Dublin Evening Mail'' says,&lt;blockquote&gt;Lady Caroline Howard has arrived at Kingstown from England.

Captain the Hon. C. Howard and Mrs. Howard have arrived at Kingstown from England.&lt;ref&gt;&quot;Viceregal Court.&quot; ''Dublin Evening Mail'' 19 January 1878, Saturday: 3 [of 4], Col. 8a [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000433/18780119/110/0003. Same print title, n.p.&lt;/ref&gt;&lt;/blockquote&gt;'''1878 July 20''', Claud John Hamilton and Carolina Chandos-Pole married.&lt;ref name=&quot;:5&quot;&gt;&quot;Lord Claud John Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p11067.htm#i110662|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;

'''1880 June 2''', Cecil Howard, 6th Earl of Wicklow and Fanny Catherine Wingfield married.&lt;ref name=&quot;:18&quot; /&gt;

'''1881 July 25, Monday''', the ''Irish Times'' says that Lady Caroline Howard and &quot;the Hon. Mrs. Howard and the Ladies Howard (2) have arrived at Kingstown from England.&quot;&lt;ref&gt;&quot;Fashionable Intelligence.&quot; ''Irish Times'' 25 July 1881, Monday: 6 [of 8, digital and print], Col. 3a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001683/18810725/124/0006. Same print title and p.&lt;/ref&gt;

'''1881 August 10, Wednesday''', the Dublin Evening Mail says that Lady Caroline Howard &quot;has left Kingstown for England.&quot;&lt;ref&gt;&quot;Fashion and Varieties.&quot; ''Dublin Evening Mail'' 10 August 1881, Wednesday: 3 [of 4], Col. 9c [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000433/18810810/046/0003. Same print and digital title, print p. is n.p.&lt;/ref&gt;

'''1881 October 22, Saturday''', Lady Caroline Howard is listed as one of the visitors staying at the Crown Hotel &quot;during the past week.&quot; The visitors listed are the following:&lt;blockquote&gt;Mr. Thomas Barber, Doctor and Mrs. Ayerst, Miss Noyce, Dr. Wilks, Mr. Nightingale, Mr. and Mrs. J. Hill, Lady Caroline Howard, the Hon. Mrs. Ross, Mr. Masters, Mr. Richardson and friend, Mr. Simpson, Mr. Wilson, &amp;c.&lt;ref&gt;&quot;Lyndhurst, Oct. 22.&quot; ''Hampshire Advertiser'' 22 October 1881, Saturday: 7 [of  8, both print and digital], Col. 2c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000495/18811022/049/0007. Print title: ''Hampshire Advertiser County Newspaper''.&lt;/ref&gt;&lt;/blockquote&gt;'''1882 March 16''', Georgiana Susan Hamilton and Edward Turnour married.&lt;ref&gt;&quot;Lady Georgiana Susan Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p1180.htm#i11791|title=Person Page|website=www.thepeerage.com|access-date=2020-10-09}}&lt;/ref&gt;

'''1883 November 20''', the marriage between Albertha Frances Anne Hamilton Spencer-Churchill and George Charles Spencer-Churchill was annulled by petition from Albertha Frances Anne Hamilton Spencer-Churchill (married in 1869).&lt;ref name=&quot;:8&quot; /&gt;

'''1891 June 2''', Ernest William Hamilton and Pamela Campbell married.&lt;ref name=&quot;:7&quot;&gt;&quot;Pamela Campbell.&quot; {{Cite web|url=http://www.thepeerage.com/p2107.htm#i21063|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;

'''1894 April 10''', Fanny Catherine Wingfield Howard, Dowager 6th Countess of Wicklow married her 2nd husband, Marcus Francis Beresford.&lt;ref name=&quot;:18&quot; /&gt;

'''1894 November 1''', James Albert Edward Hamilton and Rosaline Cecilia Caroline Bingham married at St. Paul's Church, Knightsbridge, in London.&lt;ref name=&quot;:14&quot;&gt;&quot;Lady Rosalind Cecilia Caroline Bingham.&quot; {{Cite web|url=https://www.thepeerage.com/p10104.htm#i101032|title=Person Page|website=www.thepeerage.com|access-date=2021-05-15}}&lt;/ref&gt;

'''1895 July 13 to August 7''', the general election of 1895. Following the election, the brother-in-law of Cecil Howard, 6th Earl of Wicklow's (brother of his first wife Francesca Chamberlayne) was unseated because of allegations of misconduct.&lt;ref&gt;{{Cite journal|date=2026-02-27|title=Thomas Chamberlayne (cricketer)|url=https://en.wikipedia.org/w/index.php?title=Thomas_Chamberlayne_(cricketer)&amp;oldid=1340809770|journal=Wikipedia|language=en}}&lt;/ref&gt;

'''1897 June 28, Monday''', according to the ''Morning Post'', James Hamilton, 2nd Duke and Maria, Duchess of Abercorn were invited to the [[Social Victorians/Diamond Jubilee Garden Party|Queen's Garden Party]], the official end of the Diamond Jubilee celebrations in London, as were James Albert Edward Hamilton, Marquis and Rosaline, Marchioness of Hamilton.&lt;ref&gt;“The Queen’s Garden Party.” ''Morning Post'' 29 June 1897, Tuesday: 4 [of 12], Cols. 1a–7c [of 7] and 5, Col. 1a–c. ''British Newspaper Archive''  ''&lt;nowiki&gt;https://www.britishnewspaperarchive.co.uk/viewer/BL/0000174/18970629/032/0004&lt;/nowiki&gt;'' and ''&lt;nowiki&gt;https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970629/032/0005&lt;/nowiki&gt;''.&lt;/ref&gt;

'''1897 July 2, Friday''', Alexandra Phyllis Hamilton attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House, as did her uncle Lord Frederick Spencer Hamilton, the Marquess of Hamilton, and a Mr. Ronald Hamilton. Besides these, probably, a Mr. and Mrs. Hamilton also attended.

'''1902''', Ralph Howard, 7th Earl of Wicklow and Lady Gladys Mary Hamilton married. (She was the daughter of James Hamilton, 2nd Duke of Abercorn.)&lt;ref name=&quot;:18&quot; /&gt;

'''1902 January 14''', Gladys Mary Hamilton and Ralph Francis Forward-Howard married.&lt;ref&gt;&quot;Lady Gladys Mary Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p2107.htm#i21066|title=Person Page|website=www.thepeerage.com|access-date=2020-10-09}}&lt;/ref&gt;

'''1933 July 11''', Claud Nigel Hamilton and Violet Ruby Ashton married.&lt;ref name=&quot;:4&quot;&gt;&quot;Captain Lord Sir Claud Nigel Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p2109.htm#i21081|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;

== Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball ==

[[File:Helen-Mary-Theresa-ne-Vane-Tempest-Stewart-Countess-of-Ilchester-when-Lady-Helen-Stewart-as-the-Archduchess-Marie-Christine-of-Austria.jpg|thumb|alt=Black-and-white photograph of a seated woman richly dressed in an historical costume with a white feather plume in her hair and a fan|Lady Helen Stewart as Arch-duchess Marie Christine of Austria. ©National Portrait Gallery, London.]]
=== Lady Alexandra Hamilton ===
Lady Alexandra Hamilton was one of the archduchesses — along with with 3 or 4 other young women — in [[Social Victorians/People/Londonderry#The Entourage of Maria Thérèse|the entourage of the Marchioness of Londonderry]], who led the Austrian procession as Marie Thérèse, Empress of the Holy Roman Empire.&lt;ref&gt;“The Ball at Devonshire House. Magnificent Spectacle. Description of the Dresses.” London ''Evening Standard'' 3 July 1897 Saturday: 3 [of 12], Cols. 1a–5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000183/18970703/015/0004.&lt;/ref&gt;{{rp|p. 3, Col. 3a}} These young women were present at the ball as the daughters of Marie Thérèse, and the young men dressed as archdukes were present as her sons. Lady Alexandra Hamilton went as &quot;Archduchess Marie-Josepha in the Archduchess Marie-Karoline and Emperor Joseph II section of the Austrian Court of Maria Theresa Quadrille.&quot;&lt;ref name=&quot;:9&quot;&gt;&quot;Fancy Dress Ball at Devonshire House.&quot; ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.&lt;/ref&gt;{{rp|p. 7, Col. 6b}} &lt;ref name=&quot;:10&quot;&gt;&quot;Ball at Devonshire House.&quot; The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.&lt;/ref&gt;

The newspapers report that the archduchesses were all dressed alike, but only one photograph exists of any of these young women in costume — that of [[Social Victorians/People/Londonderry#Helen Mary Theresa Vane-Tempest-Stewart|Helen Mary Theresa Vane-Tempest-Stewart]] (which is shown, right). The newspaper descriptions are on her page, with her portrait in costume, but they apply to all the archduchesses.

=== Lord Frederick Hamilton ===
[[File:Lord Frederick Spencer Hamilton Vanity Fair 1895-02-07.jpg|thumb|left|alt=Colored drawing of a man in a suit, his hands in his pockets, facing to the right|Lord Frederick Hamilton, ''Vanity Fair'', by &quot;Spy,&quot; 7 February 1895]]
Lord Frederick Spencer Hamilton was 6th son and 13th child of the 1st Duke of Abercorn. No photograph of him in costume exists.

He is shown (at left) as he looked in 7 February 1895 in a Spy caricature in ''Vanity Fair''. This caricature portrait, by Leslie Ward (&quot;Spy&quot;) is called ''The Pall Mall Magazine'' and is Number 647 in Vanity Fair's &quot;Statesmen&quot; series.&lt;ref name=&quot;:16&quot;&gt;{{Cite journal|date=2024-01-14|title=List of Vanity Fair (British magazine) caricatures (1895–1899)|url=https://en.wikipedia.org/w/index.php?title=List_of_Vanity_Fair_(British_magazine)_caricatures_(1895%E2%80%931899)&amp;oldid=1195518024|journal=Wikipedia|language=en}}&lt;/ref&gt; He was editor of the ''Pall Mall Gazette'' 1896–1900.&lt;ref&gt;{{Cite journal|date=2023-09-23|title=Lord Frederick Spencer Hamilton|url=https://en.wikipedia.org/w/index.php?title=Lord_Frederick_Spencer_Hamilton&amp;oldid=1176655264|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Lord_Frederick_Spencer_Hamilton.&lt;/ref&gt;

For the ball, Lord Frederick Hamilton was dressed
*as a &quot;gentleman of the Court of Queen Elizabeth,&quot; wearing &quot;crimson cloth of gold with jewelled belt.&quot;&lt;ref name=&quot;:15&quot;&gt;“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.&lt;/ref&gt;{{rp|p. 36, Col. 3b}}
*as a &quot;Gentleman of the Court of Queen Elizabeth. Costume of crimson and cloth of g [sic] with jewelled belt.&quot;&lt;ref name=&quot;:9&quot; /&gt;{{rp|p. 8, Col. 1b}}
*&quot;in crimson cloth of gold and jeweled belt.&quot;&lt;ref&gt;&quot;Duchess of Devonshire's Fancy Ball. A Brilliant Spectacle. Some of the Dresses.&quot; London ''Daily News'' Saturday 3 July 1897: 5 [of 10], Col. 6a–6, Col. 1b. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000051/18970703/024/0005 and http://www.britishnewspaperarchive.co.uk/viewer/BL/0000051/18970703/024/0006.&lt;/ref&gt;{{rp|p. 5, Col. 7a}}
*&quot;as a gentleman of the court of Queen Elizabeth, was dressed in a costume of crimson cloth-of-gold, with a jewelled belt.&quot;&lt;ref name=&quot;:11&quot;&gt;“The Devonshire House Ball. A Brilliant Gathering.” The ''Pall Mall Gazette'' 3 July 1897, Saturday: 7 [of 10], Col. 2a–3a. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000098/18970703/019/0007.&lt;/ref&gt;

==== Memoirs ====

* Hamilton, Frederic [sic] Spencer. ''My Yesterdays'' (3 vols.). Hodder and Stoughton, 1920.
*# ''The Days Before Yesterday''. The Internet Archive has this: https://archive.org/details/daysbeforeyester00hamiuoft/page/n5/mode/2up.
*# ''Vanished Pomps of Yesterday''. The Internet Archive has this: https://archive.org/details/vanishedpompsofy028823mbp.
*# ''Here, There and Everywhere''. The Internet Archive has this: https://archive.org/details/herethereeverywh0000hami.

[[File:James Hamilton 3rd Duke of Abercorn.png|thumb|alt=Old colored drawing of a man in a 19th-century officer's uniform of the 1st Life Guards with white gloves, a red stripe down the side of his pants and unbuttoned jacket and a hat, holding a white or silver sword under his left arm, facing 1/4 to his right|&quot;He will be the 3rd Duke&quot; (James Hamilton, Marquis of Hamilton), ''Vanity Fair'' 16 February 1899]]
=== James Hamilton, Marquess of Hamilton ===
James Hamilton, Marquis of Hamilton was dressed in a &quot;black velvet tunic; breeches and cloak trimmed jet; large hat, feathers, wig, sword, &amp;c., of the period&quot; of Charles II.&lt;ref name=&quot;:15&quot; /&gt;{{rp|34, Col. 3a}} No photograph of him in costume exists.

A caricature portrait (right) called ''He will be the 3rd Duke'' (James Hamilton, Marquess of Hamilton) by &quot;Hadge&quot; appeared in the 16 February 1899 issue of ''Vanity Fair'', as Number 739 in its &quot;Men of the Day&quot; series,&lt;ref name=&quot;:16&quot; /&gt; giving a sense of what he looked like at about the time of the ball.

In 1892 Hamilton joined the 1st Life Guards, so the uniform he is wearing in this portrait is likely that of an officer of the 1st Life Guards.&lt;ref&gt;{{Cite journal|date=2024-01-12|title=James Hamilton, 3rd Duke of Abercorn|url=https://en.wikipedia.org/w/index.php?title=James_Hamilton,_3rd_Duke_of_Abercorn&amp;oldid=1195216640|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/James_Hamilton,_3rd_Duke_of_Abercorn.&lt;/ref&gt;

James Hamilton's wife Lady Rosalind Hamilton is not reported as having been present at the ball, perhaps because she was pregnant with her second child and gave birth in August, five weeks later, so she was around 8 months pregnant.

=== Ronald Hamilton ===
Mr. Ronald Hamilton, possibly Ronald James Hamilton, was dressed as a &quot;Gentleman of the Court of Queen Elizabeth, in black velvet trimmed with jet.&quot;&lt;ref name=&quot;:9&quot; /&gt;{{rp|p. 8, Col. 1c}}

== Demographics ==

=== Nationality ===
*The title Duke of Abercorn is in the peerage of Ireland; the Marquess of Hamilton is in the peerage of the U.K.

=== Residences ===

==== The Hon. Mrs. Sarah Howard and the Earls of Wicklow ====

* Shelton Abbey, Arklow, Co. Wicklow (east coast of Ireland) (until 1951)&lt;ref&gt;{{Cite journal|date=2026-06-30|title=Shelton Abbey Prison|url=https://en.wikipedia.org/w/index.php?title=Shelton_Abbey_Prison&amp;oldid=1361924427|journal=Wikipedia|language=en}}&lt;/ref&gt;

== Family ==
*James Hamilton, 1st Duke of Abercorn (21 January 1811 – 31 October 1885)&lt;ref name=&quot;:0&quot; /&gt;
*Louisa Russell Hamilton (– March 1905)
#Lady '''Harriet Georgiana Louisa Hamilton''' Anson (6 July 1834 – 23 April 1913)
#Lady Beatrix Frances Hamilton Lambton (21 July 1835 – 21 January 1871)
#Lady Louisa Jane Hamilton Scott (26 August 1836 – 16 March 1912)
#Lord '''James Hamilton, 2nd Duke of Abercorn''' (24 August 1838 – 3 January 1913)
#Lady Katherine Elizabeth Hamilton Edgcumbe (9 January 1840 – 3 September 1874)
#Lady Georgiana Susan Hamilton Turnour (7 July 1841 – 23 March 1913)
#Lord '''Claud John Hamilton''' (20 February 1843 – 26 January 1925)
#Rt. Hon. Lord Sir '''George Francis Hamilton''' (17 December 1845 – 22 September 1927)
#Lady Albertha Frances Anne Hamilton Spencer-Churchill (29 July 1847 – 7 January 1932)
#Lord Ronald Douglas Hamilton (17 March 1849 – DVP&lt;ref&gt;{{Cite journal|date=2020-07-27|title=James Hamilton, 2nd Duke of Abercorn|url=https://en.wikipedia.org/w/index.php?title=James_Hamilton,_2nd_Duke_of_Abercorn&amp;oldid=969822724|journal=Wikipedia|language=en}}&lt;/ref&gt; 6 November 1867)
#Lady Maud Evelyn Hamilton Petty-Fitzmaurice, the [[Social Victorians/People/Lansdowne | Marchioness of Lansdowne]] (17 December 1850 – 21 October 1932)&lt;ref name=&quot;:1&quot; /&gt;
#Lord Cosmo Hamilton (16 April 1853 – 16 April 1853)
#Lord '''Frederick Spencer Hamilton''' (13 October 1856 – 11 August 1928)
#Lord '''Ernest William Hamilton''' (5 September 1858 – 14 December 1939)


*Harriet Georgiana Louisa Hamilton Anson (6 July 1834 – 23 April 1913)&lt;ref name=&quot;:2&quot; /&gt;
*Thomas George Anson, 2nd Earl of Lichfield (15 August 1825 – 7 January 1892)
#Lady Evelyn Anson ( – 2 July 1895)
#Thomas Francis Anson, 3rd Earl of Lichfield (31 January 1856 – 29 July 1918)
#Hon. Sir George Augustus Anson (22 December 1857 – 25 May 1947)
#Major Hon. Henry James Anson (29 December 1858 – 26 February 1904)
#Lady Florence Beatrice Anson (1860 – 25 September 1946)
#Hon. Frederic William Anson (4 February 1862 – 2 April 1917)
#Hon. Claud Anson (11 January 1864 – 25 December 1947)
#Lady Beatrice Anson (1865 – 15 December 1919)
#Hon. Francis Anson (7 March 1867 – 13 April 1928)
#Lady Mary Maud Anson (1869 – 22 September 1961)
#Lady Edith Anson (1870 – 8 October 1932)
#Hon. William Anson (19 April 1872 – 22 June 1926)
#Hon. Alfred Anson (15 April 1876 – 25 March 1944)


*James Hamilton, 2nd Duke of Abercorn (24 August 1838 – 3 January 1913)&lt;ref name=&quot;:12&quot; /&gt;
*Maria Anna Curzon-Howe Hamilton (23 July 1848 – 10 May 1929)&lt;ref name=&quot;:3&quot; /&gt;
#James Albert Edward Hamilton, 3rd Duke of Abercorn (30 November 1869 – 12 September 1953)
#Claud Penn Alexander Hamilton (18 October 1871 – 18 October 1871)
#Charlie Hamilton (10 April 1874 – 10 April 1874)
#'''Alexandra Phyllis Hamilton''' (23 January 1876 – 10 October 1918)
#Claud Francis Hamilton (25 October 1878 – 25 December 1878)
#Gladys Mary Hamilton Forward-Howard (10 December 1880 – 12 March 1917)
#Arthur John Hamilton (20 August 1883 – 6 November 1914)
#(unnamed son) Hamilton (31 October 1886 – 31 October 1886)
#Claud Nigel Hamilton (10 November 1889 – 22 August 1975)&lt;ref name=&quot;:4&quot; /&gt;


* '''James Albert Edward Hamilton''', Marquess of Hamilton and 3rd Duke of Abercorn (30 November 1869 – 12 September 1953)&lt;ref name=&quot;:13&quot; /&gt;
* Lady Rosalind Cecilia Caroline Bingham (26 February 1869 – 18 January 1958)&lt;ref name=&quot;:14&quot; /&gt;
*# Lady Mary Cecilia Rhodesia Hamilton (21 January 1896 – 5 September 1984)
*# Lady Cynthia Elinor Beatrix Hamilton (16 August 1897 – 4 December 1972)
*# Lady Katharine Hamilton (25 February 1900 – 28 April 1985)
*# James Edward Hamilton, 4th Duke of Abercorn (29 February 1904 – 4 June 1979)
*# Captain Lord Claud David Hamilton (13 February 1907 – 15 February 1968)


*Claud John Hamilton (20 February 1843 – 26 January 1925)&lt;ref name=&quot;:5&quot; /&gt;
*Carolina Chandos-Pole Hamilton (19 July 1857 – 21 September 1911)&lt;ref&gt;&quot;Carolina Chandos-Pole.&quot; {{Cite web|url=http://www.thepeerage.com/p11067.htm#i110663|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;
#Colonel Gilbert Claud Hamilton (21 April 1879 – 30 March 1943)
#Ida Hamilton (23 July 1883 – November 1970)


*George Francis Hamilton (17 December 1845 – 22 September 1927)&lt;ref name=&quot;:6&quot; /&gt;
*Lady Maud Caroline Lascelles Hamilton (1846 – 14 April 1938)
#'''Ronald James Hamilton''' (26 September 1872 – 22 January 1958)
#Anthony George Hamilton (17 December 1874 – 11 July 1936)
#Robert Cecil Hamilton (31 January 1882 – 31 July 1947)


*Ernest William Hamilton (5 September 1858 – 14 December 1939)&lt;ref&gt;&quot;Lord Ernest William Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p2107.htm#i21062|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;
*Pamela Campbell Hamilton ( – 11 May 1931)&lt;ref name=&quot;:7&quot; /&gt;
#Guy Ernest Frederick Hamilton (11 November 1894 – 23 November 1914)
#Mary Brenda Hamilton (28 March 1897 – 14 March 1985)
#Jean Barbara Hamilton (6 September 1898 – 2 November 1989)
#John George Peter Hamilton (15 October 1900 – 17 June 1967)


=== Earls of Wicklow ===

* Charles Hamilton (1772 – 29 September 1857)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p2139.htm#i21387|title=Charles Hamilton. Person Page #2139|website=www.thepeerage.com|access-date=2026-06-19}}&lt;/ref&gt;
* Marianne '''Caroline Tighe''' ( – 29 July 1861)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p62375.htm#i623745|title=Marianne Caroline Tighe. Person Page #62375|website=www.thepeerage.com|access-date=2026-06-19}}&lt;/ref&gt;
*# '''Sarah Hamilton''' (1805&lt;ref name=&quot;:17&quot; /&gt; – 13 March 1892)
*# Caroline Elizabeth Hamilton ( – 31 May 1909)
*# Mary Hamilton
*# Charles William Hamilton (1 April 1802 – 16 February 1880)
*# William Tighe Hamilton (31 March 1807 – )
*# Frederick John Henry Fownes Hamilton (27 July 1816 – 1893)


* Rev. Hon. Francis Howard (12 January 1797 – 16 February 1857)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p2140.htm#i21391|title=Rev. Hon. Francis Howard. Person Page #2140|website=www.thepeerage.com|access-date=2026-06-19}}&lt;/ref&gt;
* Frances Beresford ( – 17 November 1833)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p3227.htm#i32266|title=Frances Beresford. Person Page #3227|website=www.thepeerage.com|access-date=2026-06-19}}&lt;/ref&gt;
*# William George Howard (25 April 1825 – 12 October 1864)
* '''Sarah Hamilton''' (1805&lt;ref name=&quot;:17&quot;&gt;{{Cite web|url=https://catalogue.nli.ie/Collection/vtls000572704|title=Tighe, Hamilton and Howard Papers,|date=1737|website=catalogue.nli.ie|language=English|access-date=2026-06-19}}&lt;/ref&gt; – 13 March 1892)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p2141.htm#i21405|title=Sarah Hamilton. Person Page #2141|website=www.thepeerage.com|access-date=2026-06-19}}&lt;/ref&gt;
*# 4 unnamed daughters [per The Peerage; The NLI has 3 daughters]
*# Lady Alice Howard
*# Lady Louisa 'Loulie' Howard
*# Lady Caroline Howard (1836–1923)&lt;ref name=&quot;:17&quot; /&gt;
*# Charles Francis Arnold Howard, '''5th Earl of Wicklow''' (5 November 1839 – 20 June 1881)
*# Cecil Ralph Howard, '''6th Earl of Wicklow''' (26 April 1842 – 24 July 1891)


* Cecil Ralph Howard, '''6th Earl of Wicklow''' (26 April 1842 – 24 July 1891)&lt;ref name=&quot;:18&quot; /&gt;
* Francesca Maria Chamberlayne ( – 1877)
*# Ralph Howard, 7th Earl of Wicklow (24 December 1877 – 11 October 1946)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p2140.htm#i21394|title=Cecil Ralph Howard, 6th Earl of Wicklow. Person Page 2140.|website=www.thepeerage.com|access-date=2026-06-28}}&lt;/ref&gt;
* Fanny Catherine Wingfield (c. 1860 – 3 February 1914)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p2139.htm#i21388|title=Fanny Catherine Wingfield. Person Page 2139.|website=www.thepeerage.com|access-date=2026-06-28}}&lt;/ref&gt;
*# Hon. Cecil Mervyn Malcolm Howard (18 November 1881 – 16 April 1882)
*# Hon. Hugh Melville Howard (28 March 1883 – 17 February 1919)
* Marcus Francis Beresford (26 December 1862 – 14 December 1896)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p3186.htm#i31858|title=Marcus Francis Beresford. Person Page #3186.|website=www.thepeerage.com|access-date=2026-06-28}}&lt;/ref&gt;

== Memoirs and Archives ==

# The Abercorn Papers: GB 0255 PRONI/D623 (found via https://iar.ie/archive/abercorn-papers). A descriptive list is available to search online at: http://www.proni.gov.uk/. The collection is arranged as follows: D623/A Correspondence D623/B Title deeds and leases D623/C Rentals, accounts and vouchers D623/D Maps, plans, surveys, inventories and valuations D623/E Photographs, illuminations, addresses and albums D623/F Material still at Baronscourt D623/G Miscellaneous
#Alexandra Phyllis Hamilton (#64 on the [[Social Victorians/1897 Fancy Dress Ball#List of People Who Attended|list of people who were present]]) attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House, as did her uncle Lord Frederick Spencer Hamilton (#84), the Marquess of Hamilton (#657), and a Mr. Ronald Hamilton (#105). Besides these, probably, a Mr. and Mrs. Hamilton also attended.

== Questions and Notes ==
#DVP = decessit vita patris, died while the father was still living
#Mr. Ronald Hamilton cannot be Frederick Hamilton's brother, who should be Lord Ronald Hamilton rather than Mr. Ronald Hamilton, and he died in 1867. He could be this Ronald Hamilton, who would be a Mr. Hamilton: http://www.thepeerage.com/p2163.htm#i21622. He was Lady Alexandra's cousin and nephew of the 1st Duke of Abercorn.
#A Mr. Hamilton is mentioned in the ''Gentlewoman'' article: &quot;Mr. Hamilton (Elizabethan costume), black velvet, trimmed gold.&quot;&lt;ref name=&quot;:15&quot; /&gt;{{rp|34, Col. 1c}} But a later reference in this same article to Mr. Ronald Hamilton matches the description in the ''Morning Post'' article, saying he wore black velvet with jet, rather than gold trim: &quot;'''Mr. Ronald Hamilton''' (gentleman of the Court of Queen Elizabeth), black velvet with jet.&quot;&lt;ref name=&quot;:15&quot; /&gt; (36, Col. 3b) I believe the other Mr. Hamilton is Mr. [[Social Victorians/People/Cole-Hamilton|Claud Cole-Hamilton]], particularly since Mrs. Hamilton was dressed as Amy Robsart and thus must be Lucy Charlewood Cole-Hamilton because of the description of her costume in the Album of photographs given to the Duchess of Devonshire later.
#Claud John Hamilton is probably who attended the social events, because the other Claud, of whatever generation either died too young or was born too late.
== Footnotes ==
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      <text bytes="37220" sha1="cm1vw2vixs28rbfm84riv3lecevktx9" xml:space="preserve">== Overview ==
The Dukedom of Abercorn is the last non-royal dukedom created. Queen Victoria created it in 1869. 

This page includes the Earl of Wicklow, the family of which married into the Abercorn family in 1816 when William Howard, 4th Earl of Wicklow married Lady Cecil Frances Hamilton — the daughter and only child of John Hamilton, 1st Marquess of Abercorn.&lt;ref&gt;{{Cite journal|date=2026-06-24|title=William Howard, 4th Earl of Wicklow|url=https://en.wikipedia.org/w/index.php?title=William_Howard,_4th_Earl_of_Wicklow&amp;oldid=1360966619|journal=Wikipedia|language=en}}&lt;/ref&gt; William Howard, 4th Earl of Wicklow was succeeded by his nephew, Charles Howard, 5th Earl of Wicklow (5 November 1839 – 20 June 1881).&lt;ref&gt;{{Cite journal|date=2024-08-26|title=Charles Howard, 5th Earl of Wicklow|url=https://en.wikipedia.org/w/index.php?title=Charles_Howard,_5th_Earl_of_Wicklow&amp;oldid=1242455245|journal=Wikipedia|language=en}}&lt;/ref&gt; Also Ralph Howard, 7th Earl of Wicklow married Lady Gladys Mary Hamilton (daughter of the 2nd Duke of Abercorn) in 1902.&lt;ref name=&quot;:18&quot;&gt;{{Cite journal|date=2025-08-05|title=Cecil Howard, 6th Earl of Wicklow|url=https://en.wikipedia.org/w/index.php?title=Cecil_Howard,_6th_Earl_of_Wicklow&amp;oldid=1304372795|journal=Wikipedia|language=en}}&lt;/ref&gt;

The National Library of Ireland has papers from Sarah Howard and her children, including Lady Caroline Howard. 

== Also Known As ==
*Family name: Hamilton
*the Duke of Abercorn
**James Hamilton, 1st Duke of Abercorn (10 August 1868 – 31 October 1885)&lt;ref name=&quot;:0&quot;&gt;&quot;James Hamilton, 1st Duke of Abercorn.&quot; {{Cite web|url=http://www.thepeerage.com/p10144.htm#i101433|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;
**James Hamilton, 2nd Duke of Abercorn (31 October 1885 – 3 January 1913)&lt;ref name=&quot;:12&quot;&gt;&quot;James Hamilton, 2nd Duke of Abercorn.&quot; {{Cite web|url=http://www.thepeerage.com/p10104.htm#i101033|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;
**James Albert Edward Hamilton, 3rd Duke of Abercorn (3 January 1913 – 12 September 1953)&lt;ref name=&quot;:13&quot;&gt;&quot;James Albert Edward Hamilton, 3rd Duke of Abercorn.&quot; {{Cite web|url=http://www.thepeerage.com/p10104.htm#i101031|title=Person Page|website=www.thepeerage.com|access-date=2020-10-09}}&lt;/ref&gt;
*the Duchess of Abercorn
**Louisa Russell Hamilton, Duchess of Abercorn (10 August 1868 – 31 October 1885)
**Maria Anna Curzon-Howe Hamilton (31 October 1885 – 3 January 1913)
*Dowager Duchess of Hamilton
**Louisa Russell Hamilton, Duchess of Abercorn (31 October 1885 – March 1905)
**Maria Anna Curzon-Howe Hamilton (3 January 1913 – )
*Subsidiary titles:
**Marquess of Hamilton (courtesy title for the heir apparent)
***James Albert Edward Hamilton, 3rd Duke of Abercorn (31 October 1885 – 12 September 1953)
**Viscount Strabane (courtesy title for the heir apparent of the Marquess of Hamilton)

== Acquaintances, Friends and Enemies ==
=== Friends ===
*The Royal Family, especially [[Social Victorians/People/Albert Edward, Prince of Wales | Albert Edward, Prince]] and [[Social Victorians/People/Alexandra, Princess of Wales | Alexandra, Princess]] of Wales, in the generation of the 2nd duke.

== Timeline ==
A lot of people are treated on this page, so this timeline will be somewhat chaotic to read. These events probably didn't directly affect every single person treated on this page, but discussions about them probably circulated through the families. The detail about Lady Caroline Howard and her mother, the Hon. Susan Howard, is to make these people, whose papers are in the National Library of Ireland, more concrete and known.

'''1832 October 25''', James Hamilton and Louisa Russell married at Gordon Castle, Fochabers, Morayshire, in Scotland.&lt;ref name=&quot;:0&quot; /&gt;

'''1854 May 23''', Beatrix Frances Hamilton and George Frederick D'Arcy Lambton married.&lt;ref&gt;&quot;Lady Beatrix Frances Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p1147.htm#i11470|title=Person Page|website=www.thepeerage.com|access-date=2020-10-09}}&lt;/ref&gt;

'''1855 April 10''', Harriet Georgiana Louisa Hamilton and Thomas George Anson married.&lt;ref name=&quot;:2&quot;&gt;&quot;Lady Harriett Georgiana Louisa Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p1034.htm#i10332|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;

'''1858 October 26''', Katherine Elizabeth Hamilton and William Henry Edgcumbe married.&lt;ref&gt;&quot;Lady Katherine Elizabeth Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p1135.htm#i11344|title=Person Page|website=www.thepeerage.com|access-date=2020-10-09}}&lt;/ref&gt;

'''1859 November 22''', Louisa Jane Hamilton and William Montagu Douglass Scott married.&lt;ref&gt;&quot;Lady Louisa Jane Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p10359.htm#i103583|title=Person Page|website=www.thepeerage.com|access-date=2020-10-09}}&lt;/ref&gt;

'''1868''', the title the Duke of Abercorn was created.&lt;ref&gt;{{Cite journal|date=2020-07-06|title=James Hamilton, 1st Duke of Abercorn|url=https://en.wikipedia.org/w/index.php?title=James_Hamilton,_1st_Duke_of_Abercorn&amp;oldid=966293304|journal=Wikipedia|language=en}}&lt;/ref&gt;

'''1869 January 7''', James Hamilton (2nd Duke) and Maria Anna Curzon-Howe married at St. George's Church, St. George Street, Hanover Square, in London.&lt;ref name=&quot;:3&quot;&gt;&quot;Lady Mary Anna Curzon.&quot; {{Cite web|url=http://www.thepeerage.com/p10104.htm#i101034|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;

'''1869 November 8''', there may have been a double wedding: Albertha Frances Anne Hamilton and George Charles Spencer-Churchill married&lt;ref name=&quot;:8&quot;&gt;&quot;Lady Albertha Frances Anne Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p10595.htm#i105942|title=Person Page|website=www.thepeerage.com|access-date=2020-10-09}}&lt;/ref&gt;, and Maud Evelyn Hamilton and Henry Petty-Fitzmaurice married&lt;ref name=&quot;:1&quot;&gt;&quot;Lady Maud Evelyn Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p1163.htm#i11629|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;.

'''1871 February 17, Friday''', Lady Caroline Howard attended a [[Social Victorians/Timeline/1870s#Birmingham Tennis Court Club Ball|ball hosted by the &quot;bachelors of the Tennis Court Club&quot; in Birmingham]].

'''1871 May 9, Tuesday''', Lady Caroline Howard, Lady Alice Howard and Lady Louisa Howard were [[Social Victorians/Timeline/1870s#9 May 1871, Tuesday, Queen's Drawing-Room|presented to Queen Victoria at a Drawing-room]] by their mother, the Hon. Mrs. Sarah Howard.

'''1871 August 31, Thursday''', The Freeman's Journal reported that &quot;The Hon. Mrs. Howard, Lady Caroline Howard and suite have arrived at the Morrisson Hotel.&quot;&lt;blockquote&gt;The following are amongst the latest arrivals at the Morrisson Hotel: — Mrs. Percival Maxwell and the Misses Maxwell and suite, Mr and Mrs Herbert Read and suite, Rev H R Heywood, and Master H A Heywood, Mr F H Downing, Mr M Neil, Mr and Mrs Herbert and suite, Mr Abbott, Mr D'Arcy, Mr and Mrs G Woods and suite.&lt;ref&gt;&quot;Fashion and Varieties.&quot; ''Freeman's Journal'' 31 August 1871, Thursday: 4 [of 4], Col. 1a [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000056/18710831/012/0004. Same print title, n.p.&lt;/ref&gt;&lt;/blockquote&gt;'''1871 November 28''', George Francis Hamilton and Maud Caroline Lascelles married.&lt;ref name=&quot;:6&quot;&gt;&quot;Rt. Hon. Lord Sir George Francis Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p1133.htm#i11323|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;

'''1872 January 5, Friday''', the Hon. Mrs. Howard and Lady Caroline Howard were reported to &quot;have arrived at Morrisson's Hotel in Dublin.&lt;ref&gt;&quot;Fashion and Varieties.&quot; ''Morning Mail'' (Dublin) 5 January 1872, Friday: 3 [of 4, digital], Col. 2c [of 10 on digital image]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0006103/18720105/067/0003. The digital image has the last 2 columns of the prior page on this page, so the citation should be to p. 2 [of 4], Col. 8c [of 8].&lt;/ref&gt; Also at the Morrisson's Hotel at this time was Sir Roland Blennerhassett, Bart., M.P.&lt;ref&gt;&quot;Fashion and Varieties.&quot; ''Dublin Evening'' Mail 5 January 1872, Friday:  3 [of 4], Col. 8b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000433/18720105/028/0003. Same print and digital title, print n.p.&lt;/ref&gt;

'''1872 March 2, Saturday''', the ''Weekly Freeman and Irish Agriculturalist'' reported that &quot;Lady Caroline Howard, Lady Louisa Howard, and the Hon Mrs Howard and suite, Shelton Abbey, have arrived at Morrisson's Hotel.&quot; Two 1-sentence paragraphs later, the paper reported that the same group had &quot;left Morrisson's Hotel for Shelton Abbey.&quot;&lt;ref&gt;&quot;Fashion and Varieties.&quot; ''Weekly Freeman's Journal'' 2 March 1872, Saturday: 7 [of 8], Col. 1a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001446/18720302/062/0007. Print title: ''Weekly Freeman and Irish Agriculturalist'', same p.&lt;/ref&gt; Shelton Abbey was the [[Social Victorians/People/Abercorn#Residences|ancestral seat and at this time the country residence]] of the Earls of Wicklow, Arklow, Co. Wicklow.

'''1873 January 14, Saturday''', &quot;Lord Dunally and suite, Hon. Mrs. Howard, Lady Alice Howard and suite, Lady Louise Howard and suite, and Lady Caroline Howard, have arrived at Morrisson's Hotel.&quot;&lt;ref&gt;&quot;Fashionable Intelligence.&quot; ''Dublin Evening Post'' 14 January 1873, Tuesday: 3 [of 4], Col. 5a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000435/18730114/049/0003. Same print and digital title, print p. is n.p.&lt;/ref&gt;

'''1874 December 15, Tuesday''', the Right Hon. Sir Michael and Lady Lucy Hicks-Beach hosted a dinner in the Chief Secretary's Lodge, suggesting that this social event might have had a political purpose. Mr. LeFanu cannot be the Irish writer Sheridan Le Fanu, who died 7 February 1873.&lt;ref&gt;{{Cite journal|date=2026-06-28|title=Sheridan Le Fanu|url=https://en.wikipedia.org/w/index.php?title=Sheridan_Le_Fanu&amp;oldid=1361491348|journal=Wikipedia|language=en}}&lt;/ref&gt; Perhaps this LeFanu is a relation, a son or brother?&lt;blockquote&gt;THE CHIEF SECRETARY’S LODGE.&lt;p&gt;

The Right Hon. Sir Michael and Lady Lucy Hicks-Beach entertained the following at dinner on Tuesday evening at the Chief Secretary’s Lodge: — Sir Dominic Corrigan, Sir Arthur and Lady Olive Guinness, Lady Mary Fortescue, the Hon. Mrs. Howard and Lady Caroline Howard, Mr. and Mrs. Percy Bernard, Colonel Henry, R.A., and Mrs. Henry; Mr. Donnelly. C.B., and Mrs. Donnelly; Mr., Mrs., and Miss lsaac; Mr. LeFanu, Colonel Forster, Colonel Hillier, and Mr. Caulfield.&lt;ref&gt;&quot;Fashionable Intelligence.&quot; ''Cork Constitution'' 17 December 1874, Thursday: 4 [of 4; n.p. in print], Col. 1a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001648/18741217/099/0004. Print title: ''The Cork Constitution''.&lt;/ref&gt;&lt;/blockquote&gt;'''1876 March 23''', Cecil Howard, 6th Earl of Wicklow and Francesca Maria Chamberlayne married.&lt;ref name=&quot;:18&quot; /&gt;

'''1877 July 25, Wednesday''', Miss Tottenham, Lady Caroline Howard, Miss Colley are reported to have arrived at Merton Lodge in Torquay.&lt;ref&gt;&quot;The Torquay Directory.&quot; ''Torquay Directory and South Devon Journal'' 25 July 1877, Wednesday: 4 [of 8], Col. 7a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001246/18770725/085/0004. Same print and digital title and p.&lt;/ref&gt;

'''1877 July 28, Saturday''', Lady Caroline Howard is listed as one of the guests at Merton Lodge in Lincombe Hill Road Middle, Torquay. Other guests listed are Miss Kelly, Mrs. Frank Webber, Miss Tottenham and Miss Colley.&lt;ref&gt;&quot;49. Lincombe Hill Road. Middle.&quot; &quot;Torquay Directory.&quot; ''Torquay Times and South Devon Advertiser'' 28 July 1877, Saturday: 2 [of 8, both print and digital], Col. 3c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001420/18770728/039/0002.&lt;/ref&gt;

'''1877 December 22, Saturday''', Sarah Howard, Lady Caroline Howard and Captain the Hon. Cecil Ralph Howard were visitors in Dagmar Terrace in Portsmouth. The following are all the people listed as visitors at Dagmar Terrace, with the odd numbering:&lt;blockquote&gt;D&lt;small&gt;AGMAR&lt;/small&gt; T&lt;small&gt;ER&lt;/small&gt;&lt;small&gt;RACE&lt;/small&gt;.

# Captain the Hon. Cecil Ralph Howard, late 60th Rifles, &amp; the Hon Mrs Howard Lady Caroline Howard
# Captain &amp; Mrs. Henderson
## [a] The Hon. Richard and Mrs. Bineham 
# [a] Captain and Mrs. Fearson and family
# Mr.and Mrs. Hall Mrs. and the Misses Buchannans
# The Rev Palms &amp; fam
# [a] Colonel Johnston [a] Mrs. Oldfield [a] Miss Flowers
# Captain Parkinson and family&lt;ref&gt;&quot;Visitors' List.&quot; ''Portsmouth Times and Naval Gazette'' 22 December 1877, Saturday: 3 [of 10, digital and print], Col. 5 [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001365/18771222/027/0003. Print title: ''Portsmouth Times and Naval Gazette, County Journal''.&lt;/ref&gt;
&lt;/blockquote&gt;'''1878 January 19, Saturday''', The ''Dublin Evening Mail'' says,&lt;blockquote&gt;Lady Caroline Howard has arrived at Kingstown from England.

Captain the Hon. C. Howard and Mrs. Howard have arrived at Kingstown from England.&lt;ref&gt;&quot;Viceregal Court.&quot; ''Dublin Evening Mail'' 19 January 1878, Saturday: 3 [of 4], Col. 8a [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000433/18780119/110/0003. Same print title, n.p.&lt;/ref&gt;&lt;/blockquote&gt;'''1878 July 20''', Claud John Hamilton and Carolina Chandos-Pole married.&lt;ref name=&quot;:5&quot;&gt;&quot;Lord Claud John Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p11067.htm#i110662|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;

'''1880 June 2''', Cecil Howard, 6th Earl of Wicklow and Fanny Catherine Wingfield married.&lt;ref name=&quot;:18&quot; /&gt;

'''1881 July 25, Monday''', the ''Irish Times'' says that Lady Caroline Howard and &quot;the Hon. Mrs. Howard and the Ladies Howard (2) have arrived at Kingstown from England.&quot;&lt;ref&gt;&quot;Fashionable Intelligence.&quot; ''Irish Times'' 25 July 1881, Monday: 6 [of 8, digital and print], Col. 3a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001683/18810725/124/0006. Same print title and p.&lt;/ref&gt;

'''1881 August 10, Wednesday''', the Dublin Evening Mail says that Lady Caroline Howard &quot;has left Kingstown for England.&quot;&lt;ref&gt;&quot;Fashion and Varieties.&quot; ''Dublin Evening Mail'' 10 August 1881, Wednesday: 3 [of 4], Col. 9c [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000433/18810810/046/0003. Same print and digital title, print p. is n.p.&lt;/ref&gt;

'''1881 October 22, Saturday''', Lady Caroline Howard is listed as one of the visitors staying at the Crown Hotel &quot;during the past week.&quot; The visitors listed are the following:&lt;blockquote&gt;Mr. Thomas Barber, Doctor and Mrs. Ayerst, Miss Noyce, Dr. Wilks, Mr. Nightingale, Mr. and Mrs. J. Hill, Lady Caroline Howard, the Hon. Mrs. Ross, Mr. Masters, Mr. Richardson and friend, Mr. Simpson, Mr. Wilson, &amp;c.&lt;ref&gt;&quot;Lyndhurst, Oct. 22.&quot; ''Hampshire Advertiser'' 22 October 1881, Saturday: 7 [of  8, both print and digital], Col. 2c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000495/18811022/049/0007. Print title: ''Hampshire Advertiser County Newspaper''.&lt;/ref&gt;&lt;/blockquote&gt;'''1882 March 16''', Georgiana Susan Hamilton and Edward Turnour married.&lt;ref&gt;&quot;Lady Georgiana Susan Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p1180.htm#i11791|title=Person Page|website=www.thepeerage.com|access-date=2020-10-09}}&lt;/ref&gt;

'''1883 November 20''', the marriage between Albertha Frances Anne Hamilton Spencer-Churchill and George Charles Spencer-Churchill was annulled by petition from Albertha Frances Anne Hamilton Spencer-Churchill (married in 1869).&lt;ref name=&quot;:8&quot; /&gt;

'''1891 June 2''', Ernest William Hamilton and Pamela Campbell married.&lt;ref name=&quot;:7&quot;&gt;&quot;Pamela Campbell.&quot; {{Cite web|url=http://www.thepeerage.com/p2107.htm#i21063|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;

'''1894 April 10''', Fanny Catherine Wingfield Howard, Dowager 6th Countess of Wicklow married her 2nd husband, Marcus Francis Beresford.&lt;ref name=&quot;:18&quot; /&gt;

'''1894 November 1''', James Albert Edward Hamilton and Rosaline Cecilia Caroline Bingham married at St. Paul's Church, Knightsbridge, in London.&lt;ref name=&quot;:14&quot;&gt;&quot;Lady Rosalind Cecilia Caroline Bingham.&quot; {{Cite web|url=https://www.thepeerage.com/p10104.htm#i101032|title=Person Page|website=www.thepeerage.com|access-date=2021-05-15}}&lt;/ref&gt;

'''1895 July 13 to August 7''', the general election of 1895. Following the election, the brother-in-law of Cecil Howard, 6th Earl of Wicklow's (brother of his first wife Francesca Chamberlayne) was unseated because of allegations of misconduct.&lt;ref&gt;{{Cite journal|date=2026-02-27|title=Thomas Chamberlayne (cricketer)|url=https://en.wikipedia.org/w/index.php?title=Thomas_Chamberlayne_(cricketer)&amp;oldid=1340809770|journal=Wikipedia|language=en}}&lt;/ref&gt;

'''1897 June 28, Monday''', according to the ''Morning Post'', James Hamilton, 2nd Duke and Maria, Duchess of Abercorn were invited to the [[Social Victorians/Diamond Jubilee Garden Party|Queen's Garden Party]], the official end of the Diamond Jubilee celebrations in London, as were James Albert Edward Hamilton, Marquis and Rosaline, Marchioness of Hamilton.&lt;ref&gt;“The Queen’s Garden Party.” ''Morning Post'' 29 June 1897, Tuesday: 4 [of 12], Cols. 1a–7c [of 7] and 5, Col. 1a–c. ''British Newspaper Archive''  ''&lt;nowiki&gt;https://www.britishnewspaperarchive.co.uk/viewer/BL/0000174/18970629/032/0004&lt;/nowiki&gt;'' and ''&lt;nowiki&gt;https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970629/032/0005&lt;/nowiki&gt;''.&lt;/ref&gt;

'''1897 July 2, Friday''', Alexandra Phyllis Hamilton attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House, as did her uncle Lord Frederick Spencer Hamilton, the Marquess of Hamilton, and a Mr. Ronald Hamilton. Besides these, probably, a Mr. and Mrs. Hamilton also attended.

'''1902''', Ralph Howard, 7th Earl of Wicklow and Lady Gladys Mary Hamilton married. (She was the daughter of James Hamilton, 2nd Duke of Abercorn.)&lt;ref name=&quot;:18&quot; /&gt;

'''1902 January 14''', Gladys Mary Hamilton and Ralph Francis Forward-Howard married.&lt;ref&gt;&quot;Lady Gladys Mary Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p2107.htm#i21066|title=Person Page|website=www.thepeerage.com|access-date=2020-10-09}}&lt;/ref&gt;

'''1933 July 11''', Claud Nigel Hamilton and Violet Ruby Ashton married.&lt;ref name=&quot;:4&quot;&gt;&quot;Captain Lord Sir Claud Nigel Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p2109.htm#i21081|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;

== Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball ==

[[File:Helen-Mary-Theresa-ne-Vane-Tempest-Stewart-Countess-of-Ilchester-when-Lady-Helen-Stewart-as-the-Archduchess-Marie-Christine-of-Austria.jpg|thumb|alt=Black-and-white photograph of a seated woman richly dressed in an historical costume with a white feather plume in her hair and a fan|Lady Helen Stewart as Arch-duchess Marie Christine of Austria. ©National Portrait Gallery, London.]]
=== Lady Alexandra Hamilton ===
Lady Alexandra Hamilton was one of the archduchesses — along with with 3 or 4 other young women — in [[Social Victorians/People/Londonderry#The Entourage of Maria Thérèse|the entourage of the Marchioness of Londonderry]], who led the Austrian procession as Marie Thérèse, Empress of the Holy Roman Empire.&lt;ref&gt;“The Ball at Devonshire House. Magnificent Spectacle. Description of the Dresses.” London ''Evening Standard'' 3 July 1897 Saturday: 3 [of 12], Cols. 1a–5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000183/18970703/015/0004.&lt;/ref&gt;{{rp|p. 3, Col. 3a}} These young women were present at the ball as the daughters of Marie Thérèse, and the young men dressed as archdukes were present as her sons. Lady Alexandra Hamilton went as &quot;Archduchess Marie-Josepha in the Archduchess Marie-Karoline and Emperor Joseph II section of the Austrian Court of Maria Theresa Quadrille.&quot;&lt;ref name=&quot;:9&quot;&gt;&quot;Fancy Dress Ball at Devonshire House.&quot; ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.&lt;/ref&gt;{{rp|p. 7, Col. 6b}} &lt;ref name=&quot;:10&quot;&gt;&quot;Ball at Devonshire House.&quot; The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.&lt;/ref&gt;

The newspapers report that the archduchesses were all dressed alike, but only one photograph exists of any of these young women in costume — that of [[Social Victorians/People/Londonderry#Helen Mary Theresa Vane-Tempest-Stewart|Helen Mary Theresa Vane-Tempest-Stewart]] (which is shown, right). The newspaper descriptions are on her page, with her portrait in costume, but they apply to all the archduchesses.

=== Lord Frederick Hamilton ===
[[File:Lord Frederick Spencer Hamilton Vanity Fair 1895-02-07.jpg|thumb|left|alt=Colored drawing of a man in a suit, his hands in his pockets, facing to the right|Lord Frederick Hamilton, ''Vanity Fair'', by &quot;Spy,&quot; 7 February 1895]]
Lord Frederick Spencer Hamilton was 6th son and 13th child of the 1st Duke of Abercorn. No photograph of him in costume exists.

He is shown (at left) as he looked in 7 February 1895 in a Spy caricature in ''Vanity Fair''. This caricature portrait, by Leslie Ward (&quot;Spy&quot;) is called ''The Pall Mall Magazine'' and is Number 647 in Vanity Fair's &quot;Statesmen&quot; series.&lt;ref name=&quot;:16&quot;&gt;{{Cite journal|date=2024-01-14|title=List of Vanity Fair (British magazine) caricatures (1895–1899)|url=https://en.wikipedia.org/w/index.php?title=List_of_Vanity_Fair_(British_magazine)_caricatures_(1895%E2%80%931899)&amp;oldid=1195518024|journal=Wikipedia|language=en}}&lt;/ref&gt; He was editor of the ''Pall Mall Gazette'' 1896–1900.&lt;ref&gt;{{Cite journal|date=2023-09-23|title=Lord Frederick Spencer Hamilton|url=https://en.wikipedia.org/w/index.php?title=Lord_Frederick_Spencer_Hamilton&amp;oldid=1176655264|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Lord_Frederick_Spencer_Hamilton.&lt;/ref&gt;

For the ball, Lord Frederick Hamilton was dressed
*as a &quot;gentleman of the Court of Queen Elizabeth,&quot; wearing &quot;crimson cloth of gold with jewelled belt.&quot;&lt;ref name=&quot;:15&quot;&gt;“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.&lt;/ref&gt;{{rp|p. 36, Col. 3b}}
*as a &quot;Gentleman of the Court of Queen Elizabeth. Costume of crimson and cloth of g [sic] with jewelled belt.&quot;&lt;ref name=&quot;:9&quot; /&gt;{{rp|p. 8, Col. 1b}}
*&quot;in crimson cloth of gold and jeweled belt.&quot;&lt;ref&gt;&quot;Duchess of Devonshire's Fancy Ball. A Brilliant Spectacle. Some of the Dresses.&quot; London ''Daily News'' Saturday 3 July 1897: 5 [of 10], Col. 6a–6, Col. 1b. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000051/18970703/024/0005 and http://www.britishnewspaperarchive.co.uk/viewer/BL/0000051/18970703/024/0006.&lt;/ref&gt;{{rp|p. 5, Col. 7a}}
*&quot;as a gentleman of the court of Queen Elizabeth, was dressed in a costume of crimson cloth-of-gold, with a jewelled belt.&quot;&lt;ref name=&quot;:11&quot;&gt;“The Devonshire House Ball. A Brilliant Gathering.” The ''Pall Mall Gazette'' 3 July 1897, Saturday: 7 [of 10], Col. 2a–3a. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000098/18970703/019/0007.&lt;/ref&gt;

==== Memoirs ====

* Hamilton, Frederic [sic] Spencer. ''My Yesterdays'' (3 vols.). Hodder and Stoughton, 1920.
*# ''The Days Before Yesterday''. The Internet Archive has this: https://archive.org/details/daysbeforeyester00hamiuoft/page/n5/mode/2up.
*# ''Vanished Pomps of Yesterday''. The Internet Archive has this: https://archive.org/details/vanishedpompsofy028823mbp.
*# ''Here, There and Everywhere''. The Internet Archive has this: https://archive.org/details/herethereeverywh0000hami.

[[File:James Hamilton 3rd Duke of Abercorn.png|thumb|alt=Old colored drawing of a man in a 19th-century officer's uniform of the 1st Life Guards with white gloves, a red stripe down the side of his pants and unbuttoned jacket and a hat, holding a white or silver sword under his left arm, facing 1/4 to his right|&quot;He will be the 3rd Duke&quot; (James Hamilton, Marquis of Hamilton), ''Vanity Fair'' 16 February 1899]]
=== James Hamilton, Marquess of Hamilton ===
James Hamilton, Marquis of Hamilton was dressed in a &quot;black velvet tunic; breeches and cloak trimmed jet; large hat, feathers, wig, sword, &amp;c., of the period&quot; of Charles II.&lt;ref name=&quot;:15&quot; /&gt;{{rp|34, Col. 3a}} No photograph of him in costume exists.

A caricature portrait (right) called ''He will be the 3rd Duke'' (James Hamilton, Marquess of Hamilton) by &quot;Hadge&quot; appeared in the 16 February 1899 issue of ''Vanity Fair'', as Number 739 in its &quot;Men of the Day&quot; series,&lt;ref name=&quot;:16&quot; /&gt; giving a sense of what he looked like at about the time of the ball.

In 1892 Hamilton joined the 1st Life Guards, so the uniform he is wearing in this portrait is likely that of an officer of the 1st Life Guards.&lt;ref&gt;{{Cite journal|date=2024-01-12|title=James Hamilton, 3rd Duke of Abercorn|url=https://en.wikipedia.org/w/index.php?title=James_Hamilton,_3rd_Duke_of_Abercorn&amp;oldid=1195216640|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/James_Hamilton,_3rd_Duke_of_Abercorn.&lt;/ref&gt;

James Hamilton's wife Lady Rosalind Hamilton is not reported as having been present at the ball, perhaps because she was pregnant with her second child and gave birth in August, five weeks later, so she was around 8 months pregnant.

=== Ronald Hamilton ===
Mr. Ronald Hamilton, possibly Ronald James Hamilton, was dressed as a &quot;Gentleman of the Court of Queen Elizabeth, in black velvet trimmed with jet.&quot;&lt;ref name=&quot;:9&quot; /&gt;{{rp|p. 8, Col. 1c}}

== Demographics ==

=== Nationality ===
*The title Duke of Abercorn is in the peerage of Ireland; the Marquess of Hamilton is in the peerage of the U.K.

=== Residences ===

==== The Hon. Mrs. Sarah Howard and the Earls of Wicklow ====

* Shelton Abbey, Arklow, Co. Wicklow (east coast of Ireland) (until 1951)&lt;ref&gt;{{Cite journal|date=2026-06-30|title=Shelton Abbey Prison|url=https://en.wikipedia.org/w/index.php?title=Shelton_Abbey_Prison&amp;oldid=1361924427|journal=Wikipedia|language=en}}&lt;/ref&gt;

== Family ==
*James Hamilton, 1st Duke of Abercorn (21 January 1811 – 31 October 1885)&lt;ref name=&quot;:0&quot; /&gt;
*Louisa Russell Hamilton (– March 1905)
#Lady '''Harriet Georgiana Louisa Hamilton''' Anson (6 July 1834 – 23 April 1913)
#Lady Beatrix Frances Hamilton Lambton (21 July 1835 – 21 January 1871)
#Lady Louisa Jane Hamilton Scott (26 August 1836 – 16 March 1912)
#Lord '''James Hamilton, 2nd Duke of Abercorn''' (24 August 1838 – 3 January 1913)
#Lady Katherine Elizabeth Hamilton Edgcumbe (9 January 1840 – 3 September 1874)
#Lady Georgiana Susan Hamilton Turnour (7 July 1841 – 23 March 1913)
#Lord '''Claud John Hamilton''' (20 February 1843 – 26 January 1925)
#Rt. Hon. Lord Sir '''George Francis Hamilton''' (17 December 1845 – 22 September 1927)
#Lady Albertha Frances Anne Hamilton Spencer-Churchill (29 July 1847 – 7 January 1932)
#Lord Ronald Douglas Hamilton (17 March 1849 – DVP&lt;ref&gt;{{Cite journal|date=2020-07-27|title=James Hamilton, 2nd Duke of Abercorn|url=https://en.wikipedia.org/w/index.php?title=James_Hamilton,_2nd_Duke_of_Abercorn&amp;oldid=969822724|journal=Wikipedia|language=en}}&lt;/ref&gt; 6 November 1867)
#Lady Maud Evelyn Hamilton Petty-Fitzmaurice, the [[Social Victorians/People/Lansdowne | Marchioness of Lansdowne]] (17 December 1850 – 21 October 1932)&lt;ref name=&quot;:1&quot; /&gt;
#Lord Cosmo Hamilton (16 April 1853 – 16 April 1853)
#Lord '''Frederick Spencer Hamilton''' (13 October 1856 – 11 August 1928)
#Lord '''Ernest William Hamilton''' (5 September 1858 – 14 December 1939)


*Harriet Georgiana Louisa Hamilton Anson (6 July 1834 – 23 April 1913)&lt;ref name=&quot;:2&quot; /&gt;
*Thomas George Anson, 2nd Earl of Lichfield (15 August 1825 – 7 January 1892)
#Lady Evelyn Anson ( – 2 July 1895)
#Thomas Francis Anson, 3rd Earl of Lichfield (31 January 1856 – 29 July 1918)
#Hon. Sir George Augustus Anson (22 December 1857 – 25 May 1947)
#Major Hon. Henry James Anson (29 December 1858 – 26 February 1904)
#Lady Florence Beatrice Anson (1860 – 25 September 1946)
#Hon. Frederic William Anson (4 February 1862 – 2 April 1917)
#Hon. Claud Anson (11 January 1864 – 25 December 1947)
#Lady Beatrice Anson (1865 – 15 December 1919)
#Hon. Francis Anson (7 March 1867 – 13 April 1928)
#Lady Mary Maud Anson (1869 – 22 September 1961)
#Lady Edith Anson (1870 – 8 October 1932)
#Hon. William Anson (19 April 1872 – 22 June 1926)
#Hon. Alfred Anson (15 April 1876 – 25 March 1944)


*James Hamilton, 2nd Duke of Abercorn (24 August 1838 – 3 January 1913)&lt;ref name=&quot;:12&quot; /&gt;
*Maria Anna Curzon-Howe Hamilton (23 July 1848 – 10 May 1929)&lt;ref name=&quot;:3&quot; /&gt;
#James Albert Edward Hamilton, 3rd Duke of Abercorn (30 November 1869 – 12 September 1953)
#Claud Penn Alexander Hamilton (18 October 1871 – 18 October 1871)
#Charlie Hamilton (10 April 1874 – 10 April 1874)
#'''Alexandra Phyllis Hamilton''' (23 January 1876 – 10 October 1918)
#Claud Francis Hamilton (25 October 1878 – 25 December 1878)
#Gladys Mary Hamilton Forward-Howard (10 December 1880 – 12 March 1917)
#Arthur John Hamilton (20 August 1883 – 6 November 1914)
#(unnamed son) Hamilton (31 October 1886 – 31 October 1886)
#Claud Nigel Hamilton (10 November 1889 – 22 August 1975)&lt;ref name=&quot;:4&quot; /&gt;


* '''James Albert Edward Hamilton''', Marquess of Hamilton and 3rd Duke of Abercorn (30 November 1869 – 12 September 1953)&lt;ref name=&quot;:13&quot; /&gt;
* Lady Rosalind Cecilia Caroline Bingham (26 February 1869 – 18 January 1958)&lt;ref name=&quot;:14&quot; /&gt;
*# Lady Mary Cecilia Rhodesia Hamilton (21 January 1896 – 5 September 1984)
*# Lady Cynthia Elinor Beatrix Hamilton (16 August 1897 – 4 December 1972)
*# Lady Katharine Hamilton (25 February 1900 – 28 April 1985)
*# James Edward Hamilton, 4th Duke of Abercorn (29 February 1904 – 4 June 1979)
*# Captain Lord Claud David Hamilton (13 February 1907 – 15 February 1968)


*Claud John Hamilton (20 February 1843 – 26 January 1925)&lt;ref name=&quot;:5&quot; /&gt;
*Carolina Chandos-Pole Hamilton (19 July 1857 – 21 September 1911)&lt;ref&gt;&quot;Carolina Chandos-Pole.&quot; {{Cite web|url=http://www.thepeerage.com/p11067.htm#i110663|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;
#Colonel Gilbert Claud Hamilton (21 April 1879 – 30 March 1943)
#Ida Hamilton (23 July 1883 – November 1970)


*George Francis Hamilton (17 December 1845 – 22 September 1927)&lt;ref name=&quot;:6&quot; /&gt;
*Lady Maud Caroline Lascelles Hamilton (1846 – 14 April 1938)
#'''Ronald James Hamilton''' (26 September 1872 – 22 January 1958)
#Anthony George Hamilton (17 December 1874 – 11 July 1936)
#Robert Cecil Hamilton (31 January 1882 – 31 July 1947)


*Ernest William Hamilton (5 September 1858 – 14 December 1939)&lt;ref&gt;&quot;Lord Ernest William Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p2107.htm#i21062|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;
*Pamela Campbell Hamilton ( – 11 May 1931)&lt;ref name=&quot;:7&quot; /&gt;
#Guy Ernest Frederick Hamilton (11 November 1894 – 23 November 1914)
#Mary Brenda Hamilton (28 March 1897 – 14 March 1985)
#Jean Barbara Hamilton (6 September 1898 – 2 November 1989)
#John George Peter Hamilton (15 October 1900 – 17 June 1967)


=== Earls of Wicklow ===

* Charles Hamilton (1772 – 29 September 1857)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p2139.htm#i21387|title=Charles Hamilton. Person Page #2139|website=www.thepeerage.com|access-date=2026-06-19}}&lt;/ref&gt;
* Marianne '''Caroline Tighe''' ( – 29 July 1861)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p62375.htm#i623745|title=Marianne Caroline Tighe. Person Page #62375|website=www.thepeerage.com|access-date=2026-06-19}}&lt;/ref&gt;
*# '''Sarah Hamilton''' (1805&lt;ref name=&quot;:17&quot; /&gt; – 13 March 1892)
*# Caroline Elizabeth Hamilton ( – 31 May 1909)
*# Mary Hamilton
*# Charles William Hamilton (1 April 1802 – 16 February 1880)
*# William Tighe Hamilton (31 March 1807 – )
*# Frederick John Henry Fownes Hamilton (27 July 1816 – 1893)


* Rev. Hon. Francis Howard (12 January 1797 – 16 February 1857)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p2140.htm#i21391|title=Rev. Hon. Francis Howard. Person Page #2140|website=www.thepeerage.com|access-date=2026-06-19}}&lt;/ref&gt;
* Frances Beresford ( – 17 November 1833)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p3227.htm#i32266|title=Frances Beresford. Person Page #3227|website=www.thepeerage.com|access-date=2026-06-19}}&lt;/ref&gt;
*# William George Howard (25 April 1825 – 12 October 1864)
* '''Sarah Hamilton''' (1805&lt;ref name=&quot;:17&quot;&gt;{{Cite web|url=https://catalogue.nli.ie/Collection/vtls000572704|title=Tighe, Hamilton and Howard Papers,|date=1737|website=catalogue.nli.ie|language=English|access-date=2026-06-19}}&lt;/ref&gt; – 13 March 1892)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p2141.htm#i21405|title=Sarah Hamilton. Person Page #2141|website=www.thepeerage.com|access-date=2026-06-19}}&lt;/ref&gt;
*# 4 unnamed daughters [per The Peerage; The NLI has 3 daughters]
*# Lady Alice Howard
*# Lady Louisa 'Loulie' Howard
*# Lady Caroline Howard (1836–1923)&lt;ref name=&quot;:17&quot; /&gt;
*# Charles Francis Arnold Howard, '''5th Earl of Wicklow''' (5 November 1839 – 20 June 1881)
*# Cecil Ralph Howard, '''6th Earl of Wicklow''' (26 April 1842 – 24 July 1891)


* Cecil Ralph Howard, '''6th Earl of Wicklow''' (26 April 1842 – 24 July 1891)&lt;ref name=&quot;:18&quot; /&gt;
* Francesca Maria Chamberlayne ( – 1877)
*# Ralph Howard, 7th Earl of Wicklow (24 December 1877 – 11 October 1946)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p2140.htm#i21394|title=Cecil Ralph Howard, 6th Earl of Wicklow. Person Page 2140.|website=www.thepeerage.com|access-date=2026-06-28}}&lt;/ref&gt;
* Fanny Catherine Wingfield (c. 1860 – 3 February 1914)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p2139.htm#i21388|title=Fanny Catherine Wingfield. Person Page 2139.|website=www.thepeerage.com|access-date=2026-06-28}}&lt;/ref&gt;
*# Hon. Cecil Mervyn Malcolm Howard (18 November 1881 – 16 April 1882)
*# Hon. Hugh Melville Howard (28 March 1883 – 17 February 1919)
* Marcus Francis Beresford (26 December 1862 – 14 December 1896)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p3186.htm#i31858|title=Marcus Francis Beresford. Person Page #3186.|website=www.thepeerage.com|access-date=2026-06-28}}&lt;/ref&gt;

== Memoirs and Archives ==

# The Abercorn Papers: GB 0255 PRONI/D623 (found via https://iar.ie/archive/abercorn-papers). A descriptive list is available to search online at: http://www.proni.gov.uk/. The collection is arranged as follows: D623/A Correspondence D623/B Title deeds and leases D623/C Rentals, accounts and vouchers D623/D Maps, plans, surveys, inventories and valuations D623/E Photographs, illuminations, addresses and albums D623/F Material still at Baronscourt D623/G Miscellaneous
#Alexandra Phyllis Hamilton (#64 on the [[Social Victorians/1897 Fancy Dress Ball#List of People Who Attended|list of people who were present]]) attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House, as did her uncle Lord Frederick Spencer Hamilton (#84), the Marquess of Hamilton (#657), and a Mr. Ronald Hamilton (#105). Besides these, probably, a Mr. and Mrs. Hamilton also attended.

== Questions and Notes ==
#DVP = decessit vita patris, died while the father was still living
#Mr. Ronald Hamilton cannot be Frederick Hamilton's brother, who should be Lord Ronald Hamilton rather than Mr. Ronald Hamilton, and he died in 1867. He could be this Ronald Hamilton, who would be a Mr. Hamilton: http://www.thepeerage.com/p2163.htm#i21622. He was Lady Alexandra's cousin and nephew of the 1st Duke of Abercorn.
#A Mr. Hamilton is mentioned in the ''Gentlewoman'' article: &quot;Mr. Hamilton (Elizabethan costume), black velvet, trimmed gold.&quot;&lt;ref name=&quot;:15&quot; /&gt;{{rp|34, Col. 1c}} But a later reference in this same article to Mr. Ronald Hamilton matches the description in the ''Morning Post'' article, saying he wore black velvet with jet, rather than gold trim: &quot;'''Mr. Ronald Hamilton''' (gentleman of the Court of Queen Elizabeth), black velvet with jet.&quot;&lt;ref name=&quot;:15&quot; /&gt; (36, Col. 3b) I believe the other Mr. Hamilton is Mr. [[Social Victorians/People/Cole-Hamilton|Claud Cole-Hamilton]], particularly since Mrs. Hamilton was dressed as Amy Robsart and thus must be Lucy Charlewood Cole-Hamilton because of the description of her costume in the Album of photographs given to the Duchess of Devonshire later.
#Claud John Hamilton is probably who attended the social events, because the other Claud, of whatever generation either died too young or was born too late.
== Footnotes ==
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      <text bytes="37217" sha1="cjou8lrbvvzgdn3633hjue28zpqzscn" xml:space="preserve">== Overview ==
The Dukedom of Abercorn is the last non-royal dukedom created. Queen Victoria created it in 1869. 

This page includes the Earl of Wicklow, the family of which married into the Abercorn family in 1816 when William Howard, 4th Earl of Wicklow married Lady Cecil Frances Hamilton — the daughter and only child of John Hamilton, 1st Marquess of Abercorn.&lt;ref&gt;{{Cite journal|date=2026-06-24|title=William Howard, 4th Earl of Wicklow|url=https://en.wikipedia.org/w/index.php?title=William_Howard,_4th_Earl_of_Wicklow&amp;oldid=1360966619|journal=Wikipedia|language=en}}&lt;/ref&gt; William Howard, 4th Earl of Wicklow was succeeded by his nephew, Charles Howard, 5th Earl of Wicklow (5 November 1839 – 20 June 1881).&lt;ref&gt;{{Cite journal|date=2024-08-26|title=Charles Howard, 5th Earl of Wicklow|url=https://en.wikipedia.org/w/index.php?title=Charles_Howard,_5th_Earl_of_Wicklow&amp;oldid=1242455245|journal=Wikipedia|language=en}}&lt;/ref&gt; Also Ralph Howard, 7th Earl of Wicklow married Lady Gladys Mary Hamilton (daughter of the 2nd Duke of Abercorn) in 1902.&lt;ref name=&quot;:18&quot;&gt;{{Cite journal|date=2025-08-05|title=Cecil Howard, 6th Earl of Wicklow|url=https://en.wikipedia.org/w/index.php?title=Cecil_Howard,_6th_Earl_of_Wicklow&amp;oldid=1304372795|journal=Wikipedia|language=en}}&lt;/ref&gt;

The National Library of Ireland has papers from Sarah Howard and her children, including Lady Caroline Howard. 

== Also Known As ==
*Family name: Hamilton
*the Duke of Abercorn
**James Hamilton, 1st Duke of Abercorn (10 August 1868 – 31 October 1885)&lt;ref name=&quot;:0&quot;&gt;&quot;James Hamilton, 1st Duke of Abercorn.&quot; {{Cite web|url=http://www.thepeerage.com/p10144.htm#i101433|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;
**James Hamilton, 2nd Duke of Abercorn (31 October 1885 – 3 January 1913)&lt;ref name=&quot;:12&quot;&gt;&quot;James Hamilton, 2nd Duke of Abercorn.&quot; {{Cite web|url=http://www.thepeerage.com/p10104.htm#i101033|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;
**James Albert Edward Hamilton, 3rd Duke of Abercorn (3 January 1913 – 12 September 1953)&lt;ref name=&quot;:13&quot;&gt;&quot;James Albert Edward Hamilton, 3rd Duke of Abercorn.&quot; {{Cite web|url=http://www.thepeerage.com/p10104.htm#i101031|title=Person Page|website=www.thepeerage.com|access-date=2020-10-09}}&lt;/ref&gt;
*the Duchess of Abercorn
**Louisa Russell Hamilton, Duchess of Abercorn (10 August 1868 – 31 October 1885)
**Maria Anna Curzon-Howe Hamilton (31 October 1885 – 3 January 1913)
*Dowager Duchess of Hamilton
**Louisa Russell Hamilton, Duchess of Abercorn (31 October 1885 – March 1905)
**Maria Anna Curzon-Howe Hamilton (3 January 1913 – )
*Subsidiary titles:
**Marquess of Hamilton (courtesy title for the heir apparent)
***James Albert Edward Hamilton, 3rd Duke of Abercorn (31 October 1885 – 12 September 1953)
**Viscount Strabane (courtesy title for the heir apparent of the Marquess of Hamilton)

== Acquaintances, Friends and Enemies ==
=== Friends ===
*The Royal Family, especially [[Social Victorians/People/Albert Edward, Prince of Wales | Albert Edward, Prince]] and [[Social Victorians/People/Alexandra, Princess of Wales | Alexandra, Princess]] of Wales, in the generation of the 2nd duke.

== Timeline ==
A lot of people are treated on this page, so this timeline will be somewhat chaotic to read. These events probably didn't directly affect every single person treated on this page, but discussions about them probably circulated through the families. The detail about Lady Caroline Howard and her mother, the Hon. Susan Howard, is to make these people, whose papers are in the National Library of Ireland, more concrete and known.

'''1832 October 25''', James Hamilton and Louisa Russell married at Gordon Castle, Fochabers, Morayshire, in Scotland.&lt;ref name=&quot;:0&quot; /&gt;

'''1854 May 23''', Beatrix Frances Hamilton and George Frederick D'Arcy Lambton married.&lt;ref&gt;&quot;Lady Beatrix Frances Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p1147.htm#i11470|title=Person Page|website=www.thepeerage.com|access-date=2020-10-09}}&lt;/ref&gt;

'''1855 April 10''', Harriet Georgiana Louisa Hamilton and Thomas George Anson married.&lt;ref name=&quot;:2&quot;&gt;&quot;Lady Harriett Georgiana Louisa Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p1034.htm#i10332|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;

'''1858 October 26''', Katherine Elizabeth Hamilton and William Henry Edgcumbe married.&lt;ref&gt;&quot;Lady Katherine Elizabeth Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p1135.htm#i11344|title=Person Page|website=www.thepeerage.com|access-date=2020-10-09}}&lt;/ref&gt;

'''1859 November 22''', Louisa Jane Hamilton and William Montagu Douglass Scott married.&lt;ref&gt;&quot;Lady Louisa Jane Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p10359.htm#i103583|title=Person Page|website=www.thepeerage.com|access-date=2020-10-09}}&lt;/ref&gt;

'''1868''', the title the Duke of Abercorn was created.&lt;ref&gt;{{Cite journal|date=2020-07-06|title=James Hamilton, 1st Duke of Abercorn|url=https://en.wikipedia.org/w/index.php?title=James_Hamilton,_1st_Duke_of_Abercorn&amp;oldid=966293304|journal=Wikipedia|language=en}}&lt;/ref&gt;

'''1869 January 7''', James Hamilton (2nd Duke) and Maria Anna Curzon-Howe married at St. George's Church, St. George Street, Hanover Square, in London.&lt;ref name=&quot;:3&quot;&gt;&quot;Lady Mary Anna Curzon.&quot; {{Cite web|url=http://www.thepeerage.com/p10104.htm#i101034|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;

'''1869 November 8''', there may have been a double wedding: Albertha Frances Anne Hamilton and George Charles Spencer-Churchill married&lt;ref name=&quot;:8&quot;&gt;&quot;Lady Albertha Frances Anne Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p10595.htm#i105942|title=Person Page|website=www.thepeerage.com|access-date=2020-10-09}}&lt;/ref&gt;, and Maud Evelyn Hamilton and Henry Petty-Fitzmaurice married&lt;ref name=&quot;:1&quot;&gt;&quot;Lady Maud Evelyn Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p1163.htm#i11629|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;.

'''1871 February 17, Friday''', Lady Caroline Howard attended a [[Social Victorians/Timeline/1870s#Birmingham Tennis Court Club Ball|ball hosted by the &quot;bachelors of the Tennis Court Club&quot; in Birmingham]].

'''1871 May 9, Tuesday''', Lady Caroline Howard, Lady Alice Howard and Lady Louisa Howard were [[Social Victorians/Timeline/1870s#9 May 1871, Tuesday, Queen's Drawing-Room|presented to Queen Victoria at a Drawing-room]] by their mother, the Hon. Mrs. Sarah Howard.

'''1871 August 31, Thursday''', The Freeman's Journal reported that &quot;The Hon. Mrs. Howard, Lady Caroline Howard and suite have arrived at the Morrisson Hotel.&quot;&lt;blockquote&gt;The following are amongst the latest arrivals at the Morrisson Hotel: — Mrs. Percival Maxwell and the Misses Maxwell and suite, Mr and Mrs Herbert Read and suite, Rev H R Heywood, and Master H A Heywood, Mr F H Downing, Mr M Neil, Mr and Mrs Herbert and suite, Mr Abbott, Mr D'Arcy, Mr and Mrs G Woods and suite.&lt;ref&gt;&quot;Fashion and Varieties.&quot; ''Freeman's Journal'' 31 August 1871, Thursday: 4 [of 4], Col. 1a [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000056/18710831/012/0004. Same print title, n.p.&lt;/ref&gt;&lt;/blockquote&gt;'''1871 November 28''', George Francis Hamilton and Maud Caroline Lascelles married.&lt;ref name=&quot;:6&quot;&gt;&quot;Rt. Hon. Lord Sir George Francis Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p1133.htm#i11323|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;

'''1872 January 5, Friday''', the Hon. Mrs. Howard and Lady Caroline Howard were reported to &quot;have arrived at Morrisson's Hotel in Dublin.&lt;ref&gt;&quot;Fashion and Varieties.&quot; ''Morning Mail'' (Dublin) 5 January 1872, Friday: 3 [of 4, digital], Col. 2c [of 10 on digital image]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0006103/18720105/067/0003. The digital image has the last 2 columns of the prior page on this page, so the citation should be to p. 2 [of 4], Col. 8c [of 8].&lt;/ref&gt; Also at the Morrisson's Hotel at this time was Sir Roland Blennerhassett, Bart., M.P.&lt;ref&gt;&quot;Fashion and Varieties.&quot; ''Dublin Evening'' Mail 5 January 1872, Friday:  3 [of 4], Col. 8b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000433/18720105/028/0003. Same print and digital title, print n.p.&lt;/ref&gt;

'''1872 March 2, Saturday''', the ''Weekly Freeman and Irish Agriculturalist'' reported that &quot;Lady Caroline Howard, Lady Louisa Howard, and the Hon Mrs Howard and suite, Shelton Abbey, have arrived at Morrisson's Hotel.&quot; Two 1-sentence paragraphs later, the paper reported that the same group had &quot;left Morrisson's Hotel for Shelton Abbey.&quot;&lt;ref&gt;&quot;Fashion and Varieties.&quot; ''Weekly Freeman's Journal'' 2 March 1872, Saturday: 7 [of 8], Col. 1a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001446/18720302/062/0007. Print title: ''Weekly Freeman and Irish Agriculturalist'', same p.&lt;/ref&gt; Shelton Abbey was the [[Social Victorians/People/Abercorn#Residences|ancestral seat and at this time the country residence]] of the Earls of Wicklow, Arklow, Co. Wicklow.

'''1873 January 14, Saturday''', &quot;Lord Dunally and suite, Hon. Mrs. Howard, Lady Alice Howard and suite, Lady Louise Howard and suite, and Lady Caroline Howard, have arrived at Morrisson's Hotel.&quot;&lt;ref&gt;&quot;Fashionable Intelligence.&quot; ''Dublin Evening Post'' 14 January 1873, Tuesday: 3 [of 4], Col. 5a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000435/18730114/049/0003. Same print and digital title, print p. is n.p.&lt;/ref&gt;

'''1874 December 15, Tuesday''', the Right Hon. Sir Michael and Lady Lucy Hicks-Beach hosted a dinner in the Chief Secretary's Lodge, suggesting that this social event might have had a political purpose. Mr. LeFanu cannot be the Irish writer Sheridan Le Fanu, who died 7 February 1873.&lt;ref&gt;{{Cite journal|date=2026-06-28|title=Sheridan Le Fanu|url=https://en.wikipedia.org/w/index.php?title=Sheridan_Le_Fanu&amp;oldid=1361491348|journal=Wikipedia|language=en}}&lt;/ref&gt; Perhaps this LeFanu is a relation, a son or brother?&lt;blockquote&gt;THE CHIEF SECRETARY’S LODGE.

The Right Hon. Sir Michael and Lady Lucy Hicks-Beach entertained the following at dinner on Tuesday evening at the Chief Secretary’s Lodge: — Sir Dominic Corrigan, Sir Arthur and Lady Olive Guinness, Lady Mary Fortescue, the Hon. Mrs. Howard and Lady Caroline Howard, Mr. and Mrs. Percy Bernard, Colonel Henry, R.A., and Mrs. Henry; Mr. Donnelly. C.B., and Mrs. Donnelly; Mr., Mrs., and Miss lsaac; Mr. LeFanu, Colonel Forster, Colonel Hillier, and Mr. Caulfield.&lt;ref&gt;&quot;Fashionable Intelligence.&quot; ''Cork Constitution'' 17 December 1874, Thursday: 4 [of 4; n.p. in print], Col. 1a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001648/18741217/099/0004. Print title: ''The Cork Constitution''.&lt;/ref&gt;&lt;/blockquote&gt;'''1876 March 23''', Cecil Howard, 6th Earl of Wicklow and Francesca Maria Chamberlayne married.&lt;ref name=&quot;:18&quot; /&gt;

'''1877 July 25, Wednesday''', Miss Tottenham, Lady Caroline Howard, Miss Colley are reported to have arrived at Merton Lodge in Torquay.&lt;ref&gt;&quot;The Torquay Directory.&quot; ''Torquay Directory and South Devon Journal'' 25 July 1877, Wednesday: 4 [of 8], Col. 7a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001246/18770725/085/0004. Same print and digital title and p.&lt;/ref&gt;

'''1877 July 28, Saturday''', Lady Caroline Howard is listed as one of the guests at Merton Lodge in Lincombe Hill Road Middle, Torquay. Other guests listed are Miss Kelly, Mrs. Frank Webber, Miss Tottenham and Miss Colley.&lt;ref&gt;&quot;49. Lincombe Hill Road. Middle.&quot; &quot;Torquay Directory.&quot; ''Torquay Times and South Devon Advertiser'' 28 July 1877, Saturday: 2 [of 8, both print and digital], Col. 3c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001420/18770728/039/0002.&lt;/ref&gt;

'''1877 December 22, Saturday''', Sarah Howard, Lady Caroline Howard and Captain the Hon. Cecil Ralph Howard were visitors in Dagmar Terrace in Portsmouth. The following are all the people listed as visitors at Dagmar Terrace, with the odd numbering:&lt;blockquote&gt;D&lt;small&gt;AGMAR&lt;/small&gt; T&lt;small&gt;ER&lt;/small&gt;&lt;small&gt;RACE&lt;/small&gt;.

# Captain the Hon. Cecil Ralph Howard, late 60th Rifles, &amp; the Hon Mrs Howard Lady Caroline Howard
# Captain &amp; Mrs. Henderson
## [a] The Hon. Richard and Mrs. Bineham 
# [a] Captain and Mrs. Fearson and family
# Mr.and Mrs. Hall Mrs. and the Misses Buchannans
# The Rev Palms &amp; fam
# [a] Colonel Johnston [a] Mrs. Oldfield [a] Miss Flowers
# Captain Parkinson and family&lt;ref&gt;&quot;Visitors' List.&quot; ''Portsmouth Times and Naval Gazette'' 22 December 1877, Saturday: 3 [of 10, digital and print], Col. 5 [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001365/18771222/027/0003. Print title: ''Portsmouth Times and Naval Gazette, County Journal''.&lt;/ref&gt;
&lt;/blockquote&gt;'''1878 January 19, Saturday''', The ''Dublin Evening Mail'' says,&lt;blockquote&gt;Lady Caroline Howard has arrived at Kingstown from England.

Captain the Hon. C. Howard and Mrs. Howard have arrived at Kingstown from England.&lt;ref&gt;&quot;Viceregal Court.&quot; ''Dublin Evening Mail'' 19 January 1878, Saturday: 3 [of 4], Col. 8a [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000433/18780119/110/0003. Same print title, n.p.&lt;/ref&gt;&lt;/blockquote&gt;'''1878 July 20''', Claud John Hamilton and Carolina Chandos-Pole married.&lt;ref name=&quot;:5&quot;&gt;&quot;Lord Claud John Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p11067.htm#i110662|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;

'''1880 June 2''', Cecil Howard, 6th Earl of Wicklow and Fanny Catherine Wingfield married.&lt;ref name=&quot;:18&quot; /&gt;

'''1881 July 25, Monday''', the ''Irish Times'' says that Lady Caroline Howard and &quot;the Hon. Mrs. Howard and the Ladies Howard (2) have arrived at Kingstown from England.&quot;&lt;ref&gt;&quot;Fashionable Intelligence.&quot; ''Irish Times'' 25 July 1881, Monday: 6 [of 8, digital and print], Col. 3a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001683/18810725/124/0006. Same print title and p.&lt;/ref&gt;

'''1881 August 10, Wednesday''', the Dublin Evening Mail says that Lady Caroline Howard &quot;has left Kingstown for England.&quot;&lt;ref&gt;&quot;Fashion and Varieties.&quot; ''Dublin Evening Mail'' 10 August 1881, Wednesday: 3 [of 4], Col. 9c [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000433/18810810/046/0003. Same print and digital title, print p. is n.p.&lt;/ref&gt;

'''1881 October 22, Saturday''', Lady Caroline Howard is listed as one of the visitors staying at the Crown Hotel &quot;during the past week.&quot; The visitors listed are the following:&lt;blockquote&gt;Mr. Thomas Barber, Doctor and Mrs. Ayerst, Miss Noyce, Dr. Wilks, Mr. Nightingale, Mr. and Mrs. J. Hill, Lady Caroline Howard, the Hon. Mrs. Ross, Mr. Masters, Mr. Richardson and friend, Mr. Simpson, Mr. Wilson, &amp;c.&lt;ref&gt;&quot;Lyndhurst, Oct. 22.&quot; ''Hampshire Advertiser'' 22 October 1881, Saturday: 7 [of  8, both print and digital], Col. 2c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000495/18811022/049/0007. Print title: ''Hampshire Advertiser County Newspaper''.&lt;/ref&gt;&lt;/blockquote&gt;'''1882 March 16''', Georgiana Susan Hamilton and Edward Turnour married.&lt;ref&gt;&quot;Lady Georgiana Susan Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p1180.htm#i11791|title=Person Page|website=www.thepeerage.com|access-date=2020-10-09}}&lt;/ref&gt;

'''1883 November 20''', the marriage between Albertha Frances Anne Hamilton Spencer-Churchill and George Charles Spencer-Churchill was annulled by petition from Albertha Frances Anne Hamilton Spencer-Churchill (married in 1869).&lt;ref name=&quot;:8&quot; /&gt;

'''1891 June 2''', Ernest William Hamilton and Pamela Campbell married.&lt;ref name=&quot;:7&quot;&gt;&quot;Pamela Campbell.&quot; {{Cite web|url=http://www.thepeerage.com/p2107.htm#i21063|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;

'''1894 April 10''', Fanny Catherine Wingfield Howard, Dowager 6th Countess of Wicklow married her 2nd husband, Marcus Francis Beresford.&lt;ref name=&quot;:18&quot; /&gt;

'''1894 November 1''', James Albert Edward Hamilton and Rosaline Cecilia Caroline Bingham married at St. Paul's Church, Knightsbridge, in London.&lt;ref name=&quot;:14&quot;&gt;&quot;Lady Rosalind Cecilia Caroline Bingham.&quot; {{Cite web|url=https://www.thepeerage.com/p10104.htm#i101032|title=Person Page|website=www.thepeerage.com|access-date=2021-05-15}}&lt;/ref&gt;

'''1895 July 13 to August 7''', the general election of 1895. Following the election, the brother-in-law of Cecil Howard, 6th Earl of Wicklow's (brother of his first wife Francesca Chamberlayne) was unseated because of allegations of misconduct.&lt;ref&gt;{{Cite journal|date=2026-02-27|title=Thomas Chamberlayne (cricketer)|url=https://en.wikipedia.org/w/index.php?title=Thomas_Chamberlayne_(cricketer)&amp;oldid=1340809770|journal=Wikipedia|language=en}}&lt;/ref&gt;

'''1897 June 28, Monday''', according to the ''Morning Post'', James Hamilton, 2nd Duke and Maria, Duchess of Abercorn were invited to the [[Social Victorians/Diamond Jubilee Garden Party|Queen's Garden Party]], the official end of the Diamond Jubilee celebrations in London, as were James Albert Edward Hamilton, Marquis and Rosaline, Marchioness of Hamilton.&lt;ref&gt;“The Queen’s Garden Party.” ''Morning Post'' 29 June 1897, Tuesday: 4 [of 12], Cols. 1a–7c [of 7] and 5, Col. 1a–c. ''British Newspaper Archive''  ''&lt;nowiki&gt;https://www.britishnewspaperarchive.co.uk/viewer/BL/0000174/18970629/032/0004&lt;/nowiki&gt;'' and ''&lt;nowiki&gt;https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970629/032/0005&lt;/nowiki&gt;''.&lt;/ref&gt;

'''1897 July 2, Friday''', Alexandra Phyllis Hamilton attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House, as did her uncle Lord Frederick Spencer Hamilton, the Marquess of Hamilton, and a Mr. Ronald Hamilton. Besides these, probably, a Mr. and Mrs. Hamilton also attended.

'''1902''', Ralph Howard, 7th Earl of Wicklow and Lady Gladys Mary Hamilton married. (She was the daughter of James Hamilton, 2nd Duke of Abercorn.)&lt;ref name=&quot;:18&quot; /&gt;

'''1902 January 14''', Gladys Mary Hamilton and Ralph Francis Forward-Howard married.&lt;ref&gt;&quot;Lady Gladys Mary Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p2107.htm#i21066|title=Person Page|website=www.thepeerage.com|access-date=2020-10-09}}&lt;/ref&gt;

'''1933 July 11''', Claud Nigel Hamilton and Violet Ruby Ashton married.&lt;ref name=&quot;:4&quot;&gt;&quot;Captain Lord Sir Claud Nigel Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p2109.htm#i21081|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;

== Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball ==

[[File:Helen-Mary-Theresa-ne-Vane-Tempest-Stewart-Countess-of-Ilchester-when-Lady-Helen-Stewart-as-the-Archduchess-Marie-Christine-of-Austria.jpg|thumb|alt=Black-and-white photograph of a seated woman richly dressed in an historical costume with a white feather plume in her hair and a fan|Lady Helen Stewart as Arch-duchess Marie Christine of Austria. ©National Portrait Gallery, London.]]
=== Lady Alexandra Hamilton ===
Lady Alexandra Hamilton was one of the archduchesses — along with with 3 or 4 other young women — in [[Social Victorians/People/Londonderry#The Entourage of Maria Thérèse|the entourage of the Marchioness of Londonderry]], who led the Austrian procession as Marie Thérèse, Empress of the Holy Roman Empire.&lt;ref&gt;“The Ball at Devonshire House. Magnificent Spectacle. Description of the Dresses.” London ''Evening Standard'' 3 July 1897 Saturday: 3 [of 12], Cols. 1a–5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000183/18970703/015/0004.&lt;/ref&gt;{{rp|p. 3, Col. 3a}} These young women were present at the ball as the daughters of Marie Thérèse, and the young men dressed as archdukes were present as her sons. Lady Alexandra Hamilton went as &quot;Archduchess Marie-Josepha in the Archduchess Marie-Karoline and Emperor Joseph II section of the Austrian Court of Maria Theresa Quadrille.&quot;&lt;ref name=&quot;:9&quot;&gt;&quot;Fancy Dress Ball at Devonshire House.&quot; ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.&lt;/ref&gt;{{rp|p. 7, Col. 6b}} &lt;ref name=&quot;:10&quot;&gt;&quot;Ball at Devonshire House.&quot; The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.&lt;/ref&gt;

The newspapers report that the archduchesses were all dressed alike, but only one photograph exists of any of these young women in costume — that of [[Social Victorians/People/Londonderry#Helen Mary Theresa Vane-Tempest-Stewart|Helen Mary Theresa Vane-Tempest-Stewart]] (which is shown, right). The newspaper descriptions are on her page, with her portrait in costume, but they apply to all the archduchesses.

=== Lord Frederick Hamilton ===
[[File:Lord Frederick Spencer Hamilton Vanity Fair 1895-02-07.jpg|thumb|left|alt=Colored drawing of a man in a suit, his hands in his pockets, facing to the right|Lord Frederick Hamilton, ''Vanity Fair'', by &quot;Spy,&quot; 7 February 1895]]
Lord Frederick Spencer Hamilton was 6th son and 13th child of the 1st Duke of Abercorn. No photograph of him in costume exists.

He is shown (at left) as he looked in 7 February 1895 in a Spy caricature in ''Vanity Fair''. This caricature portrait, by Leslie Ward (&quot;Spy&quot;) is called ''The Pall Mall Magazine'' and is Number 647 in Vanity Fair's &quot;Statesmen&quot; series.&lt;ref name=&quot;:16&quot;&gt;{{Cite journal|date=2024-01-14|title=List of Vanity Fair (British magazine) caricatures (1895–1899)|url=https://en.wikipedia.org/w/index.php?title=List_of_Vanity_Fair_(British_magazine)_caricatures_(1895%E2%80%931899)&amp;oldid=1195518024|journal=Wikipedia|language=en}}&lt;/ref&gt; He was editor of the ''Pall Mall Gazette'' 1896–1900.&lt;ref&gt;{{Cite journal|date=2023-09-23|title=Lord Frederick Spencer Hamilton|url=https://en.wikipedia.org/w/index.php?title=Lord_Frederick_Spencer_Hamilton&amp;oldid=1176655264|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Lord_Frederick_Spencer_Hamilton.&lt;/ref&gt;

For the ball, Lord Frederick Hamilton was dressed
*as a &quot;gentleman of the Court of Queen Elizabeth,&quot; wearing &quot;crimson cloth of gold with jewelled belt.&quot;&lt;ref name=&quot;:15&quot;&gt;“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.&lt;/ref&gt;{{rp|p. 36, Col. 3b}}
*as a &quot;Gentleman of the Court of Queen Elizabeth. Costume of crimson and cloth of g [sic] with jewelled belt.&quot;&lt;ref name=&quot;:9&quot; /&gt;{{rp|p. 8, Col. 1b}}
*&quot;in crimson cloth of gold and jeweled belt.&quot;&lt;ref&gt;&quot;Duchess of Devonshire's Fancy Ball. A Brilliant Spectacle. Some of the Dresses.&quot; London ''Daily News'' Saturday 3 July 1897: 5 [of 10], Col. 6a–6, Col. 1b. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000051/18970703/024/0005 and http://www.britishnewspaperarchive.co.uk/viewer/BL/0000051/18970703/024/0006.&lt;/ref&gt;{{rp|p. 5, Col. 7a}}
*&quot;as a gentleman of the court of Queen Elizabeth, was dressed in a costume of crimson cloth-of-gold, with a jewelled belt.&quot;&lt;ref name=&quot;:11&quot;&gt;“The Devonshire House Ball. A Brilliant Gathering.” The ''Pall Mall Gazette'' 3 July 1897, Saturday: 7 [of 10], Col. 2a–3a. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000098/18970703/019/0007.&lt;/ref&gt;

==== Memoirs ====

* Hamilton, Frederic [sic] Spencer. ''My Yesterdays'' (3 vols.). Hodder and Stoughton, 1920.
*# ''The Days Before Yesterday''. The Internet Archive has this: https://archive.org/details/daysbeforeyester00hamiuoft/page/n5/mode/2up.
*# ''Vanished Pomps of Yesterday''. The Internet Archive has this: https://archive.org/details/vanishedpompsofy028823mbp.
*# ''Here, There and Everywhere''. The Internet Archive has this: https://archive.org/details/herethereeverywh0000hami.

[[File:James Hamilton 3rd Duke of Abercorn.png|thumb|alt=Old colored drawing of a man in a 19th-century officer's uniform of the 1st Life Guards with white gloves, a red stripe down the side of his pants and unbuttoned jacket and a hat, holding a white or silver sword under his left arm, facing 1/4 to his right|&quot;He will be the 3rd Duke&quot; (James Hamilton, Marquis of Hamilton), ''Vanity Fair'' 16 February 1899]]
=== James Hamilton, Marquess of Hamilton ===
James Hamilton, Marquis of Hamilton was dressed in a &quot;black velvet tunic; breeches and cloak trimmed jet; large hat, feathers, wig, sword, &amp;c., of the period&quot; of Charles II.&lt;ref name=&quot;:15&quot; /&gt;{{rp|34, Col. 3a}} No photograph of him in costume exists.

A caricature portrait (right) called ''He will be the 3rd Duke'' (James Hamilton, Marquess of Hamilton) by &quot;Hadge&quot; appeared in the 16 February 1899 issue of ''Vanity Fair'', as Number 739 in its &quot;Men of the Day&quot; series,&lt;ref name=&quot;:16&quot; /&gt; giving a sense of what he looked like at about the time of the ball.

In 1892 Hamilton joined the 1st Life Guards, so the uniform he is wearing in this portrait is likely that of an officer of the 1st Life Guards.&lt;ref&gt;{{Cite journal|date=2024-01-12|title=James Hamilton, 3rd Duke of Abercorn|url=https://en.wikipedia.org/w/index.php?title=James_Hamilton,_3rd_Duke_of_Abercorn&amp;oldid=1195216640|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/James_Hamilton,_3rd_Duke_of_Abercorn.&lt;/ref&gt;

James Hamilton's wife Lady Rosalind Hamilton is not reported as having been present at the ball, perhaps because she was pregnant with her second child and gave birth in August, five weeks later, so she was around 8 months pregnant.

=== Ronald Hamilton ===
Mr. Ronald Hamilton, possibly Ronald James Hamilton, was dressed as a &quot;Gentleman of the Court of Queen Elizabeth, in black velvet trimmed with jet.&quot;&lt;ref name=&quot;:9&quot; /&gt;{{rp|p. 8, Col. 1c}}

== Demographics ==

=== Nationality ===
*The title Duke of Abercorn is in the peerage of Ireland; the Marquess of Hamilton is in the peerage of the U.K.

=== Residences ===

==== The Hon. Mrs. Sarah Howard and the Earls of Wicklow ====

* Shelton Abbey, Arklow, Co. Wicklow (east coast of Ireland) (until 1951)&lt;ref&gt;{{Cite journal|date=2026-06-30|title=Shelton Abbey Prison|url=https://en.wikipedia.org/w/index.php?title=Shelton_Abbey_Prison&amp;oldid=1361924427|journal=Wikipedia|language=en}}&lt;/ref&gt;

== Family ==
*James Hamilton, 1st Duke of Abercorn (21 January 1811 – 31 October 1885)&lt;ref name=&quot;:0&quot; /&gt;
*Louisa Russell Hamilton (– March 1905)
#Lady '''Harriet Georgiana Louisa Hamilton''' Anson (6 July 1834 – 23 April 1913)
#Lady Beatrix Frances Hamilton Lambton (21 July 1835 – 21 January 1871)
#Lady Louisa Jane Hamilton Scott (26 August 1836 – 16 March 1912)
#Lord '''James Hamilton, 2nd Duke of Abercorn''' (24 August 1838 – 3 January 1913)
#Lady Katherine Elizabeth Hamilton Edgcumbe (9 January 1840 – 3 September 1874)
#Lady Georgiana Susan Hamilton Turnour (7 July 1841 – 23 March 1913)
#Lord '''Claud John Hamilton''' (20 February 1843 – 26 January 1925)
#Rt. Hon. Lord Sir '''George Francis Hamilton''' (17 December 1845 – 22 September 1927)
#Lady Albertha Frances Anne Hamilton Spencer-Churchill (29 July 1847 – 7 January 1932)
#Lord Ronald Douglas Hamilton (17 March 1849 – DVP&lt;ref&gt;{{Cite journal|date=2020-07-27|title=James Hamilton, 2nd Duke of Abercorn|url=https://en.wikipedia.org/w/index.php?title=James_Hamilton,_2nd_Duke_of_Abercorn&amp;oldid=969822724|journal=Wikipedia|language=en}}&lt;/ref&gt; 6 November 1867)
#Lady Maud Evelyn Hamilton Petty-Fitzmaurice, the [[Social Victorians/People/Lansdowne | Marchioness of Lansdowne]] (17 December 1850 – 21 October 1932)&lt;ref name=&quot;:1&quot; /&gt;
#Lord Cosmo Hamilton (16 April 1853 – 16 April 1853)
#Lord '''Frederick Spencer Hamilton''' (13 October 1856 – 11 August 1928)
#Lord '''Ernest William Hamilton''' (5 September 1858 – 14 December 1939)


*Harriet Georgiana Louisa Hamilton Anson (6 July 1834 – 23 April 1913)&lt;ref name=&quot;:2&quot; /&gt;
*Thomas George Anson, 2nd Earl of Lichfield (15 August 1825 – 7 January 1892)
#Lady Evelyn Anson ( – 2 July 1895)
#Thomas Francis Anson, 3rd Earl of Lichfield (31 January 1856 – 29 July 1918)
#Hon. Sir George Augustus Anson (22 December 1857 – 25 May 1947)
#Major Hon. Henry James Anson (29 December 1858 – 26 February 1904)
#Lady Florence Beatrice Anson (1860 – 25 September 1946)
#Hon. Frederic William Anson (4 February 1862 – 2 April 1917)
#Hon. Claud Anson (11 January 1864 – 25 December 1947)
#Lady Beatrice Anson (1865 – 15 December 1919)
#Hon. Francis Anson (7 March 1867 – 13 April 1928)
#Lady Mary Maud Anson (1869 – 22 September 1961)
#Lady Edith Anson (1870 – 8 October 1932)
#Hon. William Anson (19 April 1872 – 22 June 1926)
#Hon. Alfred Anson (15 April 1876 – 25 March 1944)


*James Hamilton, 2nd Duke of Abercorn (24 August 1838 – 3 January 1913)&lt;ref name=&quot;:12&quot; /&gt;
*Maria Anna Curzon-Howe Hamilton (23 July 1848 – 10 May 1929)&lt;ref name=&quot;:3&quot; /&gt;
#James Albert Edward Hamilton, 3rd Duke of Abercorn (30 November 1869 – 12 September 1953)
#Claud Penn Alexander Hamilton (18 October 1871 – 18 October 1871)
#Charlie Hamilton (10 April 1874 – 10 April 1874)
#'''Alexandra Phyllis Hamilton''' (23 January 1876 – 10 October 1918)
#Claud Francis Hamilton (25 October 1878 – 25 December 1878)
#Gladys Mary Hamilton Forward-Howard (10 December 1880 – 12 March 1917)
#Arthur John Hamilton (20 August 1883 – 6 November 1914)
#(unnamed son) Hamilton (31 October 1886 – 31 October 1886)
#Claud Nigel Hamilton (10 November 1889 – 22 August 1975)&lt;ref name=&quot;:4&quot; /&gt;


* '''James Albert Edward Hamilton''', Marquess of Hamilton and 3rd Duke of Abercorn (30 November 1869 – 12 September 1953)&lt;ref name=&quot;:13&quot; /&gt;
* Lady Rosalind Cecilia Caroline Bingham (26 February 1869 – 18 January 1958)&lt;ref name=&quot;:14&quot; /&gt;
*# Lady Mary Cecilia Rhodesia Hamilton (21 January 1896 – 5 September 1984)
*# Lady Cynthia Elinor Beatrix Hamilton (16 August 1897 – 4 December 1972)
*# Lady Katharine Hamilton (25 February 1900 – 28 April 1985)
*# James Edward Hamilton, 4th Duke of Abercorn (29 February 1904 – 4 June 1979)
*# Captain Lord Claud David Hamilton (13 February 1907 – 15 February 1968)


*Claud John Hamilton (20 February 1843 – 26 January 1925)&lt;ref name=&quot;:5&quot; /&gt;
*Carolina Chandos-Pole Hamilton (19 July 1857 – 21 September 1911)&lt;ref&gt;&quot;Carolina Chandos-Pole.&quot; {{Cite web|url=http://www.thepeerage.com/p11067.htm#i110663|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;
#Colonel Gilbert Claud Hamilton (21 April 1879 – 30 March 1943)
#Ida Hamilton (23 July 1883 – November 1970)


*George Francis Hamilton (17 December 1845 – 22 September 1927)&lt;ref name=&quot;:6&quot; /&gt;
*Lady Maud Caroline Lascelles Hamilton (1846 – 14 April 1938)
#'''Ronald James Hamilton''' (26 September 1872 – 22 January 1958)
#Anthony George Hamilton (17 December 1874 – 11 July 1936)
#Robert Cecil Hamilton (31 January 1882 – 31 July 1947)


*Ernest William Hamilton (5 September 1858 – 14 December 1939)&lt;ref&gt;&quot;Lord Ernest William Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p2107.htm#i21062|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;
*Pamela Campbell Hamilton ( – 11 May 1931)&lt;ref name=&quot;:7&quot; /&gt;
#Guy Ernest Frederick Hamilton (11 November 1894 – 23 November 1914)
#Mary Brenda Hamilton (28 March 1897 – 14 March 1985)
#Jean Barbara Hamilton (6 September 1898 – 2 November 1989)
#John George Peter Hamilton (15 October 1900 – 17 June 1967)


=== Earls of Wicklow ===

* Charles Hamilton (1772 – 29 September 1857)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p2139.htm#i21387|title=Charles Hamilton. Person Page #2139|website=www.thepeerage.com|access-date=2026-06-19}}&lt;/ref&gt;
* Marianne '''Caroline Tighe''' ( – 29 July 1861)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p62375.htm#i623745|title=Marianne Caroline Tighe. Person Page #62375|website=www.thepeerage.com|access-date=2026-06-19}}&lt;/ref&gt;
*# '''Sarah Hamilton''' (1805&lt;ref name=&quot;:17&quot; /&gt; – 13 March 1892)
*# Caroline Elizabeth Hamilton ( – 31 May 1909)
*# Mary Hamilton
*# Charles William Hamilton (1 April 1802 – 16 February 1880)
*# William Tighe Hamilton (31 March 1807 – )
*# Frederick John Henry Fownes Hamilton (27 July 1816 – 1893)


* Rev. Hon. Francis Howard (12 January 1797 – 16 February 1857)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p2140.htm#i21391|title=Rev. Hon. Francis Howard. Person Page #2140|website=www.thepeerage.com|access-date=2026-06-19}}&lt;/ref&gt;
* Frances Beresford ( – 17 November 1833)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p3227.htm#i32266|title=Frances Beresford. Person Page #3227|website=www.thepeerage.com|access-date=2026-06-19}}&lt;/ref&gt;
*# William George Howard (25 April 1825 – 12 October 1864)
* '''Sarah Hamilton''' (1805&lt;ref name=&quot;:17&quot;&gt;{{Cite web|url=https://catalogue.nli.ie/Collection/vtls000572704|title=Tighe, Hamilton and Howard Papers,|date=1737|website=catalogue.nli.ie|language=English|access-date=2026-06-19}}&lt;/ref&gt; – 13 March 1892)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p2141.htm#i21405|title=Sarah Hamilton. Person Page #2141|website=www.thepeerage.com|access-date=2026-06-19}}&lt;/ref&gt;
*# 4 unnamed daughters [per The Peerage; The NLI has 3 daughters]
*# Lady Alice Howard
*# Lady Louisa 'Loulie' Howard
*# Lady Caroline Howard (1836–1923)&lt;ref name=&quot;:17&quot; /&gt;
*# Charles Francis Arnold Howard, '''5th Earl of Wicklow''' (5 November 1839 – 20 June 1881)
*# Cecil Ralph Howard, '''6th Earl of Wicklow''' (26 April 1842 – 24 July 1891)


* Cecil Ralph Howard, '''6th Earl of Wicklow''' (26 April 1842 – 24 July 1891)&lt;ref name=&quot;:18&quot; /&gt;
* Francesca Maria Chamberlayne ( – 1877)
*# Ralph Howard, 7th Earl of Wicklow (24 December 1877 – 11 October 1946)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p2140.htm#i21394|title=Cecil Ralph Howard, 6th Earl of Wicklow. Person Page 2140.|website=www.thepeerage.com|access-date=2026-06-28}}&lt;/ref&gt;
* Fanny Catherine Wingfield (c. 1860 – 3 February 1914)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p2139.htm#i21388|title=Fanny Catherine Wingfield. Person Page 2139.|website=www.thepeerage.com|access-date=2026-06-28}}&lt;/ref&gt;
*# Hon. Cecil Mervyn Malcolm Howard (18 November 1881 – 16 April 1882)
*# Hon. Hugh Melville Howard (28 March 1883 – 17 February 1919)
* Marcus Francis Beresford (26 December 1862 – 14 December 1896)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p3186.htm#i31858|title=Marcus Francis Beresford. Person Page #3186.|website=www.thepeerage.com|access-date=2026-06-28}}&lt;/ref&gt;

== Memoirs and Archives ==

# The Abercorn Papers: GB 0255 PRONI/D623 (found via https://iar.ie/archive/abercorn-papers). A descriptive list is available to search online at: http://www.proni.gov.uk/. The collection is arranged as follows: D623/A Correspondence D623/B Title deeds and leases D623/C Rentals, accounts and vouchers D623/D Maps, plans, surveys, inventories and valuations D623/E Photographs, illuminations, addresses and albums D623/F Material still at Baronscourt D623/G Miscellaneous
#Alexandra Phyllis Hamilton (#64 on the [[Social Victorians/1897 Fancy Dress Ball#List of People Who Attended|list of people who were present]]) attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House, as did her uncle Lord Frederick Spencer Hamilton (#84), the Marquess of Hamilton (#657), and a Mr. Ronald Hamilton (#105). Besides these, probably, a Mr. and Mrs. Hamilton also attended.

== Questions and Notes ==
#DVP = decessit vita patris, died while the father was still living
#Mr. Ronald Hamilton cannot be Frederick Hamilton's brother, who should be Lord Ronald Hamilton rather than Mr. Ronald Hamilton, and he died in 1867. He could be this Ronald Hamilton, who would be a Mr. Hamilton: http://www.thepeerage.com/p2163.htm#i21622. He was Lady Alexandra's cousin and nephew of the 1st Duke of Abercorn.
#A Mr. Hamilton is mentioned in the ''Gentlewoman'' article: &quot;Mr. Hamilton (Elizabethan costume), black velvet, trimmed gold.&quot;&lt;ref name=&quot;:15&quot; /&gt;{{rp|34, Col. 1c}} But a later reference in this same article to Mr. Ronald Hamilton matches the description in the ''Morning Post'' article, saying he wore black velvet with jet, rather than gold trim: &quot;'''Mr. Ronald Hamilton''' (gentleman of the Court of Queen Elizabeth), black velvet with jet.&quot;&lt;ref name=&quot;:15&quot; /&gt; (36, Col. 3b) I believe the other Mr. Hamilton is Mr. [[Social Victorians/People/Cole-Hamilton|Claud Cole-Hamilton]], particularly since Mrs. Hamilton was dressed as Amy Robsart and thus must be Lucy Charlewood Cole-Hamilton because of the description of her costume in the Album of photographs given to the Duchess of Devonshire later.
#Claud John Hamilton is probably who attended the social events, because the other Claud, of whatever generation either died too young or was born too late.
== Footnotes ==
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    <title>Social Victorians/Timeline/1870s</title>
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      <text bytes="18667" sha1="4er8ulyoemr6q769kbe9zuvradobakr" 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)

=== 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.

B&lt;small&gt;ACHELORS'&lt;/small&gt; B&lt;small&gt;ALL&lt;/small&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;/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===
'''24 May 1871, Wednesday''': Derby Day. Baron Rothschild's Favonius won. The Prince of Wales attended.

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
February
March
April
===May===
'''28 May 1873, Wednesday''': Derby Day

June
July
August
September
October
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==
{{reflist}}</text>
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      <text bytes="20745" sha1="l40fz5rn78a04azasgkdexs65qqsegs" 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)

=== 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.

B&lt;small&gt;ACHELORS'&lt;/small&gt; B&lt;small&gt;ALL&lt;/small&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;/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;'''24 May 1871, Wednesday''': Derby Day. Baron Rothschild's Favonius won. The Prince of Wales attended.

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
February
March
April
===May===
'''28 May 1873, Wednesday''': Derby Day

June
July
August
September
October
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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    <title>WikiJournal Preprints/Mental health in Sri Lanka</title>
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| journal = WikiJournal of Medicine      &lt;!-- WikiJournal of Medicine, Science, or Humanities --&gt;
| last1 = Azeez
| orcid1 = 0009-0007-9202-4614
| first1 = Aaqib
| last2 = 
| first2 = 
| last3 = 
| first3 = 
| last4 = 
| first4 = &lt;!-- up to 9 authors can be added in this above format --&gt;
| et_al = &lt;!-- if there are &gt;9 authors, hyperlink to the list here --&gt;
| affiliation1 = Old Dominion University
| correspondence1 = aaqib.azeez@yahoo.com
| affiliations = institutes / affiliations
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| keywords = &lt;!-- up to 6 keywords --&gt;
| license = &lt;!-- default is CC-BY --&gt;
| abstract = This is a narrative review.
}}

[abstract will be put in after the paper has been completed] 

== Introduction ==
Mental health continues to be a critically relevant topic as the island nation has experienced decades of [[w:Black_July|violent ethnic conflict]], terrorist attacks, war crimes, and economic disruptions. Sri Lanka continues to recover from a [[w:Sri_Lankan_economic_crisis_(2019–2024)|severe economic crisis (2019 - 2024)]], a [[w:Sri_Lankan_civil_war|nearly 30-year civil war ending in 2009]], a [[w:2019_Sri_Lanka_Easter_bombings|2019 terrorist attack]], and continues to face the ripple effects of the [[w:2004_Boxing_Day_tsunami|2004 Boxing Day tsunami]]. The exact effect these major events have had on mental health in the country is &quot;unknown&quot;, but the statistics remain alarming despite a declining trend.

Suicide rates in the country during the mid-1990s were the second-highest in the world with ingesting toxic products being the main suicide method. Despite the decline in suicide numbers since then—possibly attributed to Sri Lanka's ban on toxic products—evidence from a 2023 study reports an upward trend in suicide through hanging from 2016 to 2021—independent of the [[w:COVID-19_pandemic_in_Sri_Lanka|COVID-19 pandemic]]. Several risk factors for suicide, such as poverty and economic instability, are still prevalent and even increasing in the country&lt;ref&gt;{{Cite journal|last=Rajapakse|first=Thilini|last2=Silva|first2=Tharuka|last3=Hettiarachchi|first3=Nirosha Madhuwanthi|last4=Gunnell|first4=David|last5=Metcalfe|first5=Chris|last6=Spittal|first6=Matthew J.|last7=Knipe|first7=Duleeka|date=2023-01-19|title=The Impact of the COVID-19 Pandemic and Lockdowns on Self-Poisoning and Suicide in Sri Lanka: An Interrupted Time Series Analysis|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC9914278/|journal=International Journal of Environmental Research and Public Health|volume=20|issue=3|pages=1833|doi=10.3390/ijerph20031833|issn=1660-4601|pmc=9914278|pmid=36767200}}&lt;/ref&gt;.

== Methods ==
A narrative review was conducted on mental health in Sri Lanka. Sources used included peer-reviewed journal articles, relevant books, historical documents, and governmental/non-governmental reports. These sources were found on Google Scholar, PubMed/PMC, Sri Lankan journals, and official Sri Lankan governmental websites displaying relevant statistics/reports. Studies included were published prior to 2026. Keywords used to initiate searches include, but not limited to were: &quot;Sri Lanka mental health&quot;, &quot;Sri Lanka civil war trauma&quot;, &quot;Sri Lanka suicide&quot;, &quot;Sri Lanka mental health ordinances&quot;, &quot;Sri Lanka religion and mental health&quot;, &quot;Sri Lanka public mental healthcare&quot;, and &quot;Sri Lanka poverty/economic crisis mental health impact.&quot; Studies that were included were relevant to the topic (Sri Lanka, South Asian mental health law, suicide, public mental health, conflict/disaster trauma, or cultural/religious practice), had full text available, and were in the English language. Non-peer-reviewed sources were primarily used to explain historical claims or contextualize non-clinical claims. ''[include date of final search when needed]'' 

==Historical Development of Mental Health Services==

Records attest to the care of the mentally ill through established hospitals in the island since the 4th century.&lt;ref name=&quot;:17&quot; /&gt; Prior to the incarceration of the mentally ill by the European colonizing forces, the mentally ill were regarded as ''Pissowetitch'', or people who had &quot;the spirit of the Gods within him&quot; and &quot;whatsoever he pronounceth, is looked upon as spoken by God himself, and the people will speak to him, as if it were the very person of God&quot;&lt;ref&gt;{{Cite web|url=https://www.gutenberg.org/files/14346/14346-h/14346-h.htm|title=An Historical Relation Of the Island Ceylon, in the East-Indies: Together, With an Account of the Detaining in Captivity the Author and divers other Englishmen now Living there, and of the Author’s Miraculous Escape.|last=Knox|first=Robert|website=www.gutenberg.org|language=en-us|access-date=2026-06-29}}&lt;/ref&gt;. With this religious understanding, Lucien de Alwis reasoned that the mentally ill in Sri Lanka were &quot;placed... at a higher social status than the mentally ill in the Western world&quot;, with this notion correlating with the unsurprising absence of evidence in there being a &quot;large scale segregation of mentally ill from society&quot;&lt;ref name=&quot;:17&quot; /&gt;. 

In the 1800s, established care for mental health began shifting primarily from indigenous practices, mainly derived from [[w:Ayurveda|Ayurveda medicine]], [[w:Siddha_medicine|Siddha medicine]], and [[w:Unani_medicine|Unani medicine]], to a Western mode by the British&lt;ref name=&quot;:17&quot; /&gt;&lt;ref name=&quot;:0&quot;&gt;Gambheera, H. (2011). [https://www.saarcpsychiatry.com/viewText?chapter=c6 The evolution of psychiatric services in Sri Lanka]. South Asian Journal of Psychiatry, 2(1), 25–27.&lt;/ref&gt;&lt;ref name=&quot;:15&quot;&gt;{{Cite book|url=https://doi.org/10.1007/978-981-96-8078-8_7|title=Social Psychiatry in Sri Lanka|last=Baminiwatta|first=Anuradha|last2=Williams|first2=Shehan|date=2025|publisher=Springer Nature|isbn=978-981-96-8078-8|editor-last=Arafat|editor-first=S. M. Yasir|location=Singapore|pages=141–158|language=en|doi=10.1007/978-981-96-8078-8_7|editor-last2=Singh|editor-first2=Amit|editor-last3=Kar|editor-first3=Sujita Kumar}}&lt;/ref&gt;. 

=== Adoption of a Western-based mental healthcare model and ordinances ===
In 1839, [[w:James_Alexander_Stewart-Mackenzie|James Alexander Stewart-Mackenzie]], the 7th Governor of British Ceylon, released the Lunacy Ordinance, authorizing municipal authorities to create lunatic asylums for the mentally ill in the country&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:2&quot;&gt;{{Cite web|url=https://mentalhealth.health.gov.lk/index.php?option=com_content&amp;view=article&amp;id=6&amp;Itemid=125&amp;lang=en|title=History - Directorate of Mental Health|website=mentalhealth.health.gov.lk|access-date=2025-05-10}}&lt;/ref&gt;. The ordinance was concerned with the legal frameworks of detaining individuals considered dangerous to others or individuals falsely presenting themselves as mentally ill, and not on medical treatments to alleviate the conditions of detained individuals. UK psychiatrist [[w:Edward_Mapother|Edward Mapother]] critiqued the ordinance during his 1937 inspection of British Ceylon's mental health institutions in a series of reports titled ''A Disgrace to a Civilised Community'', remarking that the ordinance &quot;[did] not seem to have contemplated treatment as a contingency to be considered&quot;&lt;ref name=&quot;:1&quot;&gt;{{Cite book|title=Permeable walls: historical perspectives on hospital and asylum visiting|date=2009|publisher=Rodopi|isbn=978-90-420-2599-8|editor-last=Mooney|editor-first=Graham|series=Clio medica|location=Amsterdam New York, NY|editor-last2=Reinarz|editor-first2=Jonathan}}&lt;/ref&gt;.

In 1840, the 1839 Ordinance was repealed and replaced by the 1840 Ordinance. The 1839 Ordinance was almost identical to the 1840 Ordinance, except the removal of two previous requirements: the requirement for official medical diagnoses of the mentally ill and the mandate to maintain adequate staff-to-patient ratios within lunatic asylums&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Alwis|first=L. A. P. de|last2=Seneviratne|first2=V. L.|last3=Mendis|first3=T. S. S.|last4=Abhayanayaka|first4=C.|date=2024-12-31|title=The development of laws related to the disposal of forensic patients in Sri Lanka: A historical review|url=https://sljpsyc.sljol.info/articles/10.4038/sljpsyc.v15i2.8569|journal=Sri Lanka Journal of Psychiatry|language=en-US|volume=15|issue=2|doi=10.4038/sljpsyc.v15i2.8569|issn=2012-6883}}&lt;/ref&gt;.

In 1873, a third Ordinance was released. It included linguistic changes, where the term, &quot;insane&quot;, was replaced with &quot;of unsound mind&quot;. The Ordinance also gave more power to medical professionals in determining insanity diagnoses, and more power to detainees in appealing their commitment to the mental asylum. Despite this Ordinance being the most comprehensive legislation on mental healthcare in the country at the time, the legal frameworks behind the detainment of the criminally insane were left identical to previous ordinances&lt;ref name=&quot;:3&quot; /&gt;.

=== Development of mental asylums ===
At the time the 1839 ordinance was released, mentally ill patients were placed either in prisons throughout the country or leprosy hospitals, such as the [[w:Hendala_Leprosy_Hospital|Hendala Leprosy Hospital]] in the Gampaha district&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt;. After the creation of the first mental asylum in Borella in 1846, patients from the Hendala Leprosy Hospital were transferred to the institute in Borella. Overcrowding soon became an issue and patients were sent to prisons across the country. [[File:Edward Mapother.jpg|thumb|A portrait taken of Edward Mapother during his time working at [[w:Maudsley_Hospital|Maudsley Hospital]] in London.
]]

As medical institutions were being made to house the mentally ill, another mental asylum was created in the [[w:Cinnamon_Gardens|Cinnamon Gardens]] area of Colombo in 1884, though this mental asylum faced overcrowding in just one year&lt;ref name=&quot;:0&quot; /&gt;. Treatment in these asylums was limited to occupational and protection therapy, failing to provide treatment for the root causes of the mental disorders.

In 1926, the Angoda Mental Hospital was established, marginally alleviating the severe overcrowding issues that were plaguing the preceding mental asylums. Despite the addition of 1,700 beds to the facility, treatment was still vastly limited and the patients were left in significantly poor conditions. 

=== Edward Mapother's 1937 inspection of British Ceylon ===
Edward Mapother was born in Dublin, Ireland, on July 12, 1881 and moved to London when he was 7 years old&lt;ref&gt;{{Cite book|title=Madness to mental illness: a history of the Royal College of Psychiatrists|last=Bewley|first=Thomas|date=2008|publisher=RCPsych Publications ; Distributed in North America by Balogh International|isbn=978-1-904671-35-0|location=London : [S.l.]}}&lt;/ref&gt;. Mapother attained his M.D. in 1908. While Mapother was the Medical Superintendent of Maudsley Hospital in London, England, he was invited to inspect British Ceylon's mental health institutions by Dr S. T. Gunasekara, the first Medical Director of British Ceylon&lt;ref name=&quot;:1&quot; /&gt;.

In Mapother's visit, he commented that the Angoda Mental Hospital had the atmosphere of &quot;a prison that is neglected and dilapidated&quot;&lt;ref name=&quot;:1&quot; /&gt;. Overcrowding was still a major issue, with the institute hosting 3,000 patients—more than double the intended capacity. Patients were sleeping on mats and were clearly out of reach of adequate treatment. Mapother also noted that only 4% of public health expenditure in the country was being set for hospitals, drawing a stark comparison to London's 25%&lt;ref name=&quot;:1&quot; /&gt;. Mapother offered a vivid and grim account of the hospital in his reports:
&lt;blockquote&gt;
The floor, roof and walls of each cell consist alike of drab cement without any attempt at colouring or decoration. High up in one wall is a small window with stout iron bars. In the floor is a large hole into which the patient may pass his motion and urine. These cells are incompletely divided from one another by a partition which does not reach the roof so that the noise and stink from any one cell may reach at least all the others of the same row. Into these empty cells I was informed that the most noisy and troublesome patients in the hospital; were turned at night completely naked. The doors of the cell contain no observation window, and considering the violent character of many of these patients there is every ground for believing that the doors are rarely opened in the night by the solitary attendant on duty. It needs little imagination to picture the suffering of any patient in an early stage of bodily illness passing a night under such conditions, a situation which must frequently arise. I am told that the noise proceeding from this building is like that on a bad night in a menagerie&lt;ref name=&quot;:0&quot; /&gt;.&lt;/blockquote&gt;Mapother proposed a series of reinforcements to the legal, institutional, and medical frameworks of mental health care in British Ceylon. This included the decentralization of the psychiatric services, a reworking of the Lunacy Ordinance to incorporate treatment into the legal framework, and the establishment of a separate service of medical professionals dedicated to psychiatry. Mapother's recommendations led to several of the best local medical professionals to be sent to London for extensive training in psychiatry, while nurses from England were sent to British Ceylon to supervise hospital operations and train local staff&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:1&quot; /&gt;.

On August 25, 1938, the Executive Committee of Health approved the strategies proposed by Mapother, though the Government was unable to fully implement all of Mapother's interventions due to the 'heavy cost'. In fact, the Government decided to forego one of his proposals, which was the suggestion of a &quot;Visiting Committee&quot;. This committee was tasked to &quot;meet at the hospital, carry out inspections, and make recommendations&quot; to the Executive Committee of Health&lt;ref name=&quot;:1&quot; /&gt;. The Government realized that deficiencies in their mental healthcare system could prove to be &quot;costly&quot; for their reputation. Mapother was reportedly enraged when he found out. Mapother intended to contact the Secretary of State regarding the &quot;distortion&quot; of his plans, but was interrupted by events preceding [[w:World_War_II|World War II]]&lt;ref name=&quot;:1&quot; /&gt;. Mapother passed away on March 20, 1940, without materializing his follow-up plans.

=== Post-Mapother developments and further innovations ===
[[File:Sri Lanka districts Colombo.svg|thumb|A map of Sri Lanka highlighting the Colombo District, where the capital is located. 
|right|250px]]Mapother's insights on the mental healthcare structure in British Ceylon proved to be the catalyst of massive renovations. In 1939, the first outpatient clinic was established in the [[w:National_Hospital_of_Sri_Lanka|National Hospital of Sri Lanka]] in Colombo. The first trained Ceylonese psychiatrists began practice in the 1940s, leading to the establishment of the first neuropsychiatric clinic in Colombo in 1943. Treatments for the mentally ill improved dramatically, as [[w:insulin_shock_therapy|insulin shock therapy]] and [[w:Electroconvulsive_therapy|cardiazol convulsive therapy]] were utilized&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Kathriarachchi|first=Samudra T.|last2=Seneviratne|first2=V. Lakmi|last3=Amarakoon|first3=Luckshika|date=2019-06|title=Development of Mental Health Care in Sri Lanka: Lessons Learned|url=https://journals.lww.com/tpsy/fulltext/2019/33020/development_of_mental_health_care_in_sri_lanka_.1.aspx|journal=Taiwanese Journal of Psychiatry|language=en-US|volume=33|issue=2|pages=55|doi=10.4103/TPSY.TPSY_15_19|issn=1028-3684}}&lt;/ref&gt;. Mapother's advocation for the decentralization of services were further honored through the 1947 establishment of a first child guidance clinic in Colombo General Hospital&lt;ref name=&quot;:0&quot; /&gt;. 

In 1948, British Ceylon was granted independence from the British after the [[w:Sri_Lankan_independence_movement|Sri Lankan independence movement]]. Changes in the mental healthcare structure were not immediate following independence, but rapid expansions of mental healthcare services were still ongoing. 

The following decades saw positive institutional developments, such as the creation of a second hospital in [[w:Mulleriyawa|Mulleriyawa]] in 1957, and the creation of a psychiatric inpatient unit in Colombo General Hospital in 1967—effectively granting the city of Colombo the luxury of hosting the top psychiatric care in the country&lt;ref name=&quot;:5&quot;&gt;{{Cite book|url=http://link.springer.com/10.1007/978-1-4899-7999-5_4|title=Mental Health System Development in Sri Lanka|last=Minas|first=Harry|last2=Mendis|first2=Jayan|last3=Hall|first3=Teresa|date=2017|publisher=Springer US|isbn=978-1-4899-7997-1|editor-last=Minas|editor-first=Harry|location=Boston, MA|pages=59–77|language=en|doi=10.1007/978-1-4899-7999-5_4|editor-last2=Lewis|editor-first2=Milton}}&lt;/ref&gt;. The 1950s was also the start of psychopharmacological innovations, with the introduction of [[w:Lithium_(medication)|lithium]] and long-acting injectable antipsychotics ([[w:Depot_injection|depot]] [[w:Antipsychotic|neuroleptics]]) in the succeeding years&lt;ref name=&quot;:4&quot; /&gt;. Additionally, the number of public psychiatrist positions increased by 400% from 1953 to 1967&lt;ref name=&quot;:5&quot; /&gt;.

After 1960, mental health services were being established beyond the capital to other cities in the country&lt;ref name=&quot;:2&quot; /&gt;.

In 1980, the [[w:Postgraduate_Institute_of_Medicine|Postgraduate Institute of Medicine]] began a program where students would enroll in a 5-year medical course and attain an MD in psychiatry, curbing the need for Sri Lankan medical students to be sent abroad to complete their training. Many of the medical students sent abroad for training never returned to Sri Lanka to practice, resulting in a &quot;1:500,000 to 1000,000&quot; ratio of psychiatrists to patients on &quot;most occasions&quot;&lt;ref name=&quot;:0&quot; /&gt;.

=== Mental Disease Ordinance of 1956 ===
In 1956, the 1873 Ordinance was revised a second time and renamed the &quot;Mental Disease Ordinance of 1956&quot;&lt;ref name=&quot;:5&quot; /&gt;&lt;ref name=&quot;:6&quot;&gt;{{Cite journal|last=Hapangama|first=Aruni|last2=Mendis|first2=Jayan|last3=Kuruppuarachchi|first3=K. a. L. A.|date=2023-02|title=Why are we still living in the past? Sri Lanka needs urgent and timely reforms of its archaic mental health laws|url=https://www.cambridge.org/core/journals/bjpsych-international/article/why-are-we-still-living-in-the-past-sri-lanka-needs-urgent-and-timely-reforms-of-its-archaic-mental-health-laws/B18B03DC962CC6F09BC6D7877E390EE4|journal=BJPsych International|language=en|volume=20|issue=1|pages=4–6|doi=10.1192/bji.2022.26|issn=2056-4740|pmc=9909436|pmid=36812028}}&lt;/ref&gt;. Another linguistic development is seen with the new revision as &quot;lunacy&quot; was replaced with &quot;mental disease&quot;&lt;ref name=&quot;:6&quot; /&gt;. The Ordinance paved the way for community-based services to be delivered to patients closer to their residences rather than solely allocating services to just hospitals. This led to the creation of a [[w:WHO|WHO]]-backed community clinic near the [[w:University_of_Colombo|University of Colombo]] in the 1970s, where the focus was to eventually ease patients in the Angoda Mental Hospital back into the general population&lt;ref name=&quot;:5&quot; /&gt;.

=== Developments from the 1990s ===
The 1990s and onwards saw further positive developments in framing the mental healthcare system, including the establishment of the [https://mentalhealth.health.gov.lk/index.php?option=com_content&amp;view=featured&amp;Itemid=101&amp;lang=en Directorate of Mental Health] in 1998. The Directorate of Mental Health is a part of the [[w:Ministry_of_Health_(Sri_Lanka)|Ministry of Health]] who is responsible for the monitoring and implementation of mental health programs across the country&lt;ref&gt;{{Cite web|url=https://mentalhealth.health.gov.lk/index.php?lang=en|title=Home - Directorate of Mental Health|website=mentalhealth.health.gov.lk|access-date=2025-05-12}}&lt;/ref&gt;.  As of 2025, the current director of the Directorate of Mental Health is Dr. Chithramalee de Silva&lt;ref name=&quot;:2&quot; /&gt;.

On November 11, 2005, the Mental Health Policy was approved by the Government of Sri Lanka, advocating for establishments of more de-centralized, community-based mental health services across the country beyond the capital (Colombo). The policy aimed to concisely define the rigorous standards needed to be completed for each respected medical professional, including psychiatrists and clinical psychologists&lt;ref&gt;{{Cite journal|last=Rajapakshe|first=Onali Bimalka Wickramaseckara|last2=Mohan|first2=Mohapradeep|last3=Singh|first3=Swaran Preet|date=2023-05|title=Development of adolescent mental health services in Sri Lanka|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC10895478/|journal=BJPsych international|volume=20|issue=2|pages=41–43|doi=10.1192/bji.2022.32|issn=2056-4740|pmc=10895478|pmid=38414998}}&lt;/ref&gt;. The policy also included a new position, the &quot;Medical Officer of Mental Health&quot;, who oversees and assists in the implementation of community-based mental health services&lt;ref name=&quot;:0&quot; /&gt;. This same year, the Sri Lankan government began implementing psychological services in state institutions, such as the military&lt;ref name=&quot;:8&quot; /&gt;.  

In 2007, the National Mental Health Advisory Council (NMHAC) was created to serve as an 'advisory' board for the Ministry of Health on what actions should be executed by the Directorate of Mental Health&lt;ref name=&quot;:7&quot;&gt;{{Cite web|url=https://mentalhealth.health.gov.lk/index.php?option=com_content&amp;view=article&amp;id=9&amp;Itemid=220&amp;lang=en|title=Introduction - Directorate of Mental Health|website=mentalhealth.health.gov.lk|access-date=2025-05-12}}&lt;/ref&gt;. 

In 2008, the Angoda Mental Hospital was restructured as the National Institute of Mental Health (NIMH)&lt;ref name=&quot;:7&quot; /&gt;. 

=== Modern-day Sri Lanka ===
[[File:Feeding Children in Sri Lanka.jpg|left|thumb|Despite the noteworthy improvements in mental healthcare services in recent decades, mental health remains a significant issue due to rising poverty. ]]
As of 2025, the Mental Health Act (mental health legislation) has been undergoing development since 2005 and is currently awaiting to be considered for the final stage of approval. This is expected to replace the 1956 Mental Health Ordinance&lt;ref name=&quot;:7&quot; /&gt;.

Currently, there are 7 tertiary care hospitals, 61 adult patient units, 3 child inpatient units, and 1 forensic unit with over 100 psychiatrists all throughout the 22 districts&lt;ref name=&quot;:4&quot; /&gt;. The [[w:Lady_Ridgeway_Hospital_for_Children|Lady Ridgeway Hospital]] in Colombo and the Sirimavo Bandaranayke Specialized Children Hospital in Kandy are tailored towards alleviating children with [[w:Learning_disability|SLD]], [[w:ADHD|ADHD]], [[w:Autism_Spectrum_Disorder|ASD]] and family support for diagnosed children. As of 2017, 22 rehabilitation centers exist through the country, including 7 alcohol rehab centers&lt;ref name=&quot;:7&quot; /&gt;.   

Despite the impressive advancements in mental healthcare in the last couple of decades, Sri Lanka still suffers significant mental health issues due to increasing poverty levels in the country. The [[w:World_Bank|World Bank]] reported that [https://www.wsws.org/en/articles/2024/04/08/eesc-a08.html the poverty levels in Sri Lanka increased from 11% in 2019 to 26% in 2024], with 60% of Sri Lankan households facing &quot;decreased incomes&quot;&lt;ref&gt;Lakhtakia, Shruti, Atapattu Mudiyanselage, Udahiruni Shashadari Atapat, Walker, Richard Ancrum. ''Sri Lanka Development Update - Bridge to Recovery (English).'' Washington, D.C.: World Bank Group. &lt;nowiki&gt;http://documents.worldbank.org/curated/en/099634104012434919&lt;/nowiki&gt;&lt;/ref&gt;. This was exacerbated by Sri Lanka's excessive foreign debt, economic troubles stemming from [[w:Gotabaya_Rajapaksa|Gotabaya Rajapaksa]]'s presidential term, the COVID-19 pandemic, and the [[w:Russian_invasion_of_Ukraine|ongoing invasion of Ukraine by Russia (2022)]].  

According to [[w:NYU|New York University]] graduate student [https://gc-cuny.academia.edu/NadiaAugustyniak Nadia Augustyniak] in her 2025 overview of Sri Lanka's public mental healthcare system, poverty-induced financial precarity remains a major obstacle to receiving access to mental healthcare services. Even though trauma from adverse weather and conflict is deleterious to mental health, issues originating from every-day struggles, especially struggles related to poverty, could arguably play a more significant role&lt;ref name=&quot;:8&quot;&gt;{{Cite journal|last=Augustyniak|first=Nadia|date=2025-06-01|title=Public mental healthcare and economic vulnerability in Sri Lanka|url=https://linkinghub.elsevier.com/retrieve/pii/S2666560324000926|journal=SSM - Mental Health|volume=7|pages=100387|doi=10.1016/j.ssmmh.2024.100387|issn=2666-5603}}&lt;/ref&gt;.  

== Impact of Conflicts, Terrorism, Political Instability &amp; Natural Disasters ==

=== Sri Lankan Civil War ===
The '''Sri Lankan Civil War''' was a domestic conflict that took place between the Sri Lankan government and the Liberation Tigers of Tamil Eelam (abbreviated as the ''LTTE),'' a militant group formed in the 1970s as a result of rising tensions between the majority Sinhalese and minority Tamil population. The group is considered a terrorist organization&lt;ref&gt;{{Cite web|url=https://www.start.umd.edu/baad/database/liberation-tigers-tamil-eelam-ltte-1998.html|title=BAAD - Liberation Tigers of Tamil Eelam (LTTE) - 1998 {{!}} START.umd.edu|website=www.start.umd.edu|access-date=2025-06-09}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://www.cfr.org/backgrounder/liberation-tigers-tamil-eelam-aka-tamil-tigers-sri-lanka-separatists|title=Liberation Tigers of Tamil Eelam (aka Tamil Tigers) (Sri Lanka, separatists) {{!}} Council on Foreign Relations|last=Bhattacharji|first=Preeti|website=www.cfr.org|language=en|access-date=2025-06-09}}&lt;/ref&gt;. The LTTE waged decades of massacres, assassinations of political figures, and suicide bombings to achieve ''[[w:Tamil_Eelam|Tamil Eelam]],'' leading to civilian displacement, infrastructure collapse, and the reduction of mental health services available in the northern region.[[File:DFID-funded, UNHCR emergency shelter tents, in the IDP camp at Menik Farm, Sri Lanka (3694081492).jpg|thumb|350x350px|An IDP camp in Menik Farm, Sri Lanka in 2009 ([https://www.bbc.com/news/world-asia-19703826 now closed]). Suicide rates in IDP camps were three times the general population.]]The civil war mainly affected the northeastern portion of the country, including the [[w:Vanni_(Sri_Lanka)|Vanni region]]. The conflict caused mass destruction to local mental healthcare facilities. Local residents described the conflict with the phrase ''varthayal varnicca mudiyathavai'', roughly translating into English as 'beyond description by words'&lt;ref name=&quot;:9&quot;&gt;{{Cite journal|last=Somasundaram|first=Daya|date=2010-07-28|title=Collective trauma in the Vanni- a qualitative inquiry into the mental health of the internally displaced due to the civil war in Sri Lanka|url=https://doi.org/10.1186/1752-4458-4-22|journal=International Journal of Mental Health Systems|language=en|volume=4|issue=1|pages=22|doi=10.1186/1752-4458-4-22|issn=1752-4458|pmc=2923106|pmid=20667090}}&lt;/ref&gt;. In 2003, only two psychiatrists were found in the region, operating on extremely limited resources and further deepening long-term trauma and mental health deterioration in the population&lt;ref name=&quot;:5&quot; /&gt;. 

In 2002, the humanitarian organization [https://www.msf.org/ Médecins Sans Frontières] (MSF) performed an investigation of mental health needs in the [[w:Vavuniya|Vavuniya]] area, the site of intense conflict during the civil war (including the [[w:1985_Vavuniya_massacre|1985 Vavuniya massacre]]), and found that many of the residents suffered from high suicide rates, alcohol abuse, domestic violence, grief, and a &quot;sense of ‘learnt helplessness’&quot;&lt;ref name=&quot;:5&quot; /&gt;. A team from the University of Konstanz in Germany found that 92% of grade school children in the region were exposed to &quot;combat, shelling, and witnessing the death of loved ones&quot;&lt;ref name=&quot;:9&quot; /&gt;. 
[[File:Tractors. Jan 2009 displacement in the Vanni.jpg|left|thumb|350x350px|Displaced civilians originating from the Kilinochchi and Mullaitivu Districts due to military campaigns by the Sri Lankan military (January 2009). Displaced civilians had to avoid both the atrocities committed by the LTTE and the Sri Lankan government.]]
Accusation of war crimes have been leveraged towards [[w:War_crimes_during_the_final_stages_of_the_Sri_Lankan_civil_war|the Sri Lankan government]]&lt;ref&gt;See also [[w:Sexual violence in the Sri Lankan civil war]].&lt;/ref&gt;. A 2009 HRW report alleged that the Sri Lankan government considered the native Tamil population residing in war zones to be &quot;siding with the LTTE and [therefore, were] treated as combatants&quot;, leading to indiscriminate shellings and massacres of civilians&lt;ref&gt;{{Cite journal|date=2009-02-19|title=War on the Displaced|url=https://www.hrw.org/report/2009/02/19/war-displaced/sri-lankan-army-and-ltte-abuses-against-civilians-vanni|journal=Human Rights Watch|language=en}}&lt;/ref&gt;. Additionally, the Vanni population also faced recruitment campaigns by the LTTE, where recruited men, women, and even children with minimal training, were utilized for war efforts.  

Over 200,000 Tamil civilians were moved into [[w:Internally_displaced_persons_in_Sri_Lanka|designated displacement camps during the war]], where conditions were abysmal&lt;ref&gt;{{Cite journal|last=Dissanayake|first=Lasith|last2=Jabir|first2=Sameeha|last3=Shepherd|first3=Thomas|last4=Helliwell|first4=Toby|last5=Selvaratnam|first5=Lavan|last6=Jayaweera|first6=Kaushalya|last7=Abeysinghe|first7=Nihal|last8=Mallen|first8=Christian|last9=Sumathipala|first9=Athula|date=2023-08-31|title=The aftermath of war; mental health, substance use and their correlates with social support and resilience among adolescents in a post-conflict region of Sri Lanka|url=https://doi.org/10.1186/s13034-023-00648-1|journal=Child and Adolescent Psychiatry and Mental Health|language=en|volume=17|issue=1|pages=101|doi=10.1186/s13034-023-00648-1|issn=1753-2000}}&lt;/ref&gt;. The suicide rate in these displacement camps were three times the community-level (2002), with a ratio of 103.5 per 10,000 compared to the Sri Lankan general population's rate of 37.5 per 10,000. Almost all suicide attempts involved poisonous substances. Other forms of violence included domestic violence and child abuse. Local health officials in Vavuniya admitted that mental health concerns were a major problem, but were unable to address these concerns due to a lack of resources and support from the government. During the [[wikipedia:Sri_Lankan_civil_war#2002_peace_process_(2002%E2%80%932006)|brief 2002 ceasefire]], the MSF implemented a &quot;community-based programme&quot; which included &quot;increasing awareness, community strengthening, reinforcing coping-strategies for long-term war-affected communities, and counselling&quot;. The MSF also advocated for restrictions of poisonous substances due to the suicide attempts, and stressed that &quot;much more [than resettlement]&quot; would need to be done to help alleviate the psychological pain the northern population had faced&lt;ref&gt;{{Cite journal|last=de Jong|first=Kaz|last2=Mulhern|first2=Maureen|last3=Ford|first3=Nathan|last4=Simpson|first4=Isabel|last5=Swan|first5=Alison|last6=van der Kam|first6=Saskia|date=2002-04|title=Psychological trauma of the civil war in Sri Lanka|url=https://linkinghub.elsevier.com/retrieve/pii/S0140673602084209|journal=The Lancet|language=en|volume=359|issue=9316|pages=1517–1518|doi=10.1016/S0140-6736(02)08420-9}}&lt;/ref&gt;. The ceasefire ended in 2006 and led to the [[w:Eelam_War_IV|final phase of the civil war]], eventually ending in 2009 with the [[w:https://en.wikipedia.org/wiki/Velupillai_Prabhakaran#Sri_Lankan_Army_Northern_offensive_and_death|death of the LTTE's leader]]. 

'''Post-war'''
[[File:Puttalam district.svg|left|thumb|Puttalam District, unlike its northern counterparts, was largely spared from the intense conflict, possibly explaining the lower rates of common mental disorders (CMDs).]]
The first district-wide cross-sectional multistage cluster sample survey was conducted in the [[w:Jaffna_District|Jaffna District]] shortly after the war ended. The study's sample included 1517 households and 2 internally displaced peoples camps. With a response rate of 92%, the study found that symptoms for PTSD were found in 7% of participants, symptoms of anxiety were found in 32.6% of participants, and symptoms of depression were found in 22.2% of participants. 2% of respondents were currently placed in internally displaced peoples camps at the time of the study, 29.5% were freshly resettled from the internally displaced peoples camps, and the rest of the participants (68.5%) were never placed into camps. In comparison to residents who were never placed into camps, participants that were actively held in camps tend to report more symptoms of PTSD, anxiety, and depression. The researchers also found that women were especially vulnerable to deteriorating mental health conditions. This was explained by two factors: women having to assume the roles of both the father and the mother in the family setting after the, either voluntary or forced, departure of the husband to war, and sexist violence&lt;ref&gt;{{Cite journal|last=Husain|first=Farah|last2=Anderson|first2=Mark|last3=Lopes Cardozo|first3=Barbara|last4=Becknell|first4=Kristin|last5=Blanton|first5=Curtis|last6=Araki|first6=Diane|last7=Kottegoda Vithana|first7=Eeshara|date=2011-08-03|title=Prevalence of War-Related Mental Health Conditions and Association With Displacement Status in Postwar Jaffna District, Sri Lanka|url=https://doi.org/10.1001/jama.2011.1052|journal=JAMA|volume=306|issue=5|pages=522–531|doi=10.1001/jama.2011.1052|issn=0098-7484}}&lt;/ref&gt;. A 2013 study on adult patients in [https://www.ncbi.nlm.nih.gov/books/NBK232631/ primary care settings] (divisional hospitals, primary medical care units) found major depression to be significantly higher in females (5.1%) than males (3.6%), bolstering the observation seen in the 2009 study&lt;ref&gt;{{Cite journal|last=Senarath|first=Upul|last2=Wickramage|first2=Kolitha|last3=Peiris|first3=Sharika Lasanthi|date=2014-03-24|title=Prevalence of depression and its associated factors among patients attending primary care settings in the post-conflict Northern Province in Sri Lanka: a cross-sectional study|url=https://doi.org/10.1186/1471-244X-14-85|journal=BMC Psychiatry|language=en|volume=14|issue=1|pages=85|doi=10.1186/1471-244X-14-85|issn=1471-244X|pmc=3987835|pmid=24661436}}&lt;/ref&gt;.

Muslims in Northern Sri Lanka during the conflict also faced violence and discrimination, most notably [[w:Expulsion_of_Muslims_from_the_Northern_Province_of_Sri_Lanka|the October 1990 expulsion of Muslims from the North to the Puttalam District or Jaffna]] and the [[w:Kattankudy_mosque_massacre|1990 Kattankudy mosque massacre]]. The only study testing the displaced Muslim population post-civil war was completed in 2011, where a cross-sectional survey of 450 internally displaced people or people born into displacement (ages 18 - 65) revealed 18.8% of the sample suffering from common mental health disorders (CMD), including [[w:Somatoform_disorder|somatoform disorder]] (14%), &quot;other depressive syndromes&quot; (7.3%), major depression (5.1%), and anxiety disorder (2.8%). The percentages found in this study for somatoform disorder and major depression were &quot;considerably higher&quot; than the national percentages, though the researchers noted that the prevalence of CMD was lower in comparison to other countries marred with conflict, including Palestine (40.3%) and Ethiopia (27.8%). The researchers explained that the lower rate of CMD may be attributed to the [[w:Puttalam_District|serenity of the post-settlement destination]], as conflict was mainly centered in the North and East. In contrast to earlier findings, this study did not observe a higher prevalence of CMDs among women, although increased rates of somatoform disorders were noted (though the researchers did not show the data behind this)&lt;ref&gt;{{Cite journal|last=Siriwardhana|first=Chesmal|last2=Adikari|first2=Anushka|last3=Pannala|first3=Gayani|last4=Siribaddana|first4=Sisira|last5=Abas|first5=Melanie|last6=Sumathipala|first6=Athula|last7=Stewart|first7=Robert|date=2013-05-22|title=Prolonged Internal Displacement and Common Mental Disorders in Sri Lanka: The COMRAID Study|url=https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0064742|journal=PLOS ONE|language=en|volume=8|issue=5|pages=e64742|doi=10.1371/journal.pone.0064742|issn=1932-6203|pmc=3661540|pmid=23717656}}&lt;/ref&gt;.

Research on the mental state of combatants has been limited, but a post-war 2009 study done between soldiers of the [[w:Sri_Lanka_Army_Special_Forces_Regiment|Special Forces]] and regular soldiers showed higher levels of exposure to traumatic events for units of the Special Forces, yet the former exhibited significantly less symptoms of CMDs compared to the latter. The authors of this study, [https://scholar.google.co.uk/citations?user=cVKEBdwAAAAJ&amp;hl=en&amp;oi=ao Raveen Hanwella] and [https://scholar.google.co.uk/citations?user=ZRj74qMAAAAJ&amp;hl=en&amp;oi=sra Varuni de Silva], offers the camaraderie of the unit as an explanation for the discrepancy&lt;ref&gt;{{Cite journal|last=Hanwella|first=Raveen|last2=de Silva|first2=Varuni|date=2012-08|title=Mental health of Special Forces personnel deployed in battle|url=https://pubmed.ncbi.nlm.nih.gov/22038567|journal=Social Psychiatry and Psychiatric Epidemiology|volume=47|issue=8|pages=1343–1351|doi=10.1007/s00127-011-0442-0|issn=1433-9285|pmid=22038567}}&lt;/ref&gt;. A follow-up study was completed by the pair (with the addition of former Director-General of the Health Services of the Sri Lanka Navy [[w:Nicholas_Jayasekera|Nicholas Jayasekera]]), where the findings were similar, though the statistically significant bridge between the two cohorts in the previous study evaporated in the follow-up study. This may be due to the significant decline in mental health problems observed in the regular unit forces, potentially reflecting resilience in the aftermath of jarring conflict&lt;ref&gt;{{Cite journal|last=Hanwella|first=Raveen|last2=Jayasekera|first2=Nicholas E. L. W.|last3=Silva|first3=Varuni A. de|date=2014-09-25|title=Mental Health Status of Sri Lanka Navy Personnel Three Years after End of Combat Operations: A Follow Up Study|url=https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0108113|journal=PLOS ONE|language=en|volume=9|issue=9|pages=e108113|doi=10.1371/journal.pone.0108113|issn=1932-6203|pmc=4177866|pmid=25254557}}&lt;/ref&gt;. Amputees or soldiers with spinal injuries exhibited drastically different numbers, with approximately 40% of nearly 100 male-veterans in a post-war 2009 study displaying PTSD-like symptoms&lt;ref&gt;{{Cite journal|last=Abeyasinghe|first=N. L.|last2=de Zoysa|first2=P.|last3=Bandara|first3=K.M.K.C.|last4=Bartholameuz|first4=N. A.|last5=Bandara|first5=J. M.U.J.|date=2012-05-01|title=The prevalence of symptoms of Post-Traumatic Stress Disorder among soldiers with amputation of a limb or spinal injury: A report from a rehabilitation centre in Sri Lanka|url=https://doi.org/10.1080/13548506.2011.608805|journal=Psychology, Health &amp; Medicine|volume=17|issue=3|pages=376–381|doi=10.1080/13548506.2011.608805|issn=1354-8506|pmid=21942815}}&lt;/ref&gt;.

About a decade after the conflict ceased, a few notable studies have emerged to help guide understanding on the longer-term mental health effects on victims of the civil war. 

From July 2019 to October 2020, a study was conducted on 585 local adolescents (ages 12-19) in the Vavuniya district revealed that despite 15.6% of the statistic having faced one or more war-related events, only 3.9% of the participants had moderate - severe depression. In addition to considerably low depression rates, only 5.7% of participants age 17+ were found to have moderate - severe hopelessness&lt;ref&gt;{{Cite journal|last=Dissanayake|first=Lasith|last2=Jabir|first2=Sameeha|last3=Shepherd|first3=Thomas|last4=Helliwell|first4=Toby|last5=Selvaratnam|first5=Lavan|last6=Jayaweera|first6=Kaushalya|last7=Abeysinghe|first7=Nihal|last8=Mallen|first8=Christian|last9=Sumathipala|first9=Athula|date=2023-08-31|title=The aftermath of war; mental health, substance use and their correlates with social support and resilience among adolescents in a post-conflict region of Sri Lanka|url=https://doi.org/10.1186/s13034-023-00648-1|journal=Child and Adolescent Psychiatry and Mental Health|language=en|volume=17|issue=1|pages=101|doi=10.1186/s13034-023-00648-1|issn=1753-2000|pmc=10472617|pmid=37653394}}&lt;/ref&gt;. The authors referenced a 2010 observation by psychiatrist [https://us.sagepub.com/en-us/nam/author/daya-somasundaram Daya Somasundaram], who noted that many Tamil IDPs exhibited &quot;remarkable resilience and post-traumatic growth&quot; after the civil war—an outcome he attributed to the close-knit, family-centered nature of Tamil communities&lt;ref&gt;{{Cite journal|last=Somasundaram|first=Daya|date=2010-07-28|title=Collective trauma in the Vanni- a qualitative inquiry into the mental health of the internally displaced due to the civil war in Sri Lanka|url=https://doi.org/10.1186/1752-4458-4-22|journal=International Journal of Mental Health Systems|volume=4|issue=1|pages=22|doi=10.1186/1752-4458-4-22|issn=1752-4458|pmc=2923106|pmid=20667090}}&lt;/ref&gt;. Findings originating from a 2019 study undertook by several faculty members from the University of Kelaniya, the University of Jaffna, the [[w:Gampaha_Wickramarachchi_University_of_Indigenous_Medicine|Gampaha Wickramarachchi University of Indigenous Medicine]], and the [https://onur.gov.lk/ Office for National Unity and Reconciliation (ONUR)] in Jaffna, found contrasting statistics. Out of 336 participants from districts that faced significant ramifications of the conflict (Jaffna, Kilinochchi, Mullaithivu, Vavuniya, and Mannar districts), 50.5% had extreme anxiety symptoms and 36.5% exhibited &quot;extremely severe&quot; symptoms of depression. 92.5% of families in the sample experienced suicidal ideation, with an observed negative correlation between trauma exposure and life satisfaction with families. Drug abuse (86.2%) and alcohol abuse (84.5%) were the two highest problematic behaviors recorded on a community-level, suggesting that the negative consequences of the civil war still persist, possibly on a substantial scale than previously recognized, in Tamil communities in the North&lt;ref&gt;{{Cite journal|last=Thamotharampillai|first=Umaharan|last2=Perera|first2=Ruwanthi|last3=Wickremasinghe|first3=Rajitha|last4=Williams|first4=Shehan|last5=Vijayasangar|first5=Thedsanamoorthy|last6=Sivatharsan|first6=Balasubramaniam|last7=Hilbert|first7=Vanceline|last8=Somasundaram|first8=Daya|date=2025-05-06|title=Collective Trauma- Psychosocial consequences of war in northern Sri Lanka 10 years on, a mixed methods study|url=https://www.sciencedirect.com/science/article/pii/S2666560325000696|journal=SSM - Mental Health|pages=100457|doi=10.1016/j.ssmmh.2025.100457|issn=2666-5603}}&lt;/ref&gt;. Further research should be conducted in this field. 

In 2019, [https://www.researchgate.net/scientific-contributions/R-M-M-Monaragala-2087692299 Dr. R. M. M. Monaragala] conducted a study on 1,845 soldiers with combat experience, finding that 3.9% of the sample suffered from PTSD. Dr. Monaragala noted that &quot;probable depression, fatigue, aggression, and family history of mental disorder&quot; were correlative of PTSD presence. He suggested that &quot;screening and psychosocial intervention&quot; were recommended avenues to alleviate CMDs of former combatants&lt;ref&gt;{{Cite journal|last=Monaragala|first=R. M. M.|date=2024-04-19|title=Exploring the effects of the past civil war in terms of the prevalence and associating factors of PTSD|url=https://sljpsyc.sljol.info/articles/10.4038/sljpsyc.v14i2.8465|journal=Sri Lanka Journal of Psychiatry|language=en-US|volume=14|issue=2|doi=10.4038/sljpsyc.v14i2.8465|issn=2012-6883}}&lt;/ref&gt;. 

=== 2004 Boxing Day Tsunami ===
The '''2004 Boxing Day Tsunami''' was a natural disaster where a tsunami spawned off a 9.2–9.3 magnitude earthquake off the coast of Aceh in Indonesia on December 26.  The tsunami greatly affected the coastlines of the country, with the death toll reaching to about 35,000 deaths. In addition, 90,000 houses were destroyed and 516,000 people were forced to migrate due to severe infrastructural damage&lt;ref name=&quot;:5&quot; /&gt;. It stands as the [http://www.china.org.cn/english/features/tsunami_relief/119821.htm worst natural disaster to have ever hit Sri Lanka].  
[[File:Tsunami relief 2004 02.jpg|thumb|300x300px|Volunteers from [[w:Royal_College,_Colombo|Royal College in Colombo]] assisting in tsunami relief efforts (Sarvodaya Headquaters, Moratuwa).]]
A survey conducted on schoolchildren (ages 8-14) in Manadkadu (Tamil-majority village in the northern coast), [[w:Kosgoda|Kosgoda]] (western coast), and [[w:Galle|Galle]] (southern coast), just a few weeks after the tsunami hit Sri Lanka, revealed that 33.8%, 13.9%, and 38.8% of children interviewed exhibited signs of PTSD (according to the DSM-IV's criteria), respectively (minus the time criteria, as the DSM-IV does not permit diagnosis of PTSD within 4 weeks of a traumatic incident). The loss of family members and exposure to previously traumatic incidents seem to highly correlate with PTSD development&lt;ref&gt;{{Cite journal|last=Neuner|first=Frank|last2=Schauer|first2=Elisabeth|last3=Catani|first3=Claudia|last4=Ruf|first4=Martina|last5=Elbert|first5=Thomas|date=2006|title=Post-tsunami stress: A study of posttraumatic stress disorder in children living in three severely affected regions in Sri Lanka|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/jts.20121|journal=Journal of Traumatic Stress|language=en|volume=19|issue=3|pages=339–347|doi=10.1002/jts.20121|issn=1573-6598}}&lt;/ref&gt;. 

Many victims in the Jaffna area suffered with &quot;[https://www.psychiatry.org/patients-families/prolonged-grief-disorder pathological grief], phobias, depression and PTSD&quot; post-tsunami. Schizophrenia in the Jaffna Tamil community, which had already suffered elevated prevalence of PTSD prior to the tsunami, had worsened—highlighting the need for specialized care in response to cumulative exposures to chronic and acute traumas. In a study published in the journal ''International Psychiatry'' (2006), Jaffna-based researchers noted that, contrary to their initial inclinations, there was not a &quot;large[r] (than expected) rise in [the] number of people&quot; seeking mental health support 3 months after the tsunami. However, 10 months after the disaster, the researchers anticipated that &quot;more psychiatric disorders&quot; would emerge due to &quot;very little rebuilding [efforts]&quot; and an apparent &quot;unfairness in the aid system&quot;.&lt;ref&gt;{{Cite journal|last=Somasundaram|first=D. J.|last2=Yoganathan|first2=S.|last3=Ganesvaran|first3=T.|date=1993-09|title=Schizophrenia in northern Sri Lanka|url=https://pubmed.ncbi.nlm.nih.gov/7828234|journal=The Ceylon Medical Journal..|volume=38|issue=3|pages=131–135|issn=0009-0875|pmid=7828234}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Danvers|first=K.|last2=Sivayokan|first2=S.|last3=Somasundaram|first3=D. J.|last4=Sivashankar|first4=R.|date=2006-07|title=Ten months on: qualitative assessment of psychosocial issues in northern Sri Lanka following the tsunami|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC6734678/|journal=International Psychiatry: Bulletin of the Board of International Affairs of the Royal College of Psychiatrists|volume=3|issue=3|pages=5–8|issn=1749-3676|pmc=6734678|pmid=31507850}}&lt;/ref&gt; 

At the February 2005 ''After the Tsunami: Mental Health Challenges to the Community for Today and Tomorrow'' conference in Thailand, [https://www.researchgate.net/profile/Chandanie-Hewage Dr. Chandanie Hewage] of the [[w:University_of_Ruhuna|University of Ruhuna]] commentated that measures taken to assist the affected were &quot;not coordinated&quot; due to poor &quot;communication systems and road [conditions].&quot; Regardless, efforts were continued by the government and health professionals to alleviate the struggles the victims were facing, including the psychological ramifications of the disaster. 

Several issues in the delivery of these services were highlighted by Dr. Hewage, including poor maintenance of health records, lack of awareness on drug consumption by the patients themselves, and shortages of health professionals. Dr. Hewage points out that personnel had &quot;little&quot; mental health training prior to the disaster, suggesting increased &quot;research&quot; and adequate &quot;provision[ing] and training of staff&quot; in the long-term&lt;ref&gt;{{Cite journal|last=Davidson|first=Jonathan R. T.|date=2006|title=Foreword. After the tsunami: mental health challenges to the community for today and tomorrow|url=https://pubmed.ncbi.nlm.nih.gov/16602809|journal=The Journal of Clinical Psychiatry|volume=67 Suppl 2|pages=3–8|issn=0160-6689|pmid=16602809}}&lt;/ref&gt;. With inadequate documentation, no systematic procedures in place, and insufficient personnel, tsunami victims with mental health concerns may not receive the services they need, further compacting neuropsychological ailments.  

In 2008 (about 3-4 years after the tsunami), researchers in the hard-hit village of [[w:Peraliya|Peraliya]] (Galle District) found that from a sample of approximately 90 adults, 25% suffered from moderate–severe PTSD, with women scoring &quot;above the cut-off for anxiety&quot; and reporting more &quot;somatic symptoms&quot;, though researchers inferred that the PTSD rate found in the study may be influenced by war or economic hardship&lt;ref&gt;{{Cite journal|last=Hollifield|first=Michael|last2=Hewage|first2=Chandanie|last3=Gunawardena|first3=Charlotte N.|last4=Kodituwakku|first4=Piyadasa|last5=Bopagoda|first5=Kalum|last6=Weerarathnege|first6=Krishantha|last7=Group|first7=International Post-Tsunami Study|date=2008-01|title=Symptoms and coping in Sri Lanka 20–21 months after the 2004 tsunami|url=https://www.cambridge.org/core/journals/the-british-journal-of-psychiatry/article/symptoms-and-coping-in-sri-lanka-2021-months-after-the-2004-tsunami/CB33752239AF362A0BFD55B3668D60B0|journal=The British Journal of Psychiatry|language=en|volume=192|issue=1|pages=39–44|doi=10.1192/bjp.bp.107.038422|issn=0007-1250}}&lt;/ref&gt;.  

=== 2019 Easter Bombings ===
The '''2019 Easter Bombings''' were a series of coordinated attacks perpetrated by the Islamic extremist group, [[w:National_Thowheeth_Jama'ath|National Thowheeth Jama'ath]], on April 21, 2019. The attack targeted three churches and three hotels in the Colombo area, killing nearly 300 people and injuring over 500. The attack was also attributed to the incompetency of the Sri Lankan government, who ignored [https://www.bbc.com/news/world-asia-48044636 multiple warnings regarding the attacks]. The attacks negatively affected the Sri Lankan Catholic community and further weakened relations between the major religious groups&lt;ref&gt;{{Cite journal|last=Jayawickreme|first=Nuwan|last2=Jayawickreme|first2=Eranda|last3=McCaffrey|first3=Amy Z.|last4=Thiruvarangan|first4=Mahendran|date=2025-06-01|title=Mental health futures in post-war Sri Lanka: Resilience, relational pluralism, and implementation pathways|url=https://www.sciencedirect.com/science/article/pii/S2666560325000775|journal=SSM - Mental Health|volume=7|pages=100465|doi=10.1016/j.ssmmh.2025.100465|issn=2666-5603}}&lt;/ref&gt;.  

In the aftermath of the attacks, professionals in the [[w:Gampaha_District|Gampaha District]] resorted to &quot;low-cost methodological&quot; responses to children and adolescents affected by the attack as a &quot;severe shortage&quot; of children and adolescent mental health experts were exposed&lt;ref&gt;{{Cite journal|last=Chandradasa|first=Miyuru|last2=Rathnayake|first2=Layani C|last3=Rowel|first3=Madushi|last4=Fernando|first4=Lalin|date=2020-06-01|title=Early phase child and adolescent psychiatry response after mass trauma: Lessons learned from the Easter Sunday attack in Sri Lanka|url=https://doi.org/10.1177/0020764020913314|journal=International Journal of Social Psychiatry|language=EN|volume=66|issue=4|pages=331–334|doi=10.1177/0020764020913314|issn=0020-7640}}&lt;/ref&gt;. In a qualitative study of 8 survivors of the attacks receiving grief counseling, [[w:University_of_Ruhuna|University of Ruhuna]] assistant professor [https://www.researchgate.net/profile/Virasha-Godakanda Virasha Godakanda] observed that 70% of the sample size expressed &quot;doubts&quot; in adequate mental health interventions from the government, reducing the quality of such services. Professor Godakanda strongly endorsed for &quot;culturally-sensitive&quot; programs, a diversity in therapeutic approaches (including nature-based therapy), and &quot;prolonged investigations&quot; to track developments in mental health resources and impacts of implemented interventions&lt;ref&gt;{{Cite journal|last=Godakanda|first=Virasha|date=2025-01-29|title=A GRIEF COUNSELING INTERVENTION AFTER THE MASS TRAUMA: LESSONS LEARNED FROM THE VICTIMS OF THE EASTER SUNDAY ATTACK IN SRI LANKA|url=https://kjmr.com.pk/kjmr/article/view/216|journal=Kashf Journal of Multidisciplinary Research|language=en|volume=2|issue=01|pages=13–32|doi=10.71146/kjmr216|issn=3007-200X}}&lt;/ref&gt;. 

A few weeks following the attacks, Muslims in Sri Lanka were subjected to [[w:2019_anti-Muslim_riots_in_Sri_Lanka|violent, coordinated riots]] masterminded by Sinhalese national forces&lt;ref&gt;{{Cite journal|last=Mujahidin|first=Muhammad Saekul|date=2023-07-03|title=Extremism and Islamophobia Against the Muslim Minority in Sri Lanka|url=https://www.ajis.org/|journal=American Journal of Islam and Society|language=en|volume=40|issue=1-2|pages=213–241|doi=10.35632/ajis.v40i1-2.3135|issn=2690-3741}}&lt;/ref&gt;. Riots were mainly centered in the [[w:Kurunegala_District|Kurunegala]], Gampaha, and [[w:Kandy_District|Kandy]] Districts. At least [https://www.aljazeera.com/news/2019/5/21/in-sri-lanka-muslims-say-sinhala-neighbours-turned-against-them one confirmed death was reported]. Calls for vague ''niqab'' and ''burqa'' bans were increasingly prominent, eventually leading to the 2021 burqa ban by the Sri Lankan government. Pakistani and Afghani refugees fleeing religious persecution in Negombo were forced to be &quot;made refugees again&quot; after local protests were orchestrated against their settlement. Islamophobic sentiment was &quot;unleashed online, in the law, and on the street&quot;&lt;ref&gt;{{Cite book|title=CARTOGRAPHIC JOURNEY OF RACE, GENDER AND POWER: global identity|date=2021|publisher=CAMBRIDGE SCHOLARS PUBLIS|isbn=978-1-5275-6965-2|location=S.l.}}&lt;/ref&gt;. Albeit its relevancy to the attacks, no in-depth mental health studies were administered on the minority Muslim population following the Easter bombings. Further research is imperative in exploring the sustained psychological effects of Islamophobia and its effect on the Muslim minority community in the aftermath of the 2019 Easter attacks. 

Literature on the impact of the 2019 Easter Bombings on mental health is limited and further research should be conducted.  

=== 2019-2024 Economic Crisis ===
The '''2019-2024 Economic Crisis''' refers to a 5 year period where the Sri Lankan economy experienced massive inflation and an abrupt hike in prices on basic, everyday items. It is the worse economic crisis the country has faced since the Sri Lankans were granted independence in 1948. Schools in Sri Lanka were forced to postpone examinations due to paper shortages. Gas shortages led to long lines at gas stations, some lasting for days, throughout the island. Shortages in electricity, cooking gas, and aviation were additional results of the economic crisis.

Healthcare workers faced a barrage of mental health during the crisis, including a lopsided work-life balance due to unprecedented demand, increased stress and mental fatigue from a lack of resources and personnel, unhealthy coping mechanisms, job dissatisfaction, and a reduction in work quality. Such effects perpetuate a self-enforcing cycle of psychologically distressed mental healthcare workers providing subpar services, affecting patients and amplifying mental health issues experienced by both the workforce and their patients&lt;ref&gt;{{Cite journal|last=Dilogini|first=S.|last2=Grace|first2=H. H.|last3=Thasika|first3=T.|date=2024|title=Exploring The Mental Health and Well-Being of Public Healthcare Workers (HCWs) Amid Economic Crisis in Sri Lanka|url=http://repo.lib.jfn.ac.lk/ujrr/handle/123456789/11092|language=en|publisher=Chartered Institute of Personnel Management}}&lt;/ref&gt;. 

Medical students from the Faculty of Medicine at the University of Colombo reported that the economic crisis forced abrupt changes in dietary consumption, increased hopelessness in the future, increased stress and anxiety, and a decrease in interest in pursuing a &quot;clinical post-graduate career&quot;&lt;ref&gt;{{Cite journal|last=Adikaranayake|first=Pesala Randika|last2=Perera|first2=Anusha Nimrod|last3=Nilaweera|first3=Akhila Imantha|last4=Fernando|first4=Desha Rajni|last5=Wijayaratne|first5=Dilushi Rowena|date=2025-07-01|title=Effects of Sri Lankan economic crisis on health, lifestyle and education of medical students in Faculty of Medicine, University of Colombo – an online survey|url=https://doi.org/10.1186/s12909-025-07506-y|journal=BMC Medical Education|language=en|volume=25|issue=1|pages=938|doi=10.1186/s12909-025-07506-y|issn=1472-6920|pmc=12211748}}&lt;/ref&gt;. 283 government-school teachers completed a web-based cross-sectional survey in April 2024, with majority of the participants reporting a severe reduction in monthly income &amp; 1/3 of participants exhibiting &quot;clinical levels of psychological distress&quot;&lt;ref&gt;{{Cite journal|last=Senevirathne|first=C. P.|last2=Senarathne|first2=D. L. P.|last3=Fernando|first3=M. S.|last4=Senevirathne|first4=S. P.|date=2025-05-28|title=Examining the economic burden and mental health distress among government school teachers in Sri Lanka: a cross-sectional study|url=https://doi.org/10.1186/s40359-025-02921-8|journal=BMC Psychology|language=en|volume=13|issue=1|pages=572|doi=10.1186/s40359-025-02921-8|issn=2050-7283}}&lt;/ref&gt;. A study published in that same year reported that out of 261 nurses working in teaching hospitals, 91.6% were forced to allocate their finances to strictly &quot;general needs&quot;, while more than 50% looked into international opportunism for employment. Notably, the study reported an overall near &quot;twofold greater&quot; rate of depression, anxiety, and stress compared to previously conducted studies on nurses&lt;ref&gt;{{Cite journal|last=Senevirathne|first=C.P|last2=Senarathne|first2=L.|last3=Fernando|first3=M.|date=2024-04-01|title=Exploring the Association Between Behavioural Modification in Response to the Prevailing Economic Crisis and Mental Health Outcomes of Nurses from Teaching Hospitals, Sri Lanka|url=https://doi.org/10.1177/23779608241272679|journal=SAGE Open Nursing|language=EN|volume=10|pages=23779608241272679|doi=10.1177/23779608241272679|issn=2377-9608|pmc=11311183}}&lt;/ref&gt;. 

The detrimental effects the crisis has had on the mental health sector reveal a concerning area of underappreciation and under compensation by the Sri Lankan government towards a critical sector for the well-being of the country. Comprehensive mental health interventions need to be prepared and ready to implement at times of national emergencies. 

== Present-Day Challenges ==

=== Ethnic tension ===
Despite the end of the Sri Lankan civil war and the introduction of pluralist policies, such as the [https://srilankaembassy.fr/sites/default/files/files/media/pdf/NationalPolicy-English.pdf 2017 National Policy on Reconciliation and Coexistence] under the Sirisena administration, tensions amongst members of the ethnic groups still persist in the country. Evidence of these tensions was found through a 2022 study conducted in the Ratnapura district, where religious leaders expressed skepticisms, through semi-structured interviews, for &quot;conflict transformation&quot;. A Tamil citizen of the Ratnapura community recounted that they were forced to &quot;hide in jungles&quot; and consume &quot;dirty water in drainage[s]&quot; due to scarcity of food and drinkable water as a result of the conflict. In certain personal accounts, ethnic conflicts appear to affect the social behavior and identity of the majority ethnic group. One Sinhala participant recounted his objection to the war-time retaliatory destruction of a shop run by a Tamil shopkeeper was met with interrogative questions about &quot;whether [he was] Sinhalese or not&quot;. Both accounts convey interethnic tensions stemming from decade-long conflicts&lt;ref&gt;Jayathilaka, Aruna &amp; Gamage, Sayuri. (2024). Role of Buddhist and Hindu Religious Leaders Role of Buddhist and Hindu Religious Leaders in the Post-War Conflict Transformation Process: A Study Based on Rathnapura District in Srilanka. ''Retrieved from'' https://gandhimargjournal.org/wp-content/uploads/2024/09/Volume-46-Issue-1-April-June-2024.pdf#page=66&lt;/ref&gt;. 

Beyond individual accounts and the official end of the civil war, the minority groups in the country continue to feel ostracized. The Sri Lankan Tamil population remains dissatisfied with the Sri Lankan government and their accountability of perpetrators of war crimes and information on the whereabouts of [[w:Enforced_disappearances_in_Sri_Lanka|thousands of enforced disappearances]] that took place from the 1980s. Additionally, rising anti-Muslim sentiment in recent years contribute to increased ethnic tensions, a stark contrast to the previous centuries of peaceful co-existence between the groups. 
[[File:Bodu Bala Sena symbol.svg|thumb|The symbol for Bodu Bala Sena, a nationalistic Sinhala Buddhist group criticized for catalyzing ethnic tensions in Sri Lanka.]]
Laws passed by the Sri Lankan government, such as the [[w:Prevention_of_Terrorism_Act_(Sri_Lanka)|Prevention of Terrorism Act]] and [[wikipedia:Anti-conversion_law#Sri_Lanka|anti-conversion laws]], have forced the United States Commission on International Religious Freedom to label Sri Lanka as a nation that &quot;[engages] or [tolerates] severe violations of religious freedom&quot; in their 2024 report. The government has been criticized by human rights organizations for &quot;disproportionately targeting religious minorities&quot;&lt;ref&gt;{{Cite journal|last=Jayawickreme|first=Nuwan|last2=Jayawickreme|first2=Eranda|last3=McCaffrey|first3=Amy Z.|last4=Thiruvarangan|first4=Mahendran|date=2025-06-01|title=Mental health futures in post-war Sri Lanka: Resilience, relational pluralism, and implementation pathways|url=https://www.sciencedirect.com/science/article/pii/S2666560325000775|journal=SSM - Mental Health|volume=7|pages=100465|doi=10.1016/j.ssmmh.2025.100465|issn=2666-5603}}&lt;/ref&gt;. Additionally, the implementation of the three dominant languages, English, Sinhala, and Tamil, across formal education and government services have been lackadaisical, narrowing opportunities of foundational social interactions between the groups. Persistent discrimination and prejudice towards minority groups can lead to an array of complex and self-deprecating mental health issues.   

Effort to mitigate ethnic tensions include strategies like [[w:Community-based_participatory_research|community-based participatory research]] (CBPR), task-sharing, and securing online mental health services in order to expand mental health services. However, the implementation of evidence-based plans has been met with difficulty due to inaccessibility, high costs, and shortages of adequately-trained personnel.  

Movements aiming for improved intra group and inter group coexistences, such as the Jaffna People’s Forum for Coexistence developed in the wake of the 2019 Easter bombings, should be emphasized on a systematic and multi-level basis, including but not limited to education, public sectors, and within communities. Pluralistic values are encouraged to be emphasized across both private and public schools to foster cultural sensitivity and tolerance. Measures should be taken against groups criticized for promoting sectarian hostility, such as the [[w:Bodu_Bala_Sena|Bodu Bala Sena]]. 

=== Poverty ===
It has been proven that poverty significantly increases the chances of developing mental illnesses. This is further amplified by possible discrimination&lt;ref&gt;{{Cite journal|last=Knifton|first=Lee|last2=Inglis|first2=Greig|date=2020-10|title=Poverty and mental health: policy, practice and research implications|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC7525587/|journal=BJPsych bulletin|volume=44|issue=5|pages=193–196|doi=10.1192/bjb.2020.78|issn=2056-4694|pmc=7525587|pmid=32744210}}&lt;/ref&gt;. Poverty also affects the ability for individuals with mental health concerns to receive the treatment they need. Due to the repercussions of the economic crisis, clients in Sri Lanka could not attend further counseling sessions&lt;ref name=&quot;:8&quot; /&gt;. Poverty from 2021 to 2022 [https://databankfiles.worldbank.org/public/ddpext_download/poverty/987B9C90-CB9F-4D93-AE8C-750588BF00QA/current/Global_POVEQ_LKA.pdf reportedly doubled], with future forecasts predicting the poverty line to &quot;remain above 25 percent&quot;. Suicide has been empirically linked to economic hardships in previous studies&lt;ref&gt;{{Cite journal|last=Kithulagoda|first=A. S.|last2=Gunasinghe|first2=U. C. M.|last3=Senevirathna|first3=J. M. M. S.|last4=Nufail|first4=A. L. M.|last5=Alahakoon|first5=A. M. S. S.|date=2025-07-16|title=An Analysis of Attempted Suicide Cases Registered at Teaching Hospital Batticaloa, Sri Lanka|url=https://bmj.sljol.info/articles/10.4038/bmj.v19i1.67|journal=Batticaloa Medical Journal|language=en-US|volume=19|issue=1|doi=10.4038/bmj.v19i1.67|issn=1800-4903}}&lt;/ref&gt;. A 2013 study done on suicidal patients in [[w:Batticaloa_Teaching_Hospital|Batticaloa Teaching Hospital]] revealed 76% of patients who attempted suicide were from rural areas while 15% were from urban areas&lt;ref&gt;{{Cite book|url=http://ir.lib.seu.ac.lk/handle/123456789/1457|title=The influence of common risk factors for the patient with attempted suicide hospitalized at the teaching hospital, Batticaloa|last=Kisokanth|first=G.|last2=Najeem|first2=M. M.|last3=Karunakaran|first3=K. E.|date=2014-08-02|publisher=South Eastern University of Sri Lanka, University Park, Oluvil #32360, Sri Lanka|isbn=978-955-627-053-2|language=en-US}}&lt;/ref&gt;. The Sri Lankan government should consider the economical impacts that poverty has on mental health and implement ways to aid poverty-stricken individuals with mental health concerns.   

=== Stigmas ===
Stigma consists of the &quot;combined effect of prejudice, ignorance and discrimination.&quot;&lt;ref name=&quot;:10&quot;&gt;{{Cite web|url=http://www.researchgate.net/publication/233990797_The_Stigma_of_Mental_Illness_in_Sri_Lanka_The_Perspectives_of_Community_Mental_Health_Workers|title=(PDF) The Stigma of Mental Illness in Sri Lanka: The Perspectives of Community Mental Health Workers|website=ResearchGate|language=en|access-date=2025-07-25}}&lt;/ref&gt;.

A 2012 interview consisting of nine participants (two doctors, three nurses, one occupational therapist, one development worker, and two volunteers) revealed a number of concerning societal viewpoints on individuals with mental health concerns. The interviews revealed that negative judgements were not only levied against the individual with the mental illness, but also the family. Families hid mentally ill family members from the public to avoid &quot;shame&quot; and possible hinderances in marriage proposals. Views that mentally ill individuals were &quot;violent&quot; served as the motivating factor behind socially isolating those with mental illness from their communities. Interviewees mentioned that individuals dealing with mental health challenges would have stones and &quot;derogatory names&quot; launched at them. A lack of community awareness regarding mental health and negative portrayals of mentally ill individuals in media exacerbates stigmatization, though the researchers commented that the media was &quot;improving&quot; in their depiction of mental illness. Beliefs that illnesses are caused by &quot;spirits&quot; can be problematic for individuals dealing with mental health issues and serves as evidence to poor mental health awareness in the country. Mental health workers themselves believed that they were being stigmatized, as mental health is reportedly not taken as seriously as physical health. Despite the intriguing perspectives provided, the small sample size and usage of snow sampling raise questionable concerns regarding the contextualization of the results&lt;ref name=&quot;:10&quot; /&gt;. 

Improving media portrayal of subjects concerning mental health and involving community members in interventions dealing with mental health issues are ways that could destigmatize mental health amongst communities in Sri Lanka. Tying collaborations between allopathic services and traditional healers instead of having these two services work individually could enhance engagement between traditional medicine and Western medicine.

=== Suicide Trends &amp; Risk Factors ===
Suicide is defined as &quot;the act of killing oneself deliberately, initiated and performed by the person concerned in the full knowledge or expectation of its fatal outcome&quot;&lt;ref name=&quot;:11&quot;&gt;{{Cite book|title=The neuroscience of suicidal behavior|last=Heeringen|first=Kees van|date=2018|publisher=Cambridge University Press|isbn=978-1-316-60290-4|series=Cambridge fundamentals of neuroscience in psychology|location=Cambridge, United Kingdom New York, NY, USA Port Melbourne, VIC, Australia New Delhi, India Singapore}}&lt;/ref&gt;. Although Sri Lanka has seen a significant reduction in suicide rates from the mid 1990s due to its banning of extremely toxic pesticide products, suicide and self harm remains a significant issue. The suicide rate per 100,000 people increased from 14.0 in 2019 to [https://www.who.int/srilanka/news/detail/06-09-2024-world-suicide-prevention-day-2024--changing-the-narrative-on-suicide 15.0 in 2022] (according to WHO). On average, 27 males per 100,000 males and 5 females per 100,000 females committed suicide in 2022&lt;ref&gt;{{Cite journal|last=Kithulagoda|first=A. S.|last2=Gunasinghe|first2=U. C. M.|last3=Senevirathna|first3=J. M. M. S.|last4=Nufail|first4=A. L. M.|last5=Alahakoon|first5=A. M. S. S.|date=2025-07-16|title=An Analysis of Attempted Suicide Cases Registered at Teaching Hospital Batticaloa, Sri Lanka|url=https://bmj.sljol.info/articles/10.4038/bmj.v19i1.67|journal=Batticaloa Medical Journal|language=en-US|volume=19|issue=1|doi=10.4038/bmj.v19i1.67|issn=1800-4903}}&lt;/ref&gt;. Hanging appears to be the most used method for suicide for both males and females, with studies revealing a steady increase in recent years&lt;ref name=&quot;:12&quot;&gt;{{Cite journal|last=Bandara|first=Piumee|last2=Wickrama|first2=Prabath|last3=Sivayokan|first3=Sambasivamoorthy|last4=Knipe|first4=Duleeka|last5=Rajapakse|first5=Thilini|date=2024-04-17|title=Reflections on the trends of suicide in Sri Lanka, 1997–2022: The need for continued vigilance|url=https://journals.plos.org/globalpublichealth/article?id=10.1371/journal.pgph.0003054|journal=PLOS Global Public Health|language=en|volume=4|issue=4|pages=e0003054|doi=10.1371/journal.pgph.0003054|issn=2767-3375|pmc=11023397|pmid=38630779}}&lt;/ref&gt;. 

From 2023 to 2024, a group of researchers from the [[w:Eastern_University,_Sri_Lanka|Eastern University in Sri Lanka]] assessed 828 patients admitted to the Teaching Hospital in [[w:Batticaloa,_Sri_Lanka|Batticaloa, Sri Lanka]] for attempted suicide. They concluded that suicide prevention programs should be attuned to younger people (ages 15 to 35 in the study), emphasize the importance of education and reducing unemployment, and increase social support in the Tamil community. Despite the relevant insights into certain aspects of an average Sri Lankan's life that could lead to suicidal ideation (ie, poverty), the results from this study suffer in external validity as 90% of the patients were Tamil and over 50% were between 16 and 25 years. In addition, correlations between suicide and unemployment rates have been questioned, with [[w:Austerity|austerity]] being a more reliable indicator of suicide rates than unemployment rates&lt;ref name=&quot;:11&quot; /&gt;. Further comprehensive studies on risk factors relating to suicide should be studied to assess correlations between unemployment rates and austerity measures.  

The WHO suggests implementing evidence-based suicide prevention programs, such as [https://www.who.int/initiatives/live-life-initiative-for-suicide-prevention LIVE LIFE], to reduce the national suicide rate&lt;ref&gt;{{Cite web|url=https://www.who.int/srilanka/news/detail/06-09-2024-world-suicide-prevention-day-2024--changing-the-narrative-on-suicide|title=World Suicide Prevention day 2024 “Changing the Narrative on Suicide”|website=www.who.int|language=en|access-date=2025-07-29}}&lt;/ref&gt;. Media depictions of suicidal methods, such as hanging, can lead to sensationalism and the media should be cautious of such displays in movies and TV shows&lt;ref name=&quot;:12&quot; /&gt;. Awareness of depression and other mental health issues can serve as a safeguard against suicidal ideation in Sri Lankan men and women.  

== Role of Religion ==
According to the last demographic report (2012), 70.2% of Sri Lankans are Buddhist, 12.6% are Hindus, 9.7% are Muslims, and 7.4% are Christians. The Theravada Buddhist community makes up the majority in several provinces throughout the country&lt;ref&gt;{{Cite web|url=https://www.state.gov/reports/2022-report-on-international-religious-freedom/sri-lanka/|title=Sri Lanka|website=United States Department of State|language=en-US|access-date=2025-08-07}}&lt;/ref&gt;. Religion, especially Theravada Buddhism, has had a significant influence on not only the historical treatment of mental health in the country, but also everyday life&lt;ref name=&quot;:15&quot; /&gt;. The [[w:Mahāvaṃsa|''Mahāvaṃsa'']] affirms hospitals treating patients suffering from mental health issues as early as the 4th century BC. Additionally, the 1700s Nayaka king [[w:Kirti_Sri_Rajasinha|Kirthi Sri Rajasinghe]] detailed the implementation of Buddhist philosophy in psychiatry&lt;ref name=&quot;:4&quot; /&gt;&lt;ref name=&quot;:17&quot;&gt;{{Cite journal|last=Alwis|first=L. A. P. De|date=2017-12-05|title=Development of civil commitment statutes (laws of involuntary detention and treatment) in Sri Lanka: a historical review|url=https://mljsl.sljol.info/articles/10.4038/mljsl.v5i1.7351|journal=Medico-Legal Journal of Sri Lanka|language=en|volume=5|issue=1|doi=10.4038/mljsl.v5i1.7351|issn=2012-8231}}&lt;/ref&gt;.

Modern-day empirical studies have attested to the usefulness of religion in mitigating stress and elevating mental health&lt;ref&gt;{{Cite book|url=https://doi.org/10.1007/978-94-007-4276-5_22|title=Religion and Mental Health|last=Schieman|first=Scott|last2=Bierman|first2=Alex|last3=Ellison|first3=Christopher G.|date=2013|publisher=Springer Netherlands|isbn=978-94-007-4276-5|editor-last=Aneshensel|editor-first=Carol S.|location=Dordrecht|pages=457–478|language=en|doi=10.1007/978-94-007-4276-5_22|editor-last2=Phelan|editor-first2=Jo C.|editor-last3=Bierman|editor-first3=Alex}}&lt;/ref&gt;. Religion has been found to be positively correlated with improved mental health, and more religious patients were concluded to have &quot;better mental health and adapt[ed] more quickly to health problems&quot; versus patients who weren't religious&lt;ref&gt;{{Cite journal|last=Koenig|first=Harold G.|date=2012|title=Religion, spirituality, and health: the research and clinical implications|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3671693/|journal=ISRN psychiatry|volume=2012|pages=278730|doi=10.5402/2012/278730|issn=2090-7966|pmc=3671693|pmid=23762764}}&lt;/ref&gt;. [https://www.researchgate.net/scientific-contributions/T-N-Wickramarathna-2247724082 Dr. Wickramarathna] of the University Psychiatry Unit (UPU) at the National Hospital of Sri Lanka (NHSL) argues that psychiatrists must strive for a balance in their approach to patients and &quot;make positive use of religion in [their] practice[s]&quot;&lt;ref&gt;{{Cite journal|last=Wickramarathna|first=T. N.|date=2022-12-31|title=Psychiatrists should stand far from the shrine: why and why not we should separate religion from psychiatry|url=https://sljpsyc.sljol.info/articles/10.4038/sljpsyc.v13i2.8397|journal=Sri Lanka Journal of Psychiatry|language=en|volume=13|issue=2|doi=10.4038/sljpsyc.v13i2.8397|issn=2012-6883}}&lt;/ref&gt;.

=== Buddhism ===
27 Sinhalese Buddhists from four Buddhist temples were selected for a series of 70-minute interviews and focus group discussions with the aim of learning the Sinhala Buddhist understanding and experience of spiritual well-being and psychological well-being. The interviewees held spiritual wellness to be the &quot;center&quot; of overall wellness, the &quot;precondition for a successful life&quot;&lt;ref name=&quot;:14&quot;&gt;{{Cite journal|last=Udayanga|first=Samitha|date=2021-06-30|title=Cultural understanding of ‘spiritual well-being’ and ‘psychological well-being’ among Sinhalese Buddhists in Sri Lanka|url=https://sljss.sljol.info/articles/10.4038/sljss.v44i1.7990|journal=Sri Lanka Journal of Social Sciences|language=en-US|volume=44|issue=1|doi=10.4038/sljss.v44i1.7990|issn=0258-9710}}&lt;/ref&gt;. Sinhala Buddhists believe that wellness cannot be achieved without spiritual tranquility. The report states that participants emphasized that spirituality &quot;cannot be directly intervened&quot; and can only be seen through &quot;[interactions] with society&quot;&lt;ref name=&quot;:14&quot; /&gt;. Despite the ''athmaya'' (soul) being &quot;unreachable&quot;, it can be &quot;intervened&quot;, or treated, through the actions of the mind and body with society&lt;ref name=&quot;:14&quot; /&gt;. One being &quot;psychologically ill&quot; can affect one's spiritual being, as the participants reported in their interviews, and can be affected through &quot;lifestyle stressors, environmental and socio-cultural causes, non-human related causes and bad-karma in the past lives&quot;&lt;ref name=&quot;:14&quot; /&gt;.  

The researchers concluded that despite Sinhala Buddhists not being able to articulately decipher the discrepancies between psychological well-being and spiritual well-being, they are able to conceptualize and maintain a culturally embedded understanding between the two, serving as reputable evidence of the integration of mental health in Sinhala Buddhist practices. However, it is important to note that these results come from a very small sample size and cannot be generalized to all Sri Lankan Buddhists.   

In addition, a 2009 study found that a belief in karma was correlated with poor health. However, an earlier study found a positive correlation between the reliance on the [[w:Karma_in_Buddhism|Buddhist concept of karma]] and trauma, inferencing Buddhist karma being a prevalent response to trauma&lt;ref&gt;{{Cite journal|last=Levy|first=Becca R.|last2=Slade|first2=Martin D.|last3=Ranasinghe|first3=Padmini|date=2009-03|title=Causal thinking after a tsunami wave: karma beliefs, pessimistic explanatory style and health among Sri Lankan survivors|url=https://pubmed.ncbi.nlm.nih.gov/19229624|journal=Journal of Religion and Health|volume=48|issue=1|pages=38–45|doi=10.1007/s10943-008-9162-5|issn=1573-6571|pmid=19229624}}&lt;/ref&gt;. Overall, the effectiveness of karma as a coping mechanism appears to be conflicted.   

Studies indicate that other practices of Buddhism seem to be utilized by individuals affected by the war. 40% of Sri Lankan Buddhists affected by the 2004 tsunami found the Buddhist ritual ''Bodhipuja'' to be helpful in dealing with traumatic experiences&lt;ref&gt;{{Cite web|url=https://jmvh.org/article/mental-health-and-the-role-of-cultural-and-religious-support-in-the-assistance-of-disabled-veterans-in-sri-lanka/|title=Mental Health and the Role of Cultural and Religious Support in the Assistance of Disabled Veterans in Sri Lanka|website=JMVH|language=en-US|access-date=2025-08-12}}&lt;/ref&gt;.    

=== Catholicism ===
Catholic counseling refers to &quot;a nuanced and holistic mental health care paradigm that intricately weaves together psychological science with the moral, spiritual, and pastoral traditions of the Catholic Church&quot;&lt;ref name=&quot;:13&quot;&gt;Perera, U. [https://www.researchgate.net/profile/Udeshini-Perera/publication/394095042_Catholic_Counselling_in_Sri_Lanka_Integrating_Faith_Psychology_and_Cultural_Healing/links/6889303af8031739e6098c79/Catholic-Counselling-in-Sri-Lanka-Integrating-Faith-Psychology-and-Cultural-Healing.pdf Catholic Counselling in Sri Lanka: Integrating Faith, Psychology, and Cultural Healing]. July 2025.&lt;/ref&gt; and aims to assimilate Catholic theology and evidence-based psychological treatment while including Sri Lankan cultural elements. This is achieved through emphasis on community cohesion and a locally-based understanding of &quot;personhood&quot;&lt;ref name=&quot;:13&quot; /&gt;. 

The origins of Catholic counseling trace back to the introduction of Roman Catholicism to the island in the 1600s, with the focus of the early Sri Lankan Catholic community being on &quot;[[w:Evangelism|evangelization]], education, and sacramental formation&quot;. Demand for counseling services in general increased due to the impacts of the Sri Lankan Civil War, where Catholic organizations (Caritas Sri Lanka, Seth Sarana, Subodhi Integral Centre (Piliyandala), etc.) established several Catholic-based trauma-informed programmes for victims of the Civil War. Programmes use group therapy, forgiveness rituals, and narrative repairs to alleviate war trauma. 

Examples of integration of Catholic virtues and counseling can be seen in [[w:Cognitive_Behavioral_Therapy|Cognitive Behavioral Therapy]] (CBT), where &quot;hope&quot; and &quot;humility&quot; are used as the frameworks for creating spiritual resilience&lt;ref name=&quot;:13&quot; /&gt;. The general Christian call of &quot;agape love and acceptance&quot; is echoed by the concept of [[w:Unconditional_positive_regard|unconditional positive regard]]. ''[[w:Lectio_Divina|Lectio Divina]]'' (Catholic prayer and meditation) and ''Marian devotions'' are integrated into therapeutic practices to achieve emotional regulation and mindfulness.   

Senior Lecturer [https://www.researchgate.net/profile/Udeshini-Perera Udeshini Perera] of the University of Colombo articulates a critical role of Catholic counseling. She claims that secular counseling fails to address the &quot;spiritual roots of distress and moral confusion&quot;. Catholic counseling fills in this gap by integrating &quot;psychological insights with a transcendent orientation, supporting lasting transformation and integrity&quot;&lt;ref name=&quot;:13&quot; /&gt;.  

As of 2025, no formal accreditation or standardized training exists for [[w:Pastoral_counseling|pastoral counselors]] in Sri Lanka, hampering the legitimacy of Catholic counseling. Udeshini Perera remarks that mental health stigma, lack of standardized training, research regarding Catholic counseling effectiveness, and acceptance of the combination of religion and science in a professional setting present challenges for Catholic pastoral counseling in the country. Additionally, Catholic psychiatry in Sri Lanka appears to be under-researched, and evidence of its empirical effects on followers appears sparse. Further research is needed in assessing the empirical effects of Catholic counseling in Sri Lanka.  

=== Islam ===
The literature on the empirical effects of Islamic-based psychotherapy in Sri Lanka is limited. Research has revealed a 2012 case study where a 21-year-old Muslim woman was experiencing episodic possession states. The patient ceased attending psychiatric services and opted for religious rituals. The patient reported, in a follow-up visit, that the possession states had been absent for 3 months since her switch to religious rituals. The woman and her family attributed the apparent improvement of her condition to religious rituals&lt;ref&gt;{{Cite journal|last=Hanwella|first=Raveen|last2=de Silva|first2=Varuni|last3=Yoosuf|first3=Alam|last4=Karunaratne|first4=Sanjeewani|last5=de Silva|first5=Pushpa|date=2012|title=Religious Beliefs, Possession States, and Spirits: Three Case Studies from Sri Lanka|url=http://www.hindawi.com/journals/crips/2012/232740/|journal=Case Reports in Psychiatry|language=en|volume=2012|pages=1–3|doi=10.1155/2012/232740|issn=2090-682X|pmc=3437272|pmid=22970398}}&lt;/ref&gt;.

Future recommendations would be to employ resources to research the foundations of Islamic psychiatry in the country, and to observe the rituals employed and their effects on patients. Studies have found that Islamic prayer can be an effective means of &quot;support and coping&quot;&lt;ref name=&quot;:15&quot; /&gt;. Seven world-wide case studies using Islamic-based psychotherapy on patients, consisting of religious rituals such as scriptural reading from the [[w:Quran|Quran]], teaching of fundamental Islamic concepts (such as ''[[w:Tawakkul|tawakkul]]''), and active implementation of contemplation (''[[w:Tadabbur|tadabbur]]''), have reported positive effects in decreasing cognitive and emotional symptoms associated with &quot;religious, obsessive-compulsive disorder, depression, agoraphobia, generalized anxiety disorder, grief, and substance use disorder.”&lt;ref&gt;{{Cite journal|last=Kurhade|first=Chhaya Shantaram|last2=Jagannathan|first2=Aarti|last3=Varambally|first3=Shivarama|last4=Shivanna|first4=Sushrutha|date=2022-01|title=Religion-based interventions for mental health disorders: A systematic review|url=https://journals.lww.com/10.4103/ijoyppp.ijoyppp_14_21|journal=Journal of Applied Consciousness Studies|language=en|volume=10|issue=1|pages=20–33|doi=10.4103/ijoyppp.ijoyppp_14_21|issn=2949-6993}}&lt;/ref&gt; Additionally, a community-based study of elderly patients in Bangalore, India receiving Islamic-based psychotherapy observed decreased exhibitions of sleep disorders, eating disorders, and emotional distress&lt;ref&gt;{{Cite journal|last=Hafeez|first=Nimin|last2=Sanjay|first2=Thittamaranahalli Varadappa|last3=Puthussery|first3=Yannick Poulose|last4=Madhusudan|first4=Muralidhar|last5=Kariyappa|first5=Poornima Muddaiah|last6=Kulkarni|first6=Sridevi|last7=Raj|first7=Lavanya|date=2023-12-31|title=Spiritual practices among elderly, prevalence, pattern and associated factors: a community-based study from rural Bengaluru, India|url=https://jccpsl.sljol.info/articles/10.4038/jccpsl.v29i4.8610|journal=Journal of the College of Community Physicians of Sri Lanka|language=en|volume=29|issue=4|doi=10.4038/jccpsl.v29i4.8610|issn=1391-3174}}&lt;/ref&gt;. 

=== Hinduism ===
Despite Hindus being 12.6% of the population of Sri Lanka, the research on Hinduism-based therapy in the country is limited. Ayurvedic medicine, a form of medicine originating from ancient India, predominated the Sri Lankan medical landscape for over 2,000 years and even had a symbiotic relationship with Sinhalese medicine, which also played a significant and influential role in the country's medical framework&lt;ref name=&quot;:0&quot; /&gt;&lt;ref&gt;{{Cite journal|last=Udayanga|first=Samitha|date=2021-06-30|title=Cultural understanding of ‘spiritual well-being’ and ‘psychological well-being’ among Sinhalese Buddhists in Sri Lanka|url=https://sljss.sljol.info/article/10.4038/sljss.v44i1.7990/|journal=Sri Lanka Journal of Social Sciences|volume=44|issue=1|pages=33|doi=10.4038/sljss.v44i1.7990|issn=2478-1169}}&lt;/ref&gt;. Despite its historical dominance, Ayurvedic medicine has been challenged against modern evidence-based medical standards&lt;ref&gt;{{Cite book|url=https://philarchive.org/rec/DOMAAT|title=Ayurveda: Ancient Tradition or Pseudoscientific Practice? A Philosophical Inquiry|last=Dominic|first=Shubham K.}}&lt;/ref&gt;.

=== Comparative synthesis ===
Taking an overarching review of the role of religion in Sri Lanka, methods to improve mental well-being are practiced by adherents of Buddhism, Hinduism, Islam, and Christianity. These methods are practiced through karma, tawakkul, hope, and humility. Additionally, these practices are implemented in traditionally-oriented mental health care, which has been reported to be preferred over psychiatric care at times. These rituals practiced across these religions indicate a common theme of psychologically integrated aspects of well-being. Interpretation of trauma is a central use in religion, with religious principles, such as karma and ''tawakkul'', serving as psychologically analogous mechanisms during times of distress.  

In terms of methodological comparisons to the studies described, qualitative interviews have documented Buddhist practices and principles, like Bodhipuja and the belief in karma, in response to traumatic events, while case studies found religious practices by other religious groups, such as a Muslim patient reading Islamic scripture and observing prayer to reduce emotional distress. Peer-reviewed sources have documented Catholic practices and principles, such as ''Lectio Divina'' and unconditional positive regard, in improving mindfulness and emotional regulation. The paper acknowledges limitations in the evaluation of certain findings, such as in Islam and Hinduism. These shortcomings, however, are a reflection of the existing literature and its deficiencies. Empirical findings indicate mental health practices are complex and are multifaceted in their effects. 

Evidently, religion serves a parallel role to psychiatric services in improving mental health. Despite its perceived benefits, the findings surrounding religions' role in mental health suffer from conflicting, and sometimes contradictory, results. Additionally, a disproportionate amount of empirical findings seem to be Buddhist-predominant, while other religions are underrepresented in the research. Regarding research barriers, the methodological approaches implemented to study the practices of religious followers vary, though much of the research was brought from qualitative or case-based studies, impeding generalizability. Another noteworthy issue is that many studies do not utilize standardized, psychiatric measures.

== Future Outlook ==
Despite significant changes to the mental health environment in Sri Lanka, the current legal framework shaping mental health in the country has not been updated since 1956. A Cambridge University Press article detailed many limitations of the Mental Disease Ordinance of 1956, including discrepancies between the legal provisions of involuntary admissions and modern practices, potential exposure to trauma through extra-legal detentions of the mentally ill, and an absence of legal guidelines addressing the restraint of violent patients&lt;ref name=&quot;:6&quot; /&gt;. Participants from Sri Lanka reported in a comparative legislative questionnaire that they felt the mental health laws were &quot;outdated&quot; and descriptions of clinical roles remained ambiguous&lt;ref name=&quot;:16&quot; /&gt;. A draft mental health legislation from 2007 includes provisions for human rights, but due to &quot;bureaucratic processes&quot; and a &quot;lack of consensus&quot;, the draft has not been officially approved.

These limitations pose challenges to the standardization of mental healthcare admissions and may impact the rights of detained patients. Detained patients may have their human rights violated due to a lack of an up-to-date legal framework, thereby impeding the identification of such violations. Additionally, with the lack of clarity on clinical roles, clinical responsibilities may not be routinely recognized and observed, leading to role confusion and potential legal ramifications&lt;ref name=&quot;:16&quot;&gt;{{Cite journal|last=Dey|first=Sangeeta|last2=Mellsop|first2=Graham|last3=Diesfeld|first3=Kate|last4=Dharmawardene|first4=Vajira|last5=Mendis|first5=Susitha|last6=Chaudhuri|first6=Sreemanti|last7=Deb|first7=Aniruddha|last8=Huq|first8=Nafisa|last9=Ahmed|first9=Helal Uddin|date=2019-10-24|title=Comparing legislation for involuntary admission and treatment of mental illness in four South Asian countries|url=https://ijmhs.biomedcentral.com/articles/10.1186/s13033-019-0322-7|journal=International Journal of Mental Health Systems|volume=13|issue=1|pages=67|doi=10.1186/s13033-019-0322-7|issn=1752-4458|pmc=6813093|pmid=31666805}}&lt;/ref&gt;. Lastly, current efforts should increase beyond just addressing poverty-centered matters, but also expand efforts to domestic violence victims and children with disabilities, as shelters and specialized services are limited&lt;ref name=&quot;:82&quot;&gt;{{Cite journal|last=Augustyniak|first=Nadia|date=2025-06-01|title=Public mental healthcare and economic vulnerability in Sri Lanka|url=https://linkinghub.elsevier.com/retrieve/pii/S2666560324000926|journal=SSM - Mental Health|volume=7|pages=100387|doi=10.1016/j.ssmmh.2024.100387|issn=2666-5603}}&lt;/ref&gt;.

Stagnation in policy development leaves Sri Lanka without a practical, up-to-date, and comprehensive mental health legislation, which could put both clinicians and patients at risk. Future reforms should include clarification on the treatment and detention process of involuntary admissions of patients and a clear delineation of clinical roles and their responsibilities. Without the necessary reforms to advance Sri Lankan mental health legislation, clinicians and vulnerable patients may suffer from a lack of comprehensive oversight. 

==Additional information==

===Acknowledgements===
Any people, organisations, or funding sources that you would like to thank.

===Competing interests===
No competing interests.

===Ethics statement===
An ethics statement, if appropriate, on any animal or human research performed should be included here or in the methods section.

==References==
{{reflist|35em}}

[[Category:Mental health]]
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* Cortés Velásquez, D., Strasser, M. et al. (2025b). ''Project Promoting Plurilingual Education (PEP) -KA220-HED- E96C9232 Survey Report. Language use in secondary and higher education : Teachers’ beliefs and practices''. PEP – Promoting Plurilingual Education. [https://doi.org/10.25592/uhhfdm.16755 https://doi.org/10.25592/uhhfdm.16755]
* Cortés Velásquez, D., Strasser, M. et al. (2025c). ''Sprachgebrauch in der Sekundar- und Hochschulbildung : Überzeugungen und Praktiken von Lehrkräften''. PEP – Promoting Plurilingual Education. [https://doi.org/10.25592/uhhfdm.16758 https://doi.org/10.25592/uhhfdm.16758 ]
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*[https://sites.google.com/view/pepproject/productions/livret-de-bonnes-pratiques-good-practices-booklet Booklet of adaptable plurilingual practices]
*[https://www.ecml.at/en/ECML-Programme/Programme-2020-2023/Mediation-in-teaching-and-assessment METLA - Mediation in teaching, learning and assessment]
*[https://www.coe.int/en/web/language-policy/plurilingualism CEFR and Plurilingualism]
*[https://carap.ecml.at/ CARAP/FREPA]
}}&lt;!------------------------
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* [[Portal:TESOL|Teaching English to speakers of other languages (TESOL)]]
* [[Portal:Translation|Translation]]
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       FEATURED CONTENT
-----------------------&gt;
{{Frame alt
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* [[Awakening to languages]]
* [[CLIL (Content and Language Integrated Learning)]]
* [[Decolonial perspective in plurilingual education]]
* [[Plurilingual education and digital technologies|Digital technologies in plurilingual education]]
* [[Deaf and hard of hearing people and mulitlingual education]]
* [[Dominant language constellation]]
* [[English as a Lingua Franca (ELF)]]
* [[Endangered languages and plurilingual education]]
* [[Heritage Language|Heritage language]]
* [[Intercomprehension]]
* [[Language biography and identity texts]]
* [[Language inclusion]]
* [[Language mediation]]
* [[Language policies: Educational and family language policies]]
* [[Language Portfolio|Language portfolio]]
* [[Linguistic landscapes in education]]
* [[Migrants, bilingualism &amp; parental involvement]]
* [[Multilingual awareness - Language awareness - Metacompetencies]]
* [[Multulingual turn]]
* [[Native language(s), L1, family language, border language(s)... and more!]]
* [[Native speakerism]]
* [[Non-formal and informal plurilingual education]]
* [[Pluralistic approach]]
* [[Plurilingualism in marginalized contexts]]
* [[Pluringualism in the CEFR]]
* [[Assessing the plurilingual competence|Plurilingual assessment - Assessing the plurilingual competence]]
* [[Assessment of the knowledge and competences of plurilingual learners|Plurilingual assessment - Assessment of the knowledge and competences of plurilingual learners]]
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* [[Plurilingual and inter/transcultural competence]]
* [[Plurilingualism and plurilingual education in the past]]
* [[Telecollaboration and plurilingualism]]
* [[Tertiary language teaching]]
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{{Frame alt
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* Cortés Velásquez, D., Strasser, M. et al. (2025a). ''L’utilisation des langues dans l’enseignement secondaire et supérieur : Croyances et pratiques des enseignants''. PEP – Promoting Plurilingual Education. [https://www.fdr.uni-hamburg.de/record/16757 https://www.fdr.uni-hamburg.de/record/16757]
* Cortés Velásquez, D., Strasser, M. et al. (2025b). ''Project Promoting Plurilingual Education (PEP) -KA220-HED- E96C9232 Survey Report. Language use in secondary and higher education : Teachers’ beliefs and practices''. PEP – Promoting Plurilingual Education. [https://doi.org/10.25592/uhhfdm.16755 https://doi.org/10.25592/uhhfdm.16755]
* Cortés Velásquez, D., Strasser, M. et al. (2025c). ''Sprachgebrauch in der Sekundar- und Hochschulbildung : Überzeugungen und Praktiken von Lehrkräften''. PEP – Promoting Plurilingual Education. [https://doi.org/10.25592/uhhfdm.16758 https://doi.org/10.25592/uhhfdm.16758 ]
}}

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{{Frame alt
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* '''[https://sites.google.com/view/pep-conference Conference - Bridging Voices in Plurilingual Education: Policies, Research and Practices]''', 23-24 october 2025, Rom. The conference was organised by Università degli Studi Roma Tre within the framework of the PEP project (Promoting plurilingual education, 2023-1-FR01-KA220-HED-000160820).
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*[https://sites.google.com/view/pepproject/productions/livret-de-bonnes-pratiques-good-practices-booklet Booklet of adaptable plurilingual practices]
*[https://www.ecml.at/en/ECML-Programme/Programme-2020-2023/Mediation-in-teaching-and-assessment METLA - Mediation in teaching, learning and assessment]
*[https://www.coe.int/en/web/language-policy/plurilingualism CEFR and Plurilingualism]
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* [[Portal:TESOL|Teaching English to speakers of other languages (TESOL)]]
* [[Portal:Translation|Translation]]
'''In the French Wikiversité''' 
*[https://fr.wikiversity.org/wiki/D%C3%A9partement:Didactique_des_langues Department of plurilingual education in the French Wikiversité]
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{{Education}}{{Course}}

== Starting Activity ==
[[File:Translanguaging in Paris.png|alt=Translanguaging in Paris|thumb|Figure 1 - Translanguaging in Paris]]
Imagine you are observing a classroom where students from different linguistic and cultural backgrounds interact.

* How do they switch between languages?
* Do they use different languages for different purposes (e.g. socializing, academic tasks)?
* How would you describe their language use: do they mix languages or use them separately? 

Look at the following examples (see Figure 1 and Figure 2): Signs where multiple languages are used (see Linguistic Landscapes in Education).

Now think about the questions below: 

* How are these languages being used and why?
* Try to think of monolingual versions of the examples. How would the communication/messages be different?[[File:Languages in Paris.png|alt=Languages in Paris|thumb|Figure 2 - Languages in Paris]]

== Objectives ==
At the end of this section, you will be able to:

* Recognize different theoretical perspectives on plurilingualism.
* Understand the implications of a translanguaging approach to language use.
* Develop a critical awareness of how plurilingualism functions in everyday communication and education.

== Keywords ==
Plurilingualism, linguistic repertoire, bilingualism, multilingualism, translanguaging, language practices, sociolinguistics, education.

== Table of Contents ==

# Introduction
# History of the Concept
# Definitions and Theoretical Perspectives
# Practical Examples
# Take-Home Messages
# Self-Assessment
# Resources to Go Further
# Bibliography

== Introduction ==
The study of plurilingualism and its implications for education has evolved significantly over the past decades. Traditionally, language use was conceptualized through monolingual ideologies, where each language was seen as a separate, countable entity. However, more recent theories, such as translanguaging, argue for a more fluid and dynamic vision of language use.

Within this evolving field, the concept of translanguaging has become particularly influential. Rather than viewing multilingual speakers as alternating between separate language systems, translanguaging perspectives emphasize that individuals draw flexibly on their entire linguistic and semiotic repertoire to communicate and make meaning. At the same time, this perspective has generated considerable debate. While some scholars argue that multilingual language use is best understood as a unitary system, others maintain that socially recognized (named) languages continue to have theoretical, cognitive, and pedagogical relevance. This debate is reflected in the distinction between the Unitary Translanguaging Theory (UTT) and the Crosslinguistic Translanguaging Theory (CTT) (Cummins, 2021).

Plurilingualism therefore refers not only to the knowledge of multiple languages, but also to the ability to mobilize and integrate diverse linguistic resources according to communicative purposes and social contexts. Understanding the different theoretical perspectives on plurilingualism is important because they influence how multilingual competence is conceptualized and how languages are taught, learned, and valued in educational settings.

== History of the Concept ==
The concept of plurilingualism has developed alongside changing perspectives on bilingualism, multilingualism, and language itself. Over time, researchers have moved from viewing languages as separate, autonomous systems to exploring more dynamic and integrated models of multilingual competence. However, these developments have also generated theoretical debate, and different perspectives continue to coexist.

=== Early views (1950s–1980s) ===
Early research on bilingualism generally conceptualized languages as separate, stable systems. Bilingual speakers were often assumed to alternate between distinct languages, each functioning independently. For example, Penfield and Roberts (1959) described bilingual speakers as having an &quot;automatic switch&quot; between languages, reflecting the view that languages were stored and accessed separately.

=== Holistic perspectives (1980s–1990s) ===
From the 1980s onwards, researchers began challenging strictly monolingual conceptions of bilingualism. Grosjean (1982, 1989) famously argued that &quot;the bilingual is not two monolinguals in one person,&quot; emphasizing that bilingual competence should be understood as a unique and integrated phenomenon. Similarly, Coste, Moore, and Zarate (1997) described plurilingual competence as a complex and composite repertoire rather than the simple addition of separate languages.

=== Dynamic approaches (2002) ===
Building on these holistic perspectives, Herdina and Jessner's Dynamic Model of Multilingualism (2002) proposed that multilingual competence is constantly evolving through interactions among languages, cognitive processes, and environmental influences. Their work highlighted that multilingual systems are dynamic rather than static.

=== Critical perspectives (2007–2009) ===
Critical sociolinguists further questioned the status of languages as natural, bounded entities. Makoni and Pennycook (2007) argued that named languages should be understood as social and political constructs rather than pre-existing objects. García (2009) extended this perspective by proposing that multilingual speakers draw on an integrated linguistic repertoire instead of operating separate language systems.

=== Translanguaging theories (2009–present) ===
Building on these developments, García (2009) reconceptualized the notion of translanguaging, a term originally introduced by Cen Williams in Wales to describe bilingual pedagogical practices. While Williams' conception referred primarily to alternating languages for learning activities (e.g., reading in one language and discussing or writing in another), García broadened the concept into a theoretical framework that views multilingual speakers as drawing flexibly on an integrated linguistic repertoire. This broader interpretation has become highly influential in multilingual education, while also giving rise to considerable theoretical debate.

More recently, however, scholars have debated how multilingual competence should be understood theoretically. Cummins (2021) distinguishes between two major perspectives. The Unitary Translanguaging Theory (UTT) proposes that multilinguals draw on a single integrated linguistic system and questions the cognitive reality of separate languages. In contrast, the Crosslinguistic Translanguaging Theory (CTT) accepts that languages are socially constructed rather than natural entities but argues that named languages remain meaningful categories for understanding multilingual competence, crosslinguistic transfer, and educational practice.

Consequently, contemporary research no longer focuses on whether multilingual speakers use all their linguistic resources—they clearly do—but rather on how these resources should be conceptualized. While UTT emphasizes a unitary linguistic repertoire, CTT maintains that socially recognized languages continue to play an important role in cognition, communication, and education. This ongoing debate has important implications for language teaching, curriculum design, and multilingual pedagogy.

== Definitions and Theoretical Perspectives ==

* '''Plurilingualism''': “the dynamic and developing linguistic repertoire of an individual user/learner” (Council of Europe, 2020, p. 30)

* '''Translanguaging''': A perspective that sees language use as fluid, where speakers move across linguistic boundaries without adhering to rigid categorizations.

According to Otheguy, García, and Reid (2015, p. 281), translanguaging refers to the deployment of multilingual speakers' full linguistic, semiotic, and multimodal repertoire &quot;without regard for watchful adherence to the socially and politically defined boundaries of named languages.&quot; 

Canagarajah (2011) describes translanguaging as the strategic use of all available linguistic resources to communicate effectively. This perspective emphasizes that multilingual speakers select linguistic forms according to their communicative needs and contexts, challenging the traditional view of languages as separate, autonomous systems. 

Similarly, Li Wei (2011) argues that multilingual speakers create translanguaging spaces, in which they draw on their full linguistic repertoires to make meaning, communicate, and construct knowledge. 

Although translanguaging has become an influential framework, its theoretical assumptions remain the subject of scholarly debate. Critics argue that the unitary conception of multilingual competence may underestimate the continuing relevance of named languages as socially recognized constructs, particularly in education, language policy, and language assessment (Cummins, 2021; Treffers-Daller, 2024).

* '''Linguistic Repertoire''': The full range of linguistic and semiotic resources an individual has access to, regardless of conventional language boundaries.

A linguistic repertoire includes resources acquired through different experiences, such as languages learned at home, in school, or independently, and speakers may have varying levels of proficiency in each. These resources can serve different communicative functions—for example, in family interactions, education, work, or identity construction—and are often used in combination. Rather than being fixed, linguistic repertoires are dynamic and continually evolve as individuals encounter new communicative situations. From this perspective, language learning is understood as a process of extending, reorganizing, and enriching one's repertoire. 

More recently, Piccardo (2017) has proposed the concept of plurilanguaging, extending the notion of languaging to emphasize the dynamic, agentive, and emergent nature of plurilingual communication and learning. Drawing on Complex Dynamic Systems Theory, she argues that language learning is not a linear process of acquiring separate languages but a recursive process in which learners actively construct meaning through mediation, awareness, and the flexible use of linguistic and semiotic resources. In this perspective, plurilanguaging can be understood as the operationalization of plurilingualism in educational contexts. 

* '''Unitary Translanguaging Theory (UTT) (Cummins, 2021):''' UTT proposes that multilingual speakers draw on a single, integrated linguistic system rather than on separate language systems. Consequently, named languages are viewed primarily as social constructs rather than as distinct cognitive entities.

* '''Crosslinguistic Translanguaging Theory (CTT) (Cummins, 2021):''' CTT recognizes the flexible use of multilingual speakers' full linguistic repertoire while maintaining that named languages remain meaningful categories for understanding multilingual competence, crosslinguistic transfer, and language education.

According to Cummins (2021), the principal difference between these perspectives lies in how they conceptualize the status of named languages. UTT argues that multilinguals operate through a unitary linguistic system, whereas CTT accepts that, although languages are socially constructed, they remain useful theoretical and pedagogical constructs. CTT therefore emphasizes the importance of crosslinguistic relationships, transfer, additive bilingualism, and explicit connections between languages in educational contexts.  

Both theories recognize that languages are shaped by social contexts rather than being fixed, natural entities, and both challenge traditional assumptions that languages should always be kept strictly separate in educational settings. They also agree that multilingual speakers flexibly mobilize their available linguistic resources according to communicative needs. However, they differ in the extent to which they attribute theoretical, cognitive, and pedagogical significance to named languages. 

== Practical Examples ==

=== Example 1: Linguistic Landscapes ===
[[File:Multilingual signs, fingerposts in Brisbane, Australia 02.jpg|alt=Multilingual signs, fingerposts in Brisbane, Australia|thumb|Figure 3 - Multilingual signs, fingerposts in Brisbane, Australia]]
Multilingual signs in public spaces demonstrate how languages coexist and interact. A street sign in Brussels may include French, Dutch, and English, reflecting social and political dimensions of plurilingualism.

=== Example 2: Classroom Translanguaging Practices ===
Students in a bilingual classroom use English for academic tasks but switch to their home language for peer discussions. This challenges traditional monolingual teaching models.

An engaging activity to promote the use of one's entire linguistic repertoire could be a plurilingual debate. Students are given a topic and asked to prepare several arguments. During the debate, they present their points while intentionally switching between the different languages they can use. 

=== Example 3: Crosslinguistic Transfer in Writing ===
One possible activity for exploiting similarities between languages is described in Kursiša &amp; Richter-Vapaatalo (2018, p. 63). Finnish learners of German compare the typical structure and expressions used in informal emails in German, English, Swedish and Finnish, identifying similarities and differences.

Link: “Mehr als Deutsch”: https://www.goethe.de/prj/dlp/de/unterrichtsmaterial/mehr_als_deutsch 

=== Example 4: Plurilingual Digital Communication ===
WhatsApp messages mixing multiple languages, emojis, and voice recordings showcase how plurilingualism is naturally integrated into daily interactions.

== Take-Home Messages ==

* Plurilingualism refers to the dynamic ability of individuals to draw on and develop their linguistic repertoire for communication and learning across different contexts. 
* Translanguaging emphasizes the flexible use of multilingual speakers' linguistic resources and challenges traditional views of languages as strictly separate systems. 
* Unitary Translanguaging Theory (UTT) and Crosslinguistic Translanguaging Theory (CTT) offer different explanations of how multilingual competence is organized. While both recognize the flexible use of linguistic resources, they differ in the role they attribute to named languages. 
* Both theories agree that languages are shaped by social contexts rather than being fixed entities and question rigid language separation in education. However, there is ongoing debate about the cognitive and pedagogical status of named languages and about the implications of translanguaging for language teaching and learning.

== Self-Assessment ==
&lt;quiz display=simple&gt;

{Which statement best aligns with Unitary Translanguaging Theory (UTT)?}
-A) Bilinguals have an automatic switch that separates languages in their minds.
+B) Multilingual speakers access their linguistic resources as part of a single cognitive system.
-C) Languages should be taught separately to avoid interference.
-D) Translanguaging only happens in informal settings.

{What is the main critique of traditional bilingual models according to translanguaging theories?}
+A) They assume bilinguals speak two completely separate languages.
-B) They support language diversity in education.
-C) They promote crosslinguistic mediation.
-D) They encourage translanguaging in the classroom.

{Which statement best aligns with CTT?}
-A) CTT rejects the idea of a unitary linguistic system. 
+B) CTT advocates for maintaining the concept of specific languages while also supporting additive bilingualism and the transfer of academic skills across languages.
-C) CTT criticizes switching between languages because it assumes the existence of two separate linguistic systems.
-D) CTT believes that languages do not exist as real entities.

&lt;/quiz&gt;

== Resources to go further ==

* '''CEFR and Plurilingualism''': https://www.coe.int/en/web/language-policy/plurilingualism
* '''CARAP/FREPA:''' https://carap.ecml.at/
* '''Research on Translanguaging''': https://www.tandfonline.com/doi/full/10.1080/14790718.2017.1400501
* '''CUNY-NYSIEB Translanguaging Resources''': The City University of New York's New York State Initiative on Emergent Bilinguals offers a comprehensive collection of materials, including articles, videos, and classroom strategies, to support the implementation of translanguaging in educational settings. https://www.cuny-nysieb.org/translanguaging-resources

== Bibliography ==

* Canagarajah, S. (2011). Translanguaging in the classroom: Emerging issues for research and pedagogy. ''International Journal of Bilingual Education and Bilingualism, 14''(3), 271–283.''Relations.'' Routledge. 
* Canagarajah, S. (2013). ''Translingual Practice: Global Englishes and Cosmopolitan'' 
* Cavallaro, C. J., &amp; Sembiante, S. F. (2020). Facilitating culturally sustaining, functional literacy practices in a middle school ESOL reading program: a design-based research study. ''Language and Education'', ''35''(2), 160–179. https://doi.org/10.1080/09500782.2020.1775244
* Cummins, J. (2021). Translanguaging: A critical analysis of theoretical claims. In P. Juvonen &amp; M. Källkvist (Eds.), ''Pedagogical Translanguaging: Theoretical, Methodological and Empirical Perspectives'' (pp. 7–36). Multilingual Matters.  https://doi.org/10.21832/9781788927383
* García, O. (2009). ''Bilingual education in the 21st century: A global perspective.'' Wiley-Blackwell, München.
* García, O., &amp; Lin, A. (2017). ''Translanguaging in Bilingual Education''. Springer.
* Kasula, A. J. (2016). Olowalu Review: Developing identity through translanguaging in a multilingual literary magazine. ''Colomb. Appl. Linguist. J., 18''(2), 109–118. 
* Kursiša, A., &amp; Richter-Vapaatalo, U. (Eds.). (2018). ''Mehr als Deutsch!'' Goethe-Institut Finnland. https://www.goethe.de/prj/dlp/de/unterrichtsmaterial/mehr_als_deutsch
* Makoni, S., &amp; Pennycook, A. (2007). ''Disinventing and Reconstituting Languages.'' Multilingual Matters.
* Otheguy, R., García, O., &amp; Reid, W. (2015). Clarifying Translanguaging and Deconstructing Named Languages. ''Applied Linguistics Review, 6(3),'' 281–307.
* Otheguy, R., García, O., &amp; Reid, W. (2015). Translanguaging and the role of language in social identity. ''International Journal of Bilingual Education and Bilingualism, 18''(3), 281–297.
* Penfield, W., &amp; Roberts, L. (1959). S''peech and Brain Mechanisms''. Princeton University Press.
* Piccardo, E. (2017). ''Plurilingualism as a catalyst for creativity in superdiverse societies: A systemic analysis''. ''Frontiers in Psychology, 8'', Article 2169. https://doi.org/10.3389/fpsyg.2017.02169
* Treffers-Daller, J. (2024). ''Unravelling translanguaging: A critical appraisal''. ''ELT Journal, 78''(1), 64–71. https://doi.org/10.1093/elt/ccad058 

==Credits==
This resource has been created by [[User:Projet PEP|Projet PEP]] ([[User talk:Projet PEP|discuss]] • [[Special:Contributions/Projet PEP|contribs]]) (Erasmus+ project, co-financed by the European Commission) :

* Diego Cortés Velásquez (Università Roma Tre) 
* Anastasia Gkaintartzi (University of Thessaly)

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== Starting Activity ==
[[File:Translanguaging in Paris.png|alt=Translanguaging in Paris|thumb|Figure 1 - Translanguaging in Paris]]
Imagine you are observing a classroom where students from different linguistic and cultural backgrounds interact.

* How do they switch between languages?
* Do they use different languages for different purposes (e.g. socializing, academic tasks)?
* How would you describe their language use: do they mix languages or use them separately? 

Look at the following examples (see Figure 1 and Figure 2): Signs where multiple languages are used (see Linguistic Landscapes in Education).

Now think about the questions below: 

* How are these languages being used and why?
* Try to think of monolingual versions of the examples. How would the communication/messages be different?[[File:Languages in Paris.png|alt=Languages in Paris|thumb|Figure 2 - Languages in Paris]]

== Objectives ==
At the end of this section, you will be able to:

* Recognize different theoretical perspectives on plurilingualism.
* Understand the implications of a translanguaging approach to language use.
* Develop a critical awareness of how plurilingualism functions in everyday communication and education.

== Keywords ==
Plurilingualism, linguistic repertoire, bilingualism, multilingualism, translanguaging, language practices, sociolinguistics, education.

== Table of Contents ==

# Introduction
# History of the Concept
# Definitions and Theoretical Perspectives
# Practical Examples
# Take-Home Messages
# Self-Assessment
# Resources to Go Further
# Bibliography

== Introduction ==
The study of plurilingualism and its implications for education has evolved significantly over the past decades. Traditionally, language use was conceptualized through monolingual ideologies, where each language was seen as a separate, countable entity. However, more recent theories, such as translanguaging, argue for a more fluid and dynamic vision of language use.

Within this evolving field, the concept of translanguaging has become particularly influential. Rather than viewing multilingual speakers as alternating between separate language systems, translanguaging perspectives emphasize that individuals draw flexibly on their entire linguistic and semiotic repertoire to communicate and make meaning. At the same time, this perspective has generated considerable debate. While some scholars argue that multilingual language use is best understood as a unitary system, others maintain that socially recognized (named) languages continue to have theoretical, cognitive, and pedagogical relevance. This debate is reflected in the distinction between the Unitary Translanguaging Theory (UTT) and the Crosslinguistic Translanguaging Theory (CTT) (Cummins, 2021).

Plurilingualism therefore refers not only to the knowledge of multiple languages, but also to the ability to mobilize and integrate diverse linguistic resources according to communicative purposes and social contexts. Understanding the different theoretical perspectives on plurilingualism is important because they influence how multilingual competence is conceptualized and how languages are taught, learned, and valued in educational settings.

== History of the Concept ==
The concept of plurilingualism has developed alongside changing perspectives on bilingualism, multilingualism, and language itself. Over time, researchers have moved from viewing languages as separate, autonomous systems to exploring more dynamic and integrated models of multilingual competence. However, these developments have also generated theoretical debate, and different perspectives continue to coexist.

=== Early views (1950s–1980s) ===
Early research on bilingualism generally conceptualized languages as separate, stable systems. Bilingual speakers were often assumed to alternate between distinct languages, each functioning independently. For example, Penfield and Roberts (1959) described bilingual speakers as having an &quot;automatic switch&quot; between languages, reflecting the view that languages were stored and accessed separately.

=== Holistic perspectives (1980s–1990s) ===
From the 1980s onwards, researchers began challenging strictly monolingual conceptions of bilingualism. Grosjean (1982, 1989) famously argued that &quot;the bilingual is not two monolinguals in one person,&quot; emphasizing that bilingual competence should be understood as a unique and integrated phenomenon. Similarly, Coste, Moore, and Zarate (1997) described plurilingual competence as a complex and composite repertoire rather than the simple addition of separate languages.

=== Dynamic approaches (2002) ===
Building on these holistic perspectives, Herdina and Jessner's Dynamic Model of Multilingualism (2002) proposed that multilingual competence is constantly evolving through interactions among languages, cognitive processes, and environmental influences. Their work highlighted that multilingual systems are dynamic rather than static.

=== Critical perspectives (2007–2009) ===
Critical sociolinguists further questioned the status of languages as natural, bounded entities. Makoni and Pennycook (2007) argued that named languages should be understood as social and political constructs rather than pre-existing objects. García (2009) extended this perspective by proposing that multilingual speakers draw on an integrated linguistic repertoire instead of operating separate language systems.

=== Translanguaging theories (2009–present) ===
Building on these developments, García (2009) reconceptualized the notion of translanguaging, a term originally introduced by Cen Williams in Wales to describe bilingual pedagogical practices. While Williams' conception referred primarily to alternating languages for learning activities (e.g., reading in one language and discussing or writing in another), García broadened the concept into a theoretical framework that views multilingual speakers as drawing flexibly on an integrated linguistic repertoire. This broader interpretation has become highly influential in multilingual education, while also giving rise to considerable theoretical debate.

More recently, however, scholars have debated how multilingual competence should be understood theoretically. Cummins (2021) distinguishes between two major perspectives. The Unitary Translanguaging Theory (UTT) proposes that multilinguals draw on a single integrated linguistic system and questions the cognitive reality of separate languages. In contrast, the Crosslinguistic Translanguaging Theory (CTT) accepts that languages are socially constructed rather than natural entities but argues that named languages remain meaningful categories for understanding multilingual competence, crosslinguistic transfer, and educational practice.

Consequently, contemporary research no longer focuses on whether multilingual speakers use all their linguistic resources—they clearly do—but rather on how these resources should be conceptualized. While UTT emphasizes a unitary linguistic repertoire, CTT maintains that socially recognized languages continue to play an important role in cognition, communication, and education. This ongoing debate has important implications for language teaching, curriculum design, and multilingual pedagogy.

== Definitions and Theoretical Perspectives ==

* '''Plurilingualism''': “the dynamic and developing linguistic repertoire of an individual user/learner” (Council of Europe, 2020, p. 30)

* '''Translanguaging''': A perspective that sees language use as fluid, where speakers move across linguistic boundaries without adhering to rigid categorizations.

According to Otheguy, García, and Reid (2015, p. 281), translanguaging refers to the deployment of multilingual speakers' full linguistic, semiotic, and multimodal repertoire &quot;without regard for watchful adherence to the socially and politically defined boundaries of named languages.&quot; 

Canagarajah (2011) describes translanguaging as the strategic use of all available linguistic resources to communicate effectively. This perspective emphasizes that multilingual speakers select linguistic forms according to their communicative needs and contexts, challenging the traditional view of languages as separate, autonomous systems. 

Similarly, Li Wei (2011) argues that multilingual speakers create translanguaging spaces, in which they draw on their full linguistic repertoires to make meaning, communicate, and construct knowledge. 

Although translanguaging has become an influential framework, its theoretical assumptions remain the subject of scholarly debate. Critics argue that the unitary conception of multilingual competence may underestimate the continuing relevance of named languages as socially recognized constructs, particularly in education, language policy, and language assessment (Cummins, 2021; Treffers-Daller, 2024).

* '''Linguistic Repertoire''': The full range of linguistic and semiotic resources an individual has access to, regardless of conventional language boundaries.

A linguistic repertoire includes resources acquired through different experiences, such as languages learned at home, in school, or independently, and speakers may have varying levels of proficiency in each. These resources can serve different communicative functions—for example, in family interactions, education, work, or identity construction—and are often used in combination. Rather than being fixed, linguistic repertoires are dynamic and continually evolve as individuals encounter new communicative situations. From this perspective, language learning is understood as a process of extending, reorganizing, and enriching one's repertoire. 

More recently, Piccardo (2017) has proposed the concept of plurilanguaging, extending the notion of languaging to emphasize the dynamic, agentive, and emergent nature of plurilingual communication and learning. Drawing on Complex Dynamic Systems Theory, she argues that language learning is not a linear process of acquiring separate languages but a recursive process in which learners actively construct meaning through mediation, awareness, and the flexible use of linguistic and semiotic resources. In this perspective, plurilanguaging can be understood as the operationalization of plurilingualism in educational contexts. 

* '''Unitary Translanguaging Theory (UTT) (Cummins, 2021):''' UTT proposes that multilingual speakers draw on a single, integrated linguistic system rather than on separate language systems. Consequently, named languages are viewed primarily as social constructs rather than as distinct cognitive entities.

* '''Crosslinguistic Translanguaging Theory (CTT) (Cummins, 2021):''' CTT recognizes the flexible use of multilingual speakers' full linguistic repertoire while maintaining that named languages remain meaningful categories for understanding multilingual competence, crosslinguistic transfer, and language education.

According to Cummins (2021), the principal difference between these perspectives lies in how they conceptualize the status of named languages. UTT argues that multilinguals operate through a unitary linguistic system, whereas CTT accepts that, although languages are socially constructed, they remain useful theoretical and pedagogical constructs. CTT therefore emphasizes the importance of crosslinguistic relationships, transfer, additive bilingualism, and explicit connections between languages in educational contexts.  

Both theories recognize that languages are shaped by social contexts rather than being fixed, natural entities, and both challenge traditional assumptions that languages should always be kept strictly separate in educational settings. They also agree that multilingual speakers flexibly mobilize their available linguistic resources according to communicative needs. However, they differ in the extent to which they attribute theoretical, cognitive, and pedagogical significance to named languages. 

== Practical Examples ==

=== Example 1: Linguistic Landscapes ===
[[File:Multilingual signs, fingerposts in Brisbane, Australia 02.jpg|alt=Multilingual signs, fingerposts in Brisbane, Australia|thumb|Figure 3 - Multilingual signs, fingerposts in Brisbane, Australia]]
Multilingual signs in public spaces demonstrate how languages coexist and interact. A street sign in Brussels may include French, Dutch, and English, reflecting social and political dimensions of plurilingualism.

=== Example 2: Classroom Translanguaging Practices ===
Students in a bilingual classroom use English for academic tasks but switch to their home language for peer discussions. This challenges traditional monolingual teaching models.

An engaging activity to promote the use of one's entire linguistic repertoire could be a plurilingual debate. Students are given a topic and asked to prepare several arguments. During the debate, they present their points while intentionally switching between the different languages they can use. 

=== Example 3: Crosslinguistic Transfer in Writing ===
One possible activity for exploiting similarities between languages is described in Kursiša &amp; Richter-Vapaatalo (2018, p. 63). Finnish learners of German compare the typical structure and expressions used in informal emails in German, English, Swedish and Finnish, identifying similarities and differences.

Link: “Mehr als Deutsch”: https://www.goethe.de/prj/dlp/de/unterrichtsmaterial/mehr_als_deutsch 

=== Example 4: Plurilingual Digital Communication ===
WhatsApp messages mixing multiple languages, emojis, and voice recordings showcase how plurilingualism is naturally integrated into daily interactions.

== Take-Home Messages ==

* Plurilingualism refers to the dynamic ability of individuals to draw on and develop their linguistic repertoire for communication and learning across different contexts. 
* Translanguaging emphasizes the flexible use of multilingual speakers' linguistic resources and challenges traditional views of languages as strictly separate systems. 
* Unitary Translanguaging Theory (UTT) and Crosslinguistic Translanguaging Theory (CTT) offer different explanations of how multilingual competence is organized. While both recognize the flexible use of linguistic resources, they differ in the role they attribute to named languages. 
* Both theories agree that languages are shaped by social contexts rather than being fixed entities and question rigid language separation in education. However, there is ongoing debate about the cognitive and pedagogical status of named languages and about the implications of translanguaging for language teaching and learning.

== Self-Assessment ==
&lt;quiz display=&quot;simple&quot;&gt;

{Which statement best aligns with Unitary Translanguaging Theory (UTT)?}
-A) Bilinguals have an automatic switch that separates languages in their minds.
+B) Multilingual speakers access their linguistic resources as part of a single cognitive system.
-C) Languages should be taught separately to avoid interference.
-D) Translanguaging only happens in informal settings.

{What is the main critique of traditional bilingual models according to translanguaging theories?}
+A) They assume bilinguals speak two completely separate languages.
-B) They support language diversity in education.
-C) They promote crosslinguistic mediation.
-D) They encourage translanguaging in the classroom.

{Which statement best aligns with CTT?}
-A) CTT rejects the idea of a unitary linguistic system. 
+B) CTT advocates for maintaining the concept of specific languages while also supporting additive bilingualism and the transfer of academic skills across languages.
-C) CTT criticizes switching between languages because it assumes the existence of two separate linguistic systems.
-D) CTT argues that languages do not exist as real entities.

&lt;/quiz&gt;

== Resources to go further ==

* '''CEFR and Plurilingualism''': https://www.coe.int/en/web/language-policy/plurilingualism
* '''CARAP/FREPA:''' https://carap.ecml.at/
* '''Research on Translanguaging''': https://www.tandfonline.com/doi/full/10.1080/14790718.2017.1400501
* '''CUNY-NYSIEB Translanguaging Resources''': The City University of New York's New York State Initiative on Emergent Bilinguals offers a comprehensive collection of materials, including articles, videos, and classroom strategies, to support the implementation of translanguaging in educational settings. https://www.cuny-nysieb.org/translanguaging-resources

== Bibliography ==

* Canagarajah, S. (2011). Translanguaging in the classroom: Emerging issues for research and pedagogy. ''International Journal of Bilingual Education and Bilingualism, 14''(3), 271–283.''Relations.'' Routledge. 
* Canagarajah, S. (2013). ''Translingual Practice: Global Englishes and Cosmopolitan'' 
* Cavallaro, C. J., &amp; Sembiante, S. F. (2020). Facilitating culturally sustaining, functional literacy practices in a middle school ESOL reading program: a design-based research study. ''Language and Education'', ''35''(2), 160–179. https://doi.org/10.1080/09500782.2020.1775244
* Cummins, J. (2021). Translanguaging: A critical analysis of theoretical claims. In P. Juvonen &amp; M. Källkvist (Eds.), ''Pedagogical Translanguaging: Theoretical, Methodological and Empirical Perspectives'' (pp. 7–36). Multilingual Matters.  https://doi.org/10.21832/9781788927383
* García, O. (2009). ''Bilingual education in the 21st century: A global perspective.'' Wiley-Blackwell, München.
* García, O., &amp; Lin, A. (2017). ''Translanguaging in Bilingual Education''. Springer.
* Kasula, A. J. (2016). Olowalu Review: Developing identity through translanguaging in a multilingual literary magazine. ''Colomb. Appl. Linguist. J., 18''(2), 109–118. 
* Kursiša, A., &amp; Richter-Vapaatalo, U. (Eds.). (2018). ''Mehr als Deutsch!'' Goethe-Institut Finnland. https://www.goethe.de/prj/dlp/de/unterrichtsmaterial/mehr_als_deutsch
* Makoni, S., &amp; Pennycook, A. (2007). ''Disinventing and Reconstituting Languages.'' Multilingual Matters.
* Otheguy, R., García, O., &amp; Reid, W. (2015). Clarifying Translanguaging and Deconstructing Named Languages. ''Applied Linguistics Review, 6(3),'' 281–307.
* Otheguy, R., García, O., &amp; Reid, W. (2015). Translanguaging and the role of language in social identity. ''International Journal of Bilingual Education and Bilingualism, 18''(3), 281–297.
* Penfield, W., &amp; Roberts, L. (1959). S''peech and Brain Mechanisms''. Princeton University Press.
* Piccardo, E. (2017). ''Plurilingualism as a catalyst for creativity in superdiverse societies: A systemic analysis''. ''Frontiers in Psychology, 8'', Article 2169. https://doi.org/10.3389/fpsyg.2017.02169
* Treffers-Daller, J. (2024). ''Unravelling translanguaging: A critical appraisal''. ''ELT Journal, 78''(1), 64–71. https://doi.org/10.1093/elt/ccad058 

==Credits==
This resource has been created by [[User:Projet PEP|Projet PEP]] ([[User talk:Projet PEP|discuss]] • [[Special:Contributions/Projet PEP|contribs]]) (Erasmus+ project, co-financed by the European Commission) :

* Diego Cortés Velásquez (Università Roma Tre) 
* Anastasia Gkaintartzi (University of Thessaly)

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{{Education}}{{Course}}

== Starting Activity ==
[[File:Translanguaging in Paris.png|alt=Translanguaging in Paris|thumb|Figure 1 - Translanguaging in Paris]]
Imagine you are observing a classroom where students from different linguistic and cultural backgrounds interact.

* How do they switch between languages?
* Do they use different languages for different purposes (e.g. socializing, academic tasks)?
* How would you describe their language use: do they mix languages or use them separately? 

Look at the following examples (see Figure 1 and Figure 2): Signs where multiple languages are used (see Linguistic Landscapes in Education).

Now think about the questions below: 

* How are these languages being used and why?
* Try to think of monolingual versions of the examples. How would the communication/messages be different?[[File:Languages in Paris.png|alt=Languages in Paris|thumb|Figure 2 - Languages in Paris]]

== Objectives ==
At the end of this section, you will be able to:

* Recognize different theoretical perspectives on plurilingualism.
* Understand the implications of a translanguaging approach to language use.
* Develop a critical awareness of how plurilingualism functions in everyday communication and education.

== Keywords ==
Plurilingualism, linguistic repertoire, bilingualism, multilingualism, translanguaging, language practices, sociolinguistics, education.

== Table of Contents ==

# Introduction
# History of the Concept
# Definitions and Theoretical Perspectives
# Practical Examples
# Take-Home Messages
# Self-Assessment
# Resources to Go Further
# Bibliography

== Introduction ==
The study of plurilingualism and its implications for education has evolved significantly over the past decades. Traditionally, language use was conceptualized through monolingual ideologies, where each language was seen as a separate, countable entity. However, more recent theories, such as translanguaging, argue for a more fluid and dynamic vision of language use.

Within this evolving field, the concept of translanguaging has become particularly influential. Rather than viewing multilingual speakers as alternating between separate language systems, translanguaging perspectives emphasize that individuals draw flexibly on their entire linguistic and semiotic repertoire to communicate and make meaning. At the same time, this perspective has generated considerable debate. While some scholars argue that multilingual language use is best understood as a unitary system, others maintain that socially recognized (named) languages continue to have theoretical, cognitive, and pedagogical relevance. This debate is reflected in the distinction between the Unitary Translanguaging Theory (UTT) and the Crosslinguistic Translanguaging Theory (CTT) (Cummins, 2021).

Plurilingualism therefore refers not only to the knowledge of multiple languages, but also to the ability to mobilize and integrate diverse linguistic resources according to communicative purposes and social contexts. Understanding the different theoretical perspectives on plurilingualism is important because they influence how multilingual competence is conceptualized and how languages are taught, learned, and valued in educational settings.

== History of the Concept ==
The concept of plurilingualism has developed alongside changing perspectives on bilingualism, multilingualism, and language itself. Over time, researchers have moved from viewing languages as separate, autonomous systems to exploring more dynamic and integrated models of multilingual competence. However, these developments have also generated theoretical debate, and different perspectives continue to coexist.

=== Early views (1950s–1980s) ===
Early research on bilingualism generally conceptualized languages as separate, stable systems. Bilingual speakers were often assumed to alternate between distinct languages, each functioning independently. For example, Penfield and Roberts (1959) described bilingual speakers as having an &quot;automatic switch&quot; between languages, reflecting the view that languages were stored and accessed separately.

=== Holistic perspectives (1980s–1990s) ===
From the 1980s onwards, researchers began challenging strictly monolingual conceptions of bilingualism. Grosjean (1982, 1989) famously argued that &quot;the bilingual is not two monolinguals in one person,&quot; emphasizing that bilingual competence should be understood as a unique and integrated phenomenon. Similarly, Coste, Moore, and Zarate (1997) described plurilingual competence as a complex and composite repertoire rather than the simple addition of separate languages.

=== Dynamic approaches (2002) ===
Building on these holistic perspectives, Herdina and Jessner's Dynamic Model of Multilingualism (2002) proposed that multilingual competence is constantly evolving through interactions among languages, cognitive processes, and environmental influences. Their work highlighted that multilingual systems are dynamic rather than static.

=== Critical perspectives (2007–2009) ===
Critical sociolinguists further questioned the status of languages as natural, bounded entities. Makoni and Pennycook (2007) argued that named languages should be understood as social and political constructs rather than pre-existing objects. García (2009) extended this perspective by proposing that multilingual speakers draw on an integrated linguistic repertoire instead of operating separate language systems.

=== Translanguaging theories (2009–present) ===
Building on these developments, García (2009) reconceptualized the notion of translanguaging, a term originally introduced by Cen Williams in Wales to describe bilingual pedagogical practices. While Williams' conception referred primarily to alternating languages for learning activities (e.g., reading in one language and discussing or writing in another), García broadened the concept into a theoretical framework that views multilingual speakers as drawing flexibly on an integrated linguistic repertoire. This broader interpretation has become highly influential in multilingual education, while also giving rise to considerable theoretical debate.

More recently, however, scholars have debated how multilingual competence should be understood theoretically. Cummins (2021) distinguishes between two major perspectives. The Unitary Translanguaging Theory (UTT) proposes that multilinguals draw on a single integrated linguistic system and questions the cognitive reality of separate languages. In contrast, the Crosslinguistic Translanguaging Theory (CTT) accepts that languages are socially constructed rather than natural entities but argues that named languages remain meaningful categories for understanding multilingual competence, crosslinguistic transfer, and educational practice.

Consequently, contemporary research no longer focuses on whether multilingual speakers use all their linguistic resources—they clearly do—but rather on how these resources should be conceptualized. While UTT emphasizes a unitary linguistic repertoire, CTT maintains that socially recognized languages continue to play an important role in cognition, communication, and education. This ongoing debate has important implications for language teaching, curriculum design, and multilingual pedagogy.

== Definitions and Theoretical Perspectives ==

* '''Plurilingualism''': “the dynamic and developing linguistic repertoire of an individual user/learner” (Council of Europe, 2020, p. 30)

* '''Translanguaging''': A perspective that sees language use as fluid, where speakers move across linguistic boundaries without adhering to rigid categorizations.

According to Otheguy, García, and Reid (2015, p. 281), translanguaging refers to the deployment of multilingual speakers' full linguistic, semiotic, and multimodal repertoire &quot;without regard for watchful adherence to the socially and politically defined boundaries of named languages.&quot; 

Canagarajah (2011) describes translanguaging as the strategic use of all available linguistic resources to communicate effectively. This perspective emphasizes that multilingual speakers select linguistic forms according to their communicative needs and contexts, challenging the traditional view of languages as separate, autonomous systems. 

Similarly, Li Wei (2011) argues that multilingual speakers create translanguaging spaces, in which they draw on their full linguistic repertoires to make meaning, communicate, and construct knowledge. 

Although translanguaging has become an influential framework, its theoretical assumptions remain the subject of scholarly debate. Critics argue that the unitary conception of multilingual competence may underestimate the continuing relevance of named languages as socially recognized constructs, particularly in education, language policy, and language assessment (Cummins, 2021; Treffers-Daller, 2024).

* '''Linguistic Repertoire''': The full range of linguistic and semiotic resources an individual has access to, regardless of conventional language boundaries.

A linguistic repertoire includes resources acquired through different experiences, such as languages learned at home, in school, or independently, and speakers may have varying levels of proficiency in each. These resources can serve different communicative functions—for example, in family interactions, education, work, or identity construction—and are often used in combination. Rather than being fixed, linguistic repertoires are dynamic and continually evolve as individuals encounter new communicative situations. From this perspective, language learning is understood as a process of extending, reorganizing, and enriching one's repertoire. 

More recently, Piccardo (2017) has proposed the concept of plurilanguaging, extending the notion of languaging to emphasize the dynamic, agentive, and emergent nature of plurilingual communication and learning. Drawing on Complex Dynamic Systems Theory, she argues that language learning is not a linear process of acquiring separate languages but a recursive process in which learners actively construct meaning through mediation, awareness, and the flexible use of linguistic and semiotic resources. In this perspective, plurilanguaging can be understood as the operationalization of plurilingualism in educational contexts. 

* '''Unitary Translanguaging Theory (UTT) (Cummins, 2021):''' UTT proposes that multilingual speakers draw on a single, integrated linguistic system rather than on separate language systems. Consequently, named languages are viewed primarily as social constructs rather than as distinct cognitive entities.

* '''Crosslinguistic Translanguaging Theory (CTT) (Cummins, 2021):''' CTT recognizes the flexible use of multilingual speakers' full linguistic repertoire while maintaining that named languages remain meaningful categories for understanding multilingual competence, crosslinguistic transfer, and language education.

According to Cummins (2021), the principal difference between these perspectives lies in how they conceptualize the status of named languages. UTT argues that multilinguals operate through a unitary linguistic system, whereas CTT accepts that, although languages are socially constructed, they remain useful theoretical and pedagogical constructs. CTT therefore emphasizes the importance of crosslinguistic relationships, transfer, additive bilingualism, and explicit connections between languages in educational contexts.  

Both theories recognize that languages are shaped by social contexts rather than being fixed, natural entities, and both challenge traditional assumptions that languages should always be kept strictly separate in educational settings. They also agree that multilingual speakers flexibly mobilize their available linguistic resources according to communicative needs. However, they differ in the extent to which they attribute theoretical, cognitive, and pedagogical significance to named languages. 

== Practical Examples ==

=== Example 1: Linguistic Landscapes ===
[[File:Multilingual signs, fingerposts in Brisbane, Australia 02.jpg|alt=Multilingual signs, fingerposts in Brisbane, Australia|thumb|Figure 3 - Multilingual signs, fingerposts in Brisbane, Australia]]
Multilingual signs in public spaces demonstrate how languages coexist and interact. A street sign in Brussels may include French, Dutch, and English, reflecting social and political dimensions of plurilingualism.

=== Example 2: Classroom Translanguaging Practices ===
Students in a bilingual classroom use English for academic tasks but switch to their home language for peer discussions. This challenges traditional monolingual teaching models.

An engaging activity to promote the use of one's entire linguistic repertoire could be a plurilingual debate. Students are given a topic and asked to prepare several arguments. During the debate, they present their points while intentionally switching between the different languages they can use. 

=== Example 3: Crosslinguistic Transfer in Writing ===
One possible activity for exploiting similarities between languages is described in Kursiša &amp; Richter-Vapaatalo (2018, p. 63). Finnish learners of German compare the typical structure and expressions used in informal emails in German, English, Swedish and Finnish, identifying similarities and differences.

Link: “Mehr als Deutsch”: https://www.goethe.de/prj/dlp/de/unterrichtsmaterial/mehr_als_deutsch 

=== Example 4: Plurilingual Digital Communication ===
WhatsApp messages mixing multiple languages, emojis, and voice recordings showcase how plurilingualism is naturally integrated into daily interactions.

== Take-Home Messages ==

* Plurilingualism refers to the dynamic ability of individuals to draw on and develop their linguistic repertoire for communication and learning across different contexts. 
* Translanguaging emphasizes the flexible use of multilingual speakers' linguistic resources and challenges traditional views of languages as strictly separate systems. 
* Unitary Translanguaging Theory (UTT) and Crosslinguistic Translanguaging Theory (CTT) offer different explanations of how multilingual competence is organized. While both recognize the flexible use of linguistic resources, they differ in the role they attribute to named languages. 
* Both theories agree that languages are shaped by social contexts rather than being fixed entities and question rigid language separation in education. However, there is ongoing debate about the cognitive and pedagogical status of named languages and about the implications of translanguaging for language teaching and learning.

== Self-Assessment ==
&lt;quiz display=&quot;simple&quot;&gt;

{Which statement best aligns with Unitary Translanguaging Theory (UTT)?}
-A) Bilinguals have an automatic switch that separates languages in their minds.
+B) Multilingual speakers access their linguistic resources as part of a single cognitive system.
-C) Languages should be taught separately to avoid interference.
-D) Translanguaging only happens in informal settings.

{What is the main critique of traditional bilingual models according to translanguaging theories?}
+A) They assume bilinguals speak two completely separate languages.
-B) They support language diversity in education.
-C) They promote crosslinguistic mediation.
-D) They encourage translanguaging in the classroom.

{Which statement best aligns with CTT?}
-A) CTT rejects the idea of a unitary linguistic system. 
+B) CTT advocates for maintaining the concept of specific languages while also supporting additive bilingualism and the transfer of academic skills across languages.
-C) CTT criticizes switching between languages because it assumes the existence of two separate linguistic systems.
-D) CTT argues that languages do not exist as real entities.

&lt;/quiz&gt;

== Resources to go further ==

* '''CEFR and Plurilingualism''': https://www.coe.int/en/web/language-policy/plurilingualism
* '''CARAP/FREPA:''' https://carap.ecml.at/
* '''Research on Translanguaging''': https://www.tandfonline.com/doi/full/10.1080/14790718.2017.1400501
* '''CUNY-NYSIEB Translanguaging Resources''': The City University of New York's New York State Initiative on Emergent Bilinguals offers a comprehensive collection of materials, including articles, videos, and classroom strategies, to support the implementation of translanguaging in educational settings. https://www.cuny-nysieb.org/translanguaging-resources

== Bibliography ==

* Canagarajah, S. (2011). Translanguaging in the classroom: Emerging issues for research and pedagogy. ''International Journal of Bilingual Education and Bilingualism, 14''(3), 271–283.''Relations.'' Routledge. 
* Canagarajah, S. (2013). ''Translingual Practice: Global Englishes and Cosmopolitan'' 
* Cavallaro, C. J., &amp; Sembiante, S. F. (2020). Facilitating culturally sustaining, functional literacy practices in a middle school ESOL reading program: a design-based research study. ''Language and Education'', ''35''(2), 160–179. https://doi.org/10.1080/09500782.2020.1775244
* Cummins, J. (2021). Translanguaging: A critical analysis of theoretical claims. In P. Juvonen &amp; M. Källkvist (Eds.), ''Pedagogical Translanguaging: Theoretical, Methodological and Empirical Perspectives'' (pp. 7–36). Multilingual Matters.  https://doi.org/10.21832/9781788927383
* García, O. (2009). ''Bilingual education in the 21st century: A global perspective.'' Wiley-Blackwell, München.
* García, O., &amp; Lin, A. (2017). ''Translanguaging in Bilingual Education''. Springer.
* Kasula, A. J. (2016). Olowalu Review: Developing identity through translanguaging in a multilingual literary magazine. ''Colomb. Appl. Linguist. J., 18''(2), 109–118. 
* Kursiša, A., &amp; Richter-Vapaatalo, U. (Eds.). (2018). ''Mehr als Deutsch!'' Goethe-Institut Finnland. https://www.goethe.de/prj/dlp/de/unterrichtsmaterial/mehr_als_deutsch
* Makoni, S., &amp; Pennycook, A. (2007). ''Disinventing and Reconstituting Languages.'' Multilingual Matters.
* Otheguy, R., García, O., &amp; Reid, W. (2015). Clarifying Translanguaging and Deconstructing Named Languages. ''Applied Linguistics Review, 6(3),'' 281–307.
* Otheguy, R., García, O., &amp; Reid, W. (2015). Translanguaging and the role of language in social identity. ''International Journal of Bilingual Education and Bilingualism, 18''(3), 281–297.
* Penfield, W., &amp; Roberts, L. (1959). S''peech and Brain Mechanisms''. Princeton University Press.
* Piccardo, E. (2017). ''Plurilingualism as a catalyst for creativity in superdiverse societies: A systemic analysis''. ''Frontiers in Psychology, 8'', Article 2169. https://doi.org/10.3389/fpsyg.2017.02169
* Treffers-Daller, J. (2024). ''Unravelling translanguaging: A critical appraisal''. ''ELT Journal, 78''(1), 64–71. https://doi.org/10.1093/elt/ccad058 

==Credits==
This resource has been created by [[User:Projet PEP|Projet PEP]] ([[User talk:Projet PEP|discuss]] • [[Special:Contributions/Projet PEP|contribs]]) (Erasmus+ project, co-financed by the European Commission) :

* Diego Cortés Velásquez (Università Roma Tre) 
* Anastasia Gkaintartzi (University of Thessaly)

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{{Education}}{{Course}}
== Starting activity ==
Try to remember your experiences over the last few weeks.

* In which situations were you confronted with more than one language?
* Please describe the circumstances in which these situations occurred (for example, at work, in your free time, at school/university, etc.).
* Which languages were involved, and how were they used?

Have you ever been faced with multiple languages in one or more of the following situations?

* When watching films, listening to podcasts, etc.
* When using apps/programmes
* On platforms such as LinkedIn, Instagram, Reddit, etc.
* In professional situations (meetings, etc.)

How did you handle/react in these situations? (Switch to another language, subtitles, GoogleTrad, ChatGPT)

What did you learn from these experiences?

* Individual words or phrases in other languages
* Special features of the language (language and/or writing system, structure of texts, etc.)
* Other aspects

== Objectives ==
At the end of this section, you should: 

* know the concepts and characteristics of formal, non-formal and informal learning;
* be able to identify their specific features within various educational and social contexts.
* understand how formal, non-formal and informal learning interact and complement each other in diverse learning contexts;
* understand how non-formal and informal learning contribute to lifelong learning, enhance transversal skills and support personal development.
* understand the importance of different forms of learning for developing multilingual competence.

== Key words ==
Informal learning, non-formal learning, learning environments

== Table of content ==

# Introduction
# History of the concept
# Definition(s)
# Formal, informal and non-formal learning within the education system
## Recognition of non-formal and informal learning
## Digital language education and informal/non-formal learning ''' '''
## Plurilingualism and informal/non-formal learning
# Take-home messages
# Self-assessment
# Further resources 
# Bibliography

== Introduction ==
The concepts of informal and non-formal emphasise that learning takes place not only in institutions such as schools, universities and adult education centres within a clearly defined framework, but also and to a large extent outside the classroom, in everyday life, etc., and plays an important role in the development of a person throughout his or her life. In plurilingual approaches, especially in Europe (e.g., CEFR) these experiences are not just supplementary but are central to a holistic understanding of learners’ language resources. 

== History of the concept ==
The term 'informal education' was coined by the American philosopher and educator John Dewey (1859–1952) during his lectures at the University of Chicago. A transcript of one of these lectures was published by Reginald D. Archambault in 1966. Dewey attributes the same effectiveness to informal education as to learning in the classroom, arguing that it must be based on social relationships outside the classroom (Archambault, 1966, as cited in Rohs, 2016, p. 7).

The American philosopher Mary Parker Follett (1868-1933) also emphasised the importance of non-institutional education and lifelong learning, without explicitly referring to the term ‘informal education/learning’.

That education is a continuous process is a truism. […] Life and education must never be separated. We must have more life in our universities, more education in our life. (Follett, 1918, p. 369)

The term ‘informal learning’ was first used by Eduard Lindeman, a pioneer in adult education, in an unpublished manuscript 1925 (as cited in Jarvis, 1987, p. 122). Like John Dewey, Lindeman believed that learning is closely linked to real-life experiences, and that education should not be confined to formal institutions. Despite these early references, the concept of informal learning was not widely explored within academia. From the 1970s onwards, however, international organisations began to recognise its importance. A significant milestone was the 1972 UNESCO report (Faure et al., 1972), which argued that rapid scientific and technological developments required a new perspective on education — one that included lifelong and informal learning as essential components. 

Coombs and Ahmed (1974) investigated how non-formal education could be used to combat rural poverty for the World Bank. By the 1990s, the concept had evolved significantly, with multiple definitions emerging, each highlighting different aspects of informal learning. Institutions such as the United Nations, the European Union and the Council of Europe published policy papers and position statements acknowledging the value of informal and non-formal learning in terms of personal development, social inclusion and employability (Harring et al., 2018; Johnson &amp; Majewska, 2022; Overwien, 2005; Rohs, 2016)

== Definitions: Overview and critical discussion ==
There is no universally accepted definition of informal and non-formal learning. Informal and non-formal learning are usually described in contrast to formal learning, based on certain criteria, such as learning taking place outside of educational institutions without leading to certification (e.g. COM(2001) 678 final, p. 32).

Informal learning is often characterised by taking place outside formal learning environments. It is usually driven by intrinsic motivation rather than the pursuit of specific goals and often associated with leisure time, family life, or even working life.

Non-formal learning is a type of learning that falls between formal and informal learning. It is typically defined by intentionality, intrinsic motivation, and learning objectives. Non-formal learning can also take place within educational institutions and may be based on a curriculum. However, it always involves a voluntary dimension.

Formal learning is structured, guided and usually follows a curriculum. Learning takes place in educational institutions and leads to certification (Johnson &amp; Majewska, 2022; Organisation for Economic Co-operation and Development - Education Policy Committee &amp; Werquin, 2007, pp. 22–25).

An overview of the possible criteria for distinguishing between different types of learning is provided by Johnson &amp; Majewska (2022, pp. 4–5). 
{| class=&quot;wikitable&quot;
|'''Formal learning'''
|'''Non-formal learning'''
| colspan=&quot;2&quot; |'''Informal learning'''
|-
|Learning is structured (e.g., linear objectives)
|Learning '''may be''' structured
| colspan=&quot;2&quot; |Learning is not structured
|-
|Learning is promoted through direct teaching behaviours
| colspan=&quot;3&quot; |Learning is promoted through indirect teaching behaviours
|-
|Learning is intended (by educator and learner)
|Learning is intended by the

'''learner'''
| colspan=&quot;2&quot; |Learning may not be intended by the learner
|-
|Learning is recognised by the learner and educator
|Learning is recognised by the

'''learner'''
| colspan=&quot;2&quot; |Learning may not be recognised by the learner
|-
| colspan=&quot;2&quot; |Motivation for learning may be extrinsic to the learner
| colspan=&quot;2&quot; |Motivation for learning is intrinsic to the learner
|-
|Learning takes place in educational institutions
|Learning '''can''' take place in educational institutions
| colspan=&quot;2&quot; |Learning can take place anywhere
|-
|Learning has a mandated dimension
| colspan=&quot;3&quot; |Learning has a voluntary dimension
|-
| colspan=&quot;2&quot; |Learning may be recognised or measured through qualifications
| colspan=&quot;2&quot; |Learning is not recognised or measured through

qualifications
|-
|Learning may primarily focus on propositional knowledge
| colspan=&quot;3&quot; |Learning may focus on both propositional and procedural

knowledge
|-
|Learning tends to have a cognitive emphasis
| colspan=&quot;3&quot; |Learning involves cognitive, emotional, social and behavioural elements
|-
|Curriculum is written down
| colspan=&quot;2&quot; |Curriculum '''may be''' written down
|Curriculum is not written down
|-
|Learning process is ‘top down’, focusing on developing specific

knowledge and skills
| colspan=&quot;3&quot; |Learning process is ‘bottom up’, focusing on the learner and their needs
|-
|Learning follows formal curriculum
| colspan=&quot;3&quot; |Learning may complement formal curricula
|-
| colspan=&quot;3&quot; |Learning may not be linked to socialisation
|Learning is often linked to socialisation
|}
(Johnson &amp; Majewska, 2022, p. 4–5.)

What recent definitions have in common is that learning is viewed from a holistic perspective, i.e. in the context of a person's entire environment and throughout their entire lifetime (Harring et al., 2018).

== Formal, informal and non-formal learning within the education system ==
In contemporary education and labour markets, the recognition of learning acquired outside formal institutions – through work, volunteering, online learning, or community engagement  – has become increasingly significant. Informal and non-formal learning contribute substantially to individuals’ competencies, yet these achievements often remain invisible in traditional qualification systems.

=== Recognition of non-formal and informal learning ===
To address this gap, several international organizations have developed frameworks and guidelines to support the validation of such learning. The European Union, for instance, has published the “European guidelines for validating non-formal and informal learning” (Cedefop, 2015), which provide a comprehensive framework for member states to implement validation systems that are accessible, fair, and learner-centered.

Similarly, UNESCO has advanced the “Guidelines for the recognition, validation and accreditation (RVA) of the outcomes of non-formal and informal learning (UNESCO Institute for Lifelong Learning, 2012), developed in collaboration with 42 member states. These guidelines aim to promote inclusive education systems and support lifelong learning by enabling individuals to receive formal recognition for diverse learning experiences.

By integrating these international frameworks into national policies, education systems can become more inclusive, flexible, and responsive to the realities of learners’ lives. Recognizing informal and non-formal learning not only enhances individual empowerment and employability but also contributes to social cohesion and economic innovation.

As a possible framework for describing language learning outside formal education and training, Benson (2011) proposes a model encompassing location, formality, pedagogy, and locus of control. Each of these aspects can be viewed as a pole on a scale. The model recognises that the increased range of language learning opportunities has brought about significant changes to learners' environments and that the boundaries between formal and informal learning are becoming increasingly blurred. 

=== Digital language education and informal/non-formal learning   ===
Some examples of wider and emerging research fields especially in digital informal and non-formal learning are learning in the digital wilds and learning with AI based tools. 

Since the 2000s, digital technology has opened up many opportunities to use languages outside the classroom. Informal learning is now a focus of language teaching and research. Sauro and Zourou (2017, p. 186) define 'language learning in the digital wilds' as 'informal language learning that takes place in digital spaces, communities, and networks that are independent of formal instructional contexts'. Similar expressions such as 'online informal learning of English (OILE)' (Sockett, 2014) or 'informal digital learning of English (IDLE)' (Lee &amp; Dressman, 2018) also refer to digital learning environments. 

Recently, AI-based tools such as large language models (LLMs) have added a new dimension to informal and non-formal language learning. Learners increasingly interact with AI chatbots (e.g., ChatGPT) outside of institutional settings in order to practise writing, rehearse dialogues, clarify vocabulary, or simulate real-life communication scenarios. These interactions often happen spontaneously, driven by learners' curiosity or needs, and blend characteristics of both informal and non-formal learning. (Guan et al., 2024)

Furthermore, outside digital learning environments, social developments such as an increasingly multilingual society are creating more opportunities for informal 'language learning beyond the classroom” (Nunan &amp; Richards, 2015), an expression often used instead of “informal language learning”.

=== Plurilingualism and informal / non-formal learning ===
The expansion of learning environments offers more opportunities to use languages in all modes: reception, production, interaction and mediation. Online resources such as podcasts and films offer a wide range of opportunities to experience language authentically. Multilingual practices are particularly popular on digital platforms, with users adopting them for specific purposes (e.g. Androutsopoulos, 2015; Lee, 2017; Ndlangamandla, 2020). Physical environments also provide many ways to expand one's linguistic repertoire. These can be used to specifically integrate non-formal learning opportunities or encourage informal learning in formal settings. Various projects have demonstrated such potential, e.g. the [https://www.linguanum.eu/productions-products Lingu@num project] for digital tasks. Examples of raising awareness of multilingualism can be seen in the [https://mlm.humanities.manchester.ac.uk/linguasnapp/index.html LinguaSnapp citizen science project], which documents the multilingual landscape of Manchester.

The Linguistic Risk-Taking Initiative is a pedagogical approach that promotes the use of the target language in daily situations outside of formal settings of instruction (Slavkov, 2023; Slavkov, 2020; Slavkov &amp; Séror, 2019). Language learners are encouraged to take linguistic risks, which are defined as “authentic everyday communicative acts that take place outside of the language classroom and involve spontaneous and meaningful second language use” (Slavkov &amp; Séror, 2019, p. 259). Since the Linguistic Risk-Taking Initiative is introduced to language learners in the classroom, but focuses on their target language use in everyday life, it can be considered a link between formal and informal learning (Cajka et al., 2023; Griffiths &amp; Slavkov, 2021). 

== Take-home messages ==

* Learning happens everywhere: Education is not limited to schools or universities. Informal and non-formal learning occur in everyday life, through digital media, social interactions, and community engagement.
* Blurred boundaries: The lines between formal, non-formal, and informal learning are increasingly fluid, especially in language education.
* Plurilingualism is a resource: Using multiple languages in various contexts (e.g., online platforms, work, leisure) enhances language competencies and cultural awareness.
* Digital environments matter: Informal digital learning (e.g., through podcasts, games, social media) plays a growing role in language acquisition.
* Recognition is key: International frameworks (e.g., from the EU and UNESCO) aim to validate and recognize learning that happens outside formal education.

== Self-assessment ==
&lt;quiz display=simple&gt;

{What best describes informal language learning?}
-A) It is structured and leads to certification.
+B) It happens during everyday activities, often spontaneously and unintended.
-C) It is always guided by a teacher.
-D) It only occurs in formal institutions.

{Which of the following is an example of non-formal learning?}
-A) Watching a movie in another language at home.
+B) Attending a language conversation group outside school.
-C) Learning a language through family conversations.
-D) Reading a textbook in school.

{What is the main goal of the EU and UNESCO guidelines on informal/non-formal learning?}
-A) To replace formal education systems.
-B) To foster self-directed learning.
+C) To validate and recognise learning outside formal institutions.
-D) To standardise learning in European education systems.
&lt;/quiz&gt;

== Further resources ==

* Lingu@num project: https://www.linguanum.eu/productions-products
* LinguaSnapp citizen science project: https://mlm.humanities.manchester.ac.uk/linguasnapp/index.html
* Recognition of further learning in Europe: https://education.ec.europa.eu/news/recognition-of-prior-learning-in-europe 
* Europass: https://europass.europa.eu/en/validation-non-formal-and-informal-learning 
* UNESCO Institute for Lifelong Learning: Recognition, validation and accreditation of non-formal and informal Learning: https://www.uil.unesco.org/en/lifelong-learning/recognition-validation-accreditation 

== Bibliography ==
Androutsopoulos, J. (2015). Networked multilingualism: Some language practices on Facebook and their implications. ''International Journal of Bilingualism'', ''19''(2), 185–205. https://doi.org/10.1177/1367006913489198

Benson, P. (2011). Language learning and teaching beyond the classroom: An introduction to the field. In P. Benson &amp; H. Reinders, Hayo (Eds.), ''Beyond the language classroom'' (pp. 7–16). Palgrave Macmillan.

Cajka, Stefanie; Ed, Griffiths, Nikolay Slavkov &amp; Eva Vetter. 2023. Linguistic risk-taking and informal language learning in Canada and Austria. In Denyze Toffoli, Geoffrey Sockett &amp; Meryl Kusyk (eds.), ''Language Learning and Leisure: Informal Language Learning in the Digital Age'' (Studies on Language Acquisition [SOLA] 66), 207-237. Berlin/Boston: De Gruyter Mouton.

Cedefop. (2015). ''European guidelines for validating non-formal and informal learning''. Publications Office. http://dx.doi.org/10.2801/378817

Coombs, P. H., Ahmed, M., International Council for Educational Development., &amp; World Bank. (1974). ''Attacking rural poverty; how nonformal education can help''. Johns Hopkins University Press. https://documents1.worldbank.org/curated/en/656871468326130937/pdf/multi-page.pdf

Faure, E., Herrera, F., Kaddoura, A.-R., Lopes, H., Petrovsky, A. V., Rahnema, M., &amp; Champion Ward, F. (1972). ''Learning to be: The world of education today and tomorrow.'' United Nations Educational, Scientific and Cultural Organization (UNESCO).

Follett, M. P. (with Cornell University Library). (1918). ''The new state, group organization the solution of popular government''. Longmans, Green and co. http://archive.org/details/cu31924016856209

Griffiths, Ed &amp; Nikolay Slavkov. 2021. Linguistic risk-taking: A bridge between the classroom and the outside world. ''Canadian Journal of Applied Linguistics'' 24(2). 127-158.

Guan, L., Li, S., &amp; Gu, M. M. (2024). AI in informal digital English learning: A meta-analysis of its effectiveness on proficiency, motivation, and self-regulation. Computers and Education: Artificial Intelligence, 7, 100323. https://doi.org/10.1016/j.caeai.2024.100323

Harring, M., Witte, M. D., &amp; Burger, T. (2018). Informelles Lernen—Eine Einführung. In M. Harring, M. D. Witte, &amp; T. Burger (Eds.), ''Handbuch informelles Lernen: Interdisziplinäre und internationale Perspektiven'' (pp. 12–25). Beltz Juventa.

Jarvis, P. (Ed.). (1987). Eduard Lindeman. In S. Brookfield, ''Twentieth century thinkers in adult &amp; continuing education'' (pp. 119–143). Croom Helm.

Johnson, M., &amp; Majewska, D. (2022, September 1). ''Formal, non-formal, and informal learning: What are they, and how can we research them?'' Cambridge University Press &amp; Assessment. https://www.repository.cam.ac.uk/handle/1810/371731 

Lee, M.-K. (2017). To be autonomous or not to be: Issues of subsuming self-determination theory into research on language learner autonomy. ''TESOL Quarterly'', ''51''(1), 220–228. https://doi.org/10.1002/tesq.343

Lee, J. S., &amp; Dressman, M. (2018). When IDLE hands make an English workshop: Informal digital learning of English and language proficiency. ''TESOL Quarterly'', ''52''(2), 435–445. https://doi.org/10.1002/tesq.422 

Lindeman, E. (1926). ''The meaning of adult education''. New Republic, inc.

Ndlangamandla, S. (2020). Language alternation in online forums: English monolingual normativity and multilingual practices. ''Scrutiny2'', ''25''(1), 43–63. https://doi.org/10.1080/18125441.2020.1802504

Nunan, D., &amp; Richards, J. C. (Eds.). (2015). ''Language learning beyond the classroom'' (1. publ.). Routledge.

Organisation for Economic Co-operation and Development - Education Policy Committee, &amp; Werquin, P. (2007, September 19). ''Terms, concepts and models for analysing the value of recognition programme''. https://one.oecd.org/document/EDU/EDPC(2007)24/en/pdf

Overwien, B. (2005). Stichwort: Informelles lernen. ''Zeitschrift für Erziehungswissenschaft'', ''8''(3), 339–355. https://doi.org/10.1007/s11618-005-0144-z

Rohs, M. (2016). Genese informellen Lernens. In M. D. Witte, T. Burger, &amp; M. Harring (Eds.), ''Handbuch informelles Lernen: Interdisziplinäre und internationale Perspektiven'' (pp. 3–38). Beltz Juventa.

Sauro, S., &amp; Zourou, K. (2017). Call for Papers for CALL in the Digital Wilds Special Issue. ''CALL in the Digital Wilds'', ''21''(1). https://scholarspace.manoa.hawaii.edu/items/c9da50ba-1f65-41d9-a3d3-f6ebad8c37ce

Slavkov, N. (2020). Where the magic happens: Fostering language learning, bilingualism and multilingualism through linguistic risk-taking. In T. Tinnefeld (Ed.), The magic of language: Productivity in linguistics and language teaching (pp. 47–70). COD. https://d-nb.info/1241778841/04

Slavkov, N. (2023). Linguistic risk-taking: A new pedagogical approach and a research program. Canadian Journal of Applied Linguistics, 26(2), 32–59. https://doi.org/10.37213/cjal.2023.33038

Slavkov, N., &amp; Séror, J. (2019). The development of the linguistic risk-taking initiative at the University of Ottawa. The Canadian Modern Language Review, 75(3), 254–272. https://doi.org/10.3138/cmlr.2018-0202

Sockett, G. (2014). ''The online informal learning of English''. Palgrave Macmillan UK. https://doi.org/10.1057/9781137414885 

UNESCO Institute for Lifelong Learning. (2012). ''UNESCO Guidelines for the recognition, validation and accreditation of the outcomes of non-formal and informal learning''. UNESCO Institute for Lifelong Learning (UIL). http://unesdoc.unesco.org/images/0021/002163/216360e.pdf

==Credits==
This resource has been created by [[User:Projet PEP|Projet PEP]] ([[User talk:Projet PEP|discuss]] • [[Special:Contributions/Projet PEP|contribs]]) (Erasmus+ project, co-financed by the European Commission) : 

* Margareta Strasser (Universität Salzburg)
* Paulina Wagner (Universität Wien)

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  <page>
    <title>WikiJournal Preprints/Pentagram map</title>
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      <text bytes="59524" sha1="gkpcetf5jaw2zd05rr8sd8t5dhm86k8" xml:space="preserve">{{Article info
| last1 = Stiegler
| orcid1 = 0009-0001-5789-6923
| first1 = Jean-Baptiste
| affiliation1 = Université Paris-Saclay
| correspondence1 = jean-baptiste.stiegler@universite-paris-saclay.fr
| journal = WikiJournal of Science
| et_al = true
| w1 = Pentagram map
| from w1 = true
| keywords = Pentagram map, Dynamical system, Projective geometry, Moduli space, Integrable systems
| license = CC-BY-SA 4.0
| submitted = 2025-12-08
| abstract = In [[w:mathematics|mathematics]], the '''pentagram map''' is a [[w:Dynamical system#Discrete dynamical system|discrete dynamical system]] acting on [[w:polygons|polygons]] in the [[w:projective plane|projective plane]]. It defines a new polygon whose vertices are obtained as the intersection points of the shortest [[w:Diagonal|diagonals]] of the initial polygon. This is a [[w:Projective linear group|projectively]] [[w:Equivariant map|equivariant]] procedure, hence it [[w:Quotient space (topology)|descends]] to the [[w:moduli space|moduli space]] of polygons and defines another dynamical system (which is also referred to as the pentagram map). It was first introduced by [[w:Richard Schwartz (mathematician)|Richard Schwartz]] in 1992.{{Sfn|Schwartz|1992}}

The pentagram map on the moduli space is famous for its [[w:Completely integrable|complete integrability]] and its link with [[w:cluster algebra|cluster algebras]].{{sfn|Gekhtman|Izosimov|2025|p=14}}

It admits many generalizations in [[w:Projective space|projective spaces]] and other settings.
}}
== Introduction ==

=== Informal definition ===

==== On polygons ====
[[File:Pentagram pentagon nolabel big.svg|alt=|thumb|300x300px|The pentagram map applied on a [[w:Convex set|convex]] [[w:pentagon|pentagon]].]]
Initially, the pentagram map was defined for [[w:convex polygon|convex polygon]]s (with at least five sides) on the [[w:euclidean plane|Euclidean plane]]. Given such a polygon &lt;math&gt;P&lt;/math&gt; with &lt;math&gt;n&lt;/math&gt; sides, one can draw the &quot;shortest [[w:diagonal|diagonal]]s&quot;, meaning the [[w:Line segment|segments]] whose endpoints are a [[w:Vertex (geometry)|vertex]] and one of its second neighbors (as in Figure 1). The intersections of the shortest diagonals are then taken as the vertices of a new &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;T(P)&lt;/math&gt;; this new polygon is the output of the pentagram map.{{Sfn|Berger|2005}}

The same construction can be done on [[w:Concave polygon|non-convex polygons]], but there are several complications. First, some consecutive short diagonals may not intersect, so one must extend the segments to [[w:Line (geometry)|lines]]. Second, the image &lt;math&gt;T(P)&lt;/math&gt; can fail to be a new &lt;math&gt;n&lt;/math&gt;-gon because some consecutive vertices could coincide. However, this [[w:Generic property|generically]] doesn't happen.{{Sfn|Ovsienko|Schwartz|Tabachnikov|p=411|2009}} Finally, it is possible that two diagonals are [[w:Parallel (geometry)|parallel]] and don't intersect on the [[w:euclidean plane|Euclidean plane]]. This is resolved by extending the Euclidean plane to the [[w:real projective plane|real projective plane]] by the addition of a [[w:line at infinity|line at infinity]], where the [[w:Vanishing point|intersection point]] lies (see Figure 3). Hence, the pentagram map is defined for generic polygons in the real projective plane.{{Sfn|Berger|2005|p=25}}

More generally, the construction of the pentagram map is well defined whenever the concepts of lines and their intersections make sense. This is encompassed by the notion of a general [[w:projective plane|projective plane]], of which the real projective plane is one example; but the pentagram map can also be considered over other [[w:Field (mathematics)|fields]], for instance the [[w:complex number|complex number]]s, which give the [[w:complex projective plane|complex projective plane]].{{Sfn|Weinreich|2022|loc=§3.1.1}}

==== On the moduli space of polygons ====
Since the pentagram map is constructed by drawing lines and marking their intersections, it [[w:Commutative property|commutes]] with any transformation that sends lines to lines. Such maps are called [[w:projective transformations|projective transformations]]. This allows to identify polygons [[w:up to|up to]] [[w:Perspectivity#Projectivity|projective transformations]]. This identification gives a [[w:Quotient space (topology)|quotient space]] (technically called a [[w:moduli space|moduli space]]) of [[w:Equivalence class|classes]] of polygons. 

The pentagram map on polygons induces another dynamical system on the moduli space,{{Sfn|Schwartz|1992|loc=§1 Projective geometry}} whose behavior differs quite a lot from the initial one.{{Efn|Compare the paragraph about the [[w:Pentagram map#Collapsing of convex polygons|collapsing of convex polygons]] and the one about [[w:Pentagram map#Complete integrability|complete integrability]].}} The dynamic is trivial for the classes of pentagons and hexagons, but this stops to be the case for polygons with more vertices.{{Efn|See the paragraph about [[w:Pentagram_map#Pentagons_and_hexagons|pentagons and hexagons]].}}

=== Historical elements ===
The pentagram map for general polygons was introduced in {{Harvard citation|Schwartz|1992}}, but the simplest case is the one of [[w:pentagons|pentagons]], hence the name &quot;[[w:pentagram|pentagram]]&quot;.{{Sfn|Marí-Beffa|2014|p=1}} Their study goes back to {{Harvard citation|Clebsch|1871}},{{Sfn|Izosimov|2022a|p=1085}} {{Harvard citation|Kasner|1928}}{{Sfn|Tabachnikov|2019}} and {{Harvard citation|Motzkin|1945}}.{{Sfn|Schwartz|2013|p=1}}

The pentagram map interacts with some classical configuration theorems of [[w:projective geometry|projective geometry]]. It provides results analogous to the ones of [[w:Pascal's theorem|Pascal's theorem]] and [[wikipedia:Brianchon's_theorem|Brianchon's theorem]].{{Sfn|Schwartz|Tabachnikov|2010}} Some specific configurations make [[w:Desargues' theorem|Desargues's theorem]] and [[w:Poncelet's porism|Poncelet's porism]] appear.{{Sfn|Berger|2005|loc=§4 and §5}}{{Efn|See the paragraph about [[w:Pentagram map#Poncelet polygons|Poncelet polygons]].}}

==Definitions and first properties==

=== Definition of the map ===
[[File:Pentagram pentagon label big.svg|alt=|thumb|300x300px|The pentagram map on a convex pentagon, with vertices labeled.]]
[[File:Pentagram on nonconvex pentagon.svg|alt=|thumb|300x300px|The pentagram map applied on a [[w:Self-intersecting polygon|self-intersecting]] (in particular, non-convex) pentagon. The vertex &lt;math&gt;w_2&lt;/math&gt; is on the [[w:line at infinity|line at infinity]], because it is the [[w:Vanishing point|intersection of two parallel lines]].]]
Let &lt;math&gt;n\geq 5&lt;/math&gt; be an integer. A polygon &lt;math&gt;P&lt;/math&gt; with &lt;math&gt;n&lt;/math&gt; sides, or &lt;math&gt;n&lt;/math&gt;-gon, is a tuple of [[w:Vertex (geometry)|vertices]] &lt;math&gt;(v_1,\dots,v_n)&lt;/math&gt; lying in some [[w:projective plane|projective plane]] &lt;math&gt;\mathbb P ^2&lt;/math&gt;,{{Efn|In the following, the figures represent polygons on the real plane, where the intuition is easier to grasp.}} where the indices are understood [[w:Modular arithmetic|modulo]] &lt;math&gt;n&lt;/math&gt;. The [[w:Dimension of an algebraic variety|dimension]] of the space of &lt;math&gt;n&lt;/math&gt;-gons is &lt;math&gt;2n&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=definition 1.1}}

Suppose that the vertices are in sufficiently [[w:general position|general position]], meaning that no consecutive triple of points are [[w:Collinearity|collinear]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=1}} Taking the intersection of two consecutive &quot;shortest&quot; [[w:diagonal|diagonal]]s{{Efn|Meaning the line between a vertex &lt;math&gt;v_k&lt;/math&gt; and a &quot;second neighbour&quot; &lt;math&gt;v_{k\pm 2}&lt;/math&gt;.}} defines a new point&lt;math display=&quot;block&quot;&gt; w_k := \overline{v_{k-1} v_{k+1}} \cap \overline{v_{k} v_{k+2}}. &lt;/math&gt;This procedure defines a new &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;T(P)=(w_1,\dots,w_n)&lt;/math&gt;, as in Figure 2.{{Sfn|Schwartz|1992|p=71}}

The labeling of the indices of &lt;math&gt;T(P)&lt;/math&gt; is not [[w:canonical|canonical]]. In most papers, a choice is made at the beginning of the paper and the formulas are tuned accordingly.{{Sfn|Izosimov|2016|loc=remark 1.5}}

The pentagram map on polygons is a [[w:birational map|birational map]] &lt;math&gt;T:(\mathbb P^2)^n&lt;/math&gt;{{nowrap|{{font|size=145%|⇢}}}}&lt;math&gt;(\mathbb P^2)^n&lt;/math&gt;. Indeed, each [[w:Homogeneous coordinates|coordinate]] of &lt;math&gt;w_k&lt;/math&gt; is given as a [[w:rational function|rational function]] of the coordinates of &lt;math&gt;v_{k-1},\dots,v_{k+2}&lt;/math&gt;, since it is defined as the intersection of lines passing by them. Moreover, the [[w:inverse map|inverse map]] is given by taking the intersections &lt;math&gt;\overline{w_{k-2} w_{k-1}} \cap \overline{w_{k} w_{k+1}} &lt;/math&gt;, which is rational for the same reason.{{Sfnp|Weinreich|2022|loc=definition 1.2}}

=== Moduli space ===
The pentagram map is defined by taking [[w:Line (geometry)|lines]] and intersections of them. The biggest [[w:Group (mathematics)|group]] which maps lines to lines is the one of [[w:projective transformations|projective transformations]] &lt;math&gt;\mathbb P \mathrm{GL}_{3}&lt;/math&gt;. Such a transformation &lt;math&gt;M&lt;/math&gt; [[w:Group action|acts]] on a polygon &lt;math&gt;P&lt;/math&gt; by sending it to &lt;math&gt;M \cdot P:=(Mv_1,\dots,Mv_n)&lt;/math&gt;. The pentagram map [[w:Commutative property|commutes]] with this action, and thereby induces another [[w:dynamical system|dynamical system]] on the [[w:moduli space|moduli space]] of projective [[w:equivalence classes|equivalence classes]] of polygons. Its [[w:Dimension of an algebraic variety|dimension]] is &lt;math&gt;2n-8&lt;/math&gt;.{{Sfn|Schwartz|1992|loc=§1 Projective geometry}}

===Twisted polygons===
[[File:Twisted heptagon.svg|alt=|thumb|300x300px|An example of twisted [[w:heptagon|heptagon]] on the real plane.]]
The pentagram map naturally generalizes on the larger space of twisted polygons (see example in Figure 4). For any integer &lt;math&gt;n\geq5&lt;/math&gt;, a twisted &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;P&lt;/math&gt; is the data of:

* a [[w:Sequence#Indexing|bi-infinite sequence]] of points &lt;math&gt;(v_k)_{k\in\mathbb Z}&lt;/math&gt; in the projective plane (called the vertices),
* a [[w:projective transformation|projective transformation]] &lt;math&gt;M \in \mathbb P \mathrm{GL}_3&lt;/math&gt; (called the [[w:monodromy|monodromy]]),

such that for any &lt;math&gt;k \in \mathbb Z&lt;/math&gt;, the property &lt;math&gt;v_{k+n}=Mv_k&lt;/math&gt; is satisfied. The dimension of the space of twisted &lt;math&gt;n&lt;/math&gt;-gons is &lt;math&gt;2n+8&lt;/math&gt;.{{Sfn|Schwartz|2008}}

When &lt;math&gt;M=\mathrm{Id}&lt;/math&gt;, this gives back the initial definition of polygons (which are said to be closed). The space of closed &lt;math&gt;n&lt;/math&gt;-gons is of [[w:codimension|codimension]] &lt;math&gt;8&lt;/math&gt; in the space of twisted ones.{{Sfn|Soloviev|2013|p=2816}}

The action of projective transformations over the space of closed polygons generalizes to the space of twisted ones (the monodromy is changed by [[w:Matrix similarity|conjugation]]). This provides again a moduli space, of dimension &lt;math&gt;2n&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=definition 1.3}}

== Collapsing of convex polygons ==

=== Exponential shrinking ===
[[File:Pentagram map convex heptagon iterate.svg|alt=|thumb|300x300px|The pentagram map iterated on a convex [[w:heptagon|heptagon]], exhibiting the convergence.]]
Let &lt;math&gt;P&lt;/math&gt; be a closed [[w:Convex polygon#Strictly convex polygon|strictly convex polygon]] lying on the real plane. One of the first results proved by Richard Schwartz it that its iterates under the pentagram map shrink [[w:Exponential growth|exponentially fast]] to a point, as illustrated in Figure 5. This follows from two facts.
# The image of a strictly convex polygon is contained in its [[w:Interior (topology)|interior]], and is also strictly convex.{{Sfn|Glick|2020|p=2818}}
# There exists a constant &lt;math&gt;0&lt; \eta_P&lt;1&lt;/math&gt;, depending on &lt;math&gt;P&lt;/math&gt;, such that for any &lt;math&gt;N \in \mathbb N&lt;/math&gt;, the diameters of the iterates verify the inequality &lt;math display=&quot;inline&quot;&gt;\operatorname{diam}(T^N(P))\leq\eta_P^N \operatorname{diam}(P). &lt;/math&gt;{{Sfn|Schwartz|1992|loc=theorem 3.1}}

Hence, by [[w:Cantor's intersection theorem#Variant in complete metric spaces|Cantor's intersection theorem]], the sequence of polygons collapses toward a point.{{Sfn|Schwartz|1992|loc=§3 Convex polygons}}

The behavior on the moduli space is very different, since the dynamics is [[w:Recurrent point|recurrent]].{{Sfn|Schwartz|2001|loc=theorem 1.1}} It is even a [[w:quasiperiodic motion|quasiperiodic motion]],{{Sfn|Ovsienko|Schwartz|Tabachnikov|2009}} as discussed in [[w:Pentagram map#Complete integrability|the section about integrability]].

=== Coordinates of the limit point ===
The limit point coordinates are found in {{Harvard citation|Glick|2020}}. They satisfy some [[w:Degree of a polynomial|degree]] 3 [[w:polynomial equation|polynomial equations]], whose coefficients are [[w:rational function|rational function]]s in the coordinates of the vertices of the starting polygon. The proof relies on the fact that the limit point must be an [[w:eigenline|eigenline]] of a certain [[w:linear operator|linear operator]] of &lt;math&gt;\mathbb R^3&lt;/math&gt;.{{sfn|Glick|2020}}

This operator was reinterpreted in {{Harvard citation|Aboud|Izosimov|2022}} as the infinitesimal monodromy of the polygon. The [[w:Pentagram map#The scaling symmetry|scaling symmetry]] is used to [[w:Deformation (mathematics)|deform]] a closed polygon &lt;math&gt;P&lt;/math&gt; into a family of twisted ones '''&lt;math&gt;(P_z)_{z\in \mathbb C^*}&lt;/math&gt;''' with monodromy &lt;math&gt;M_z&lt;/math&gt;. The infinitesimal monodromy is defined to be:{{sfn|Aboud|Izosimov|2022}}
&lt;math display=&quot;block&quot;&gt;\left.\frac{dM_z}{dz}\right|_{z=1}.&lt;/math&gt;

=== Generalization ===
The collapsing of polygons may also happen in some [[w:Pentagram map#Generalizations|generalization of the pentagram map]], when considering some specific configurations of polygons in the real plane. The coordinates of the collapse point are given by a formula analogous to the one for the original pentagram map.{{Sfn|Schwartz|2026}}

== Periodic orbits on the moduli space ==
For some configurations of closed polygons, the iterate of the pentagram map will send &lt;math&gt;P&lt;/math&gt; to a projectively equivalent polygon (up to some shift of the indices). This means that, on the moduli space, the orbit of the class of &lt;math&gt;P&lt;/math&gt; is [[w:Periodic orbit|periodic]].

===Pentagons and hexagons===
[[File:penta hexagon.svg|300px|thumb|The outward hexagon is projectively equivalent to the inward one, with respect to their labeling.]]The following two facts are proved by checking [[w:cross-ratio|cross-ratio]] equalities, so they are true for polygons in any [[w:projective plane|projective plane]] (not just the [[w:Real projective plane|real one]]).{{Sfn|Schwartz|1992|loc=§2 Pentagons and hexagons}}

The pentagram map &lt;math&gt;T&lt;/math&gt; is the identity on the moduli space of [[w:pentagon|pentagon]]s.{{Sfn|Schwartz|1992|loc=theorem 2.1}}{{Sfn|Clebsch|1871}}{{Sfn|Motzkin|1945}} The second iterate &lt;math&gt;T^2&lt;/math&gt; is the identity on the space of labeled [[w:hexagon|hexagon]]s, up to a shift of labeling (see Figure 6).{{Sfn|Schwartz|1992|loc=theorem 2.3}} This phenomenon doesn't generalize to generic polygons with at least seven sides, for which the motion is [[w:Quasiperiodic motion|quasi-periodic]].{{Sfn|Tupan|2022}}

==== Generalization ====
The result about pentagons and hexagons generalizes to some [[w:Pentagram map#Generalizations|higher pentagram maps]] in &lt;math&gt;\mathbb P ^k&lt;/math&gt;, for polygons with &lt;math&gt;k+3&lt;/math&gt; or &lt;math&gt;2k+2&lt;/math&gt; sides. The proof uses a generalization of the [[w:Gale transform|Gale transform]].{{Sfn|Dirdak|2024}}

=== Poncelet polygons ===
A polygon is said to be Poncelet{{Efn|The name comes from [[w:Jean-Victor Poncelet|Jean-Victor Poncelet]] and [[w:Poncelet porism|his porism]].{{Sfn|Izosimov|2022a|p=1085}}}} if it is [[w:Inscribed figure|inscribed]] in a [[w:Conic section|conic]] and circumscribed about another one.{{Sfn|Schwartz|2015|loc=|p=433}}{{Efn|In particular, pentagons are Poncelet since [[w:five points determine a conic|five points determine a conic]].{{Sfn|Schwartz|2015|loc=|p=433}}}} For a convex Poncelet &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;P&lt;/math&gt; lying on the [[w:real projective plane|real projective plane]], the polygon &lt;math&gt;T^2(P)&lt;/math&gt; is projectively equivalent to &lt;math&gt;P&lt;/math&gt;.{{Sfn|Schwartz|2015|loc=theorem 1.1}} In fact, when &lt;math&gt;n&lt;/math&gt; is odd, the converse is also true.{{Sfn|Izosimov|2022a|loc=corollary 1.1}}

However, this converse statement is no longer true when the polygons are considered over the [[w:complex projective plane|complex projective plane]].{{Sfn|Izosimov|2022a|loc=remark 1.3}}

==Coordinates for the moduli space==
The moduli space can be described by different [[w:Coordinate_system|coordinate systems]]. The following ones are practical to explicit the dynamic, as presented in the next section.

=== Corner coordinates ===
[[File:Corner coordinates big.svg|thumb|300x300px|The geometric construction of the points defining the corner invariants.]]
Define the [[w:cross-ratio|cross-ratio]] of four [[w:Collinearity|collinear]] points to be

: &lt;math&gt; [a,b,c,d]=\frac{(a-b)(c-d)}{(a-c)(b-d)}. &lt;/math&gt;

The corner invariants are a system of coordinates on the space of twisted polygons, constructed by taking intersections as in Figure 7.{{Sfn|Schwartz|2001|loc=figure 2}} The left and right invariants are respectively defined{{Efn|The ordering of the vertices in the cross-ratios can differ from a paper to another one, which slightly changes the formulas in the following sections.}} as the following cross-ratios:

: &lt;math&gt;x_k:=[v_{k-2},v_{k-1},\overline{v_{k-2}v_{k-1}}\cap\overline{v_{k}v_{k+1}},\overline{v_{k-2}v_{k-1}}\cap\overline{v_{k+1}v_{k+2}}],&lt;/math&gt;

: &lt;math&gt;y_k:=[\overline{v_{k+1}v_{k+2}}\cap\overline{v_{k-2}v_{k-1}}, \overline{v_{k+1}v_{k+2}}\cap\overline{v_{k-1}v_{k}},v_{k+1},v_{k+2}].&lt;/math&gt;

Since the cross-ratio is [[w:Cross-ratio#Projective geometry|projective invariant]], the sequences &lt;math&gt;(x_k)_{k \in \mathbb Z}&lt;/math&gt; and &lt;math&gt;(y_k)_{k \in \mathbb Z}&lt;/math&gt; associated to a twisted &lt;math&gt;n&lt;/math&gt;-gon are &lt;math&gt;n&lt;/math&gt; periodic.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=415}}

The corner invariants are elements of &lt;math&gt;\mathbb{P}^1\smallsetminus\{0,1,\infty\}&lt;/math&gt;, and they realize an [[w:Isomorphism_of_varieties|isomorphism of variety]] between the moduli space of twisted &lt;math&gt;n&lt;/math&gt;-gons and &lt;math&gt;(\mathbb{P}^1\smallsetminus\{0,1,\infty\})^{2n}&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=theorem 3.6}}

===ab-coordinates===

There is a second set of coordinates for the moduli space of twisted &lt;math&gt;n&lt;/math&gt;-gons defined over a [[w:Field (mathematics)|field]] &lt;math&gt;F&lt;/math&gt; satisfying &lt;math&gt;\mathrm{SL}_3(F)\cong \mathbb P\mathrm{GL}_3(F)&lt;/math&gt;,{{Sfn|Weinreich|2022|loc=remark 3.8}} and such that &lt;math&gt;n&lt;/math&gt; is not divisible by &lt;math&gt;3&lt;/math&gt;.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=section 4.1}}

The vertices &lt;math&gt;v_k&lt;/math&gt; in the [[w:projective plane|projective plane]] &lt;math&gt;\mathbb P^2(F)&lt;/math&gt; can be [[w:Lift (mathematics)|lifted]] to [[w:Vector space|vectors]] &lt;math&gt;V_k&lt;/math&gt; in the [[w:affine space|affine space]] &lt;math&gt;F^3&lt;/math&gt; so that each consecutive triple of vectors spans a [[w:parallelepiped|parallelepiped]] having [[w:determinant|determinant]] equal to &lt;math&gt;1&lt;/math&gt;. This leads to the relation{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=equation 4.1}}

: &lt;math&gt;V_{k+3} = a_k V_{k+2} + b_k V_{k+1} + V_k.&lt;/math&gt;

This bring out an analogy between twisted polygons and solutions of third order linear [[w:ordinary differential equations|ordinary differential equations]], normalized to have unit [[w:Wronskian|Wronskian]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=remark 6.6}}

They are linked to the corner coordinates by:{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=lemma 4.5}}
: &lt;math&gt;x_k=\frac{a_{k-2}}{b_{k-2}b_{k-1}},&lt;/math&gt;
: &lt;math&gt;y_k=-\frac{b_{k-1}}{a_{k-2}a_{k-1}}.&lt;/math&gt;

==Formulas on the moduli space==

===As a birational map ===
The pentagram map is a [[w:birational map|birational map]] on the moduli space, because it can be decomposed as the [[w:Function composition|composition]] of two [[w:Birational geometry|birational]] [[w:Involution (mathematics)|involutions]].{{Sfn|Schwartz|2008|loc=§1.2 The Pentagram Map}} The corner invariants change in the following way:{{Sfn|Ovsienko|Schwartz|loc=lemma 2.4|Tabachnikov|2010}}

: &lt;math&gt;x_k'=x_k\frac{1-x_{k-1} y_{k-1}}{1-x_{k+1}y_{k+1}},&lt;/math&gt;

: &lt;math&gt;y_k'=y_{k+1}\frac{1-x_{k+2} y_{k+2}}{1-x_k y_k}.&lt;/math&gt;

=== The scaling symmetry ===
The [[w:multiplicative group|multiplicative group]] &lt;math&gt;F\smallsetminus\{0\}&lt;/math&gt; [[w:One-parameter group|acts]] on the moduli space in the following way:

: &lt;math&gt;R_s\cdot(x_1,\dots,x_n,y_1,\dots,y_n)=(sx_1,\dots,sx_n,s^{-1}y_1,\dots,s^{-1}y_n),&lt;/math&gt;

where &lt;math&gt;R&lt;/math&gt; is called the scaling action and &lt;math&gt;s&lt;/math&gt; is the scaling parameter. This action commutes with the pentagram map on the moduli space (as presented in the previous formulas). This property is called the scaling symmetry, and is instrumental in proving the [[w:Pentagram map#Complete integrability|complete integrability]] of the dynamics.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=corollary 2.5}}

==Invariant structures==

===Monodromy invariants===

The monodromy invariants, introduced in {{Harvard citation|Schwartz|2008}}, are a collection of [[w:Function (mathematics)|functions]] on the [[w:moduli space|moduli space]] that are invariant under the pentagram map.{{Sfn|Schwartz|2008|loc=theorem 1.2}} The simplest example of them are 
:&lt;math&gt; O_n= x_1x_2\cdots x_{n}, \quad E_n = y_1y_2\cdots y_n. &lt;/math&gt;
The other monodromy invariants can be retrieved through different points of view: through the [[w:Pentagram map#The scaling symmetry|scaling symmetry]], as [[w:Combinatorics|combinatorial]] objects, or as some [[w:determinant|determinant]]s.{{Sfn|Schwartz|Tabachnikov|2011|loc=§2 The Monodromy Invariants}} The one involving scaling symmetry is presented here.

Let &lt;math&gt;M\in \mathrm{GL}_3&lt;/math&gt; be a [[w:Lift (mathematics)|lift]] of the monodromy of a twisted &lt;math&gt;n&lt;/math&gt;-gon. The quantities

: &lt;math&gt;\Omega_1=\frac{\operatorname{trace}^3(M)}{\det(M)}, \quad \Omega_2=\frac{\operatorname{trace}^3(M^{-1})}{\det(M^{-1})},&lt;/math&gt;

are independent of the choice of lift and are invariant under [[w:Matrix similarity|conjugation]], so they are well defined for the projective class of the polygon. They are invariant under the pentagram map, since the monodromy matrix doesn't change.{{Sfn|Schwartz|Tabachnikov|2011|loc=|p=5}} Now, the quantities

: &lt;math&gt;\tilde{\Omega}_1=O_n^2E_n\Omega_1, \quad \tilde{\Omega}_2=O_nE_n^2\Omega_2,&lt;/math&gt;

have the same properties, but turn out to be polynomials in the corner invariants.{{Efn|Some papers consider the cube roots of this functions, but it doesn't change the following definitions of the monodromy invariants.}} They can be written as{{Sfn|Schwartz|Tabachnikov|2011|loc=|p=5}}

: &lt;math&gt;
\tilde{\Omega}_1=\biggl(\sum_{k=0}^{\lfloor n/2\rfloor}O_k\biggr)^3, \quad
\tilde{\Omega}_2=\biggl(\sum_{k=0}^{\lfloor n/2\rfloor}E_k\biggr)^3,
&lt;/math&gt;

where each &lt;math&gt;O_k&lt;/math&gt; and &lt;math&gt;E_k&lt;/math&gt; are [[w:homogeneous polynomial|homogeneous polynomial]]s respectively of weight &lt;math&gt;k&lt;/math&gt; and &lt;math&gt;-k&lt;/math&gt;,{{Sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=11}} meaning they change under the [[w:Pentagram map#The scaling symmetry|rescaling action]] on variables by{{Sfn|Schwartz|Tabachnikov|2011|p=5}}

: &lt;math&gt; R_s(O_k)= s^k O_k, \quad R_s(E_k)= s^{-k} E_k. &lt;/math&gt;

The quantities &lt;math&gt;O_1,\dots,O_{\lfloor n/2 \rfloor},O_n, E_1,\dots,E_{\lfloor n/2 \rfloor},E_n,&lt;/math&gt; are unchanged by the dynamics, and are called the monodromy invariants. Moreover, they are [[w:algebraically independent|algebraically independent]].{{Sfn|Schwartz|2008|loc=theorem 1.2}}

==== Polygons on conics ====
Whenever &lt;math&gt;P&lt;/math&gt; is [[w:Inscribed figure|inscribed]] on a [[w:conic section|conic section]], one has &lt;math&gt;O_k(P)=E_k(P)&lt;/math&gt; for all &lt;math&gt;k&lt;/math&gt;.{{Sfn|Schwartz|Tabachnikov|2011|loc=theorem 1.1}} Moreover, if &lt;math&gt;P&lt;/math&gt; is circumscribed about another conic,{{Efn|See the paragraph about [[w:Pentagram map#Poncelet polygons|Poncelet polygons]].}} then its monodromy invariants are characterized by the pair of conics.{{Sfn|Schwartz|2015|loc=theorem 1.2}} For such odd-gons, the translation on the [[w:Jacobian variety|Jacobian variety]]{{Efn|See the paragraph about [[w:Pentagram map#Algebro-geometric integrability|algebraic integrability]].}} is restricted to the [[w:Prym variety|Prym variety]] (which is a half-dimensional torus in the Jacobian).{{Sfn|Izosimov|2016|loc=theorem 1.3}}

===Poisson bracket===

An invariant [[w:Poisson bracket|Poisson bracket]] on the space of twisted polygons was found in {{Harvard citation|Ovsienko|Schwartz|Tabachnikov|2010}}. The monodromy invariants [[w:Poisson bracket#Constants of motion|commute]] with respect to it:
&lt;math display=&quot;block&quot;&gt; \{O_i,O_j\}=\{O_i,E_j\}=\{E_i,E_j\}=0 &lt;/math&gt;for all &lt;math&gt;i,j&lt;/math&gt;.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}}

The Poisson bracket is defined in terms of the corner coordinates by:
&lt;math display=&quot;block&quot;&gt; \begin{align}
 \{x_i,x_{i\pm1}\} &amp;= \mp x_i x_{i+1}, \\
 \{y_i,y_{i\pm 1}\} &amp;= \mp y_i y_{i+1}, \\
 \{x_i,x_j\} &amp;= \{y_i,y_j\} = \{x_i,y_j\} = 0
\end{align}&lt;/math&gt;for all other &lt;math&gt; i,j.&lt;/math&gt;{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=equation 2.16}}

=== The spectral curve ===

Let &lt;math&gt;\zeta&lt;/math&gt; be an element of the [[w:multiplicative group|multiplicative group]] and &lt;math&gt;P_\zeta&lt;/math&gt; be the polygon obtained by applying the [[w:Pentagram map#The scaling symmetry|rescaling action]] &lt;math&gt;R_\zeta&lt;/math&gt; on &lt;math&gt;P&lt;/math&gt;.  A [[w:Lax matrix|Lax matrix]] &lt;math&gt;\hat{T}(\zeta) \in \mathrm{GL}_3&lt;/math&gt; is a lift of the monodromy of &lt;math&gt;P_\zeta&lt;/math&gt; satisfying a [[w:Lax pair#Zero-curvature equation|zero-curvature equation]].{{Sfn|Weinreich|2022|loc=§5 The Lax representation}} Then, the spectral function is the [[w:Bivariate polynomial|bivariate]] [[w:characteristic polynomial|characteristic polynomial]]
&lt;math display=&quot;block&quot;&gt; Q(\lambda,\zeta) := \det(\lambda\operatorname{Id}-\hat{T}(\zeta)),&lt;/math&gt;or some renormalization of it. The [[w:spectral curve|spectral curve]] is the [[w:Projective variety#projective completion|projective completion]] of the [[w:Algebraic curve|affine curve]] defined by the equation &lt;math&gt;Q(\lambda,\zeta)=0&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=§6. The geometry of the spectral curve}} It is invariant under the pentagram map, and the monodromy invariants appear as the [[w:coefficient|coefficient]]s of &lt;math&gt;Q&lt;/math&gt;.{{Sfn|Soloviev|2013|loc=theorem 6.4}} Its [[w:geometric genus|geometric genus]] is &lt;math&gt;n-1&lt;/math&gt; if &lt;math&gt;n&lt;/math&gt; is odd, and &lt;math&gt;n-2&lt;/math&gt; if &lt;math&gt;n&lt;/math&gt; is even.{{Sfn|Weinreich|2022|p=|loc=theorem 6.4}}

It was first introduced in {{Harv|Soloviev|2013|ps=|p=}} for his proof of [[w:Pentagram map#Algebro-geometric integrability|algebro-geometric integrability]].{{sfn|Soloviev|2013}}

==Complete integrability==
The pentagram map on the moduli space has been proved to be a [[w:completely integrable|completely integrable]] [[w:discrete dynamical system|discrete dynamical system]], both in the [[w:Integrable system#Hamiltonian systems and Liouville integrability|Arnold-Liouville]]{{Efn|Over the [[w:real number|real number]]s.}} and the [[w:Integrable system#Complete integrability over the complex numbers|algebro-geometric]]{{Efn|Over [[w:algebraically closed field|algebraically closed field]]s of [[w:Characteristic (algebra)|characteristic]] different from 2.}} senses. In any case, this means that the moduli space is [[w:almost everywhere|almost everywhere]] [[w:Foliation|foliated]] by [[w:Torus#Flat torus|flat tori]] (or in the algebraic setting, [[w:Abelian variety|Abelian varieties]]), where the motion is a [[w:Translation (geometry)|translation]]. This [[w:Generic property|generically]] induces a [[w:quasiperiodic motion|quasiperiodic motion]] on the corresponding torus.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2009}}

===Arnold–Liouville integrability===

The proof of the integrability of the pentagram map on a real twisted polygon was achieved in {{Harvard citation|Ovsienko|Schwartz|Tabachnikov|2010}}. This is done by noticing that the monodromy invariants &lt;math&gt;O_n&lt;/math&gt; and &lt;math&gt;E_n&lt;/math&gt; are [[w:Casimir invariant|Casimir invariant]]s for the bracket, meaning (in this context) that&lt;math display=&quot;block&quot;&gt; \{O_n,f\}=\{E_n,f\} = 0 &lt;/math&gt;for all functions &lt;math&gt;f&lt;/math&gt;.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}} When &lt;math&gt;n&lt;/math&gt; is even, this is also true for the monodromy invariants &lt;math&gt;O_{\lfloor n/2 \rfloor }&lt;/math&gt; and &lt;math&gt;E_{\lfloor n/2 \rfloor }&lt;/math&gt;.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}}

This allows to consider the Casimir [[w:level set|level set]], where each Casimir has a specified value. Because of [[w:Sard's theorem|Sard's theorem]], any generic level set is a [[w:smooth manifold|smooth manifold]].{{Sfn|Schwartz|2017|p=44}} They form a [[w:foliation|foliation]] in [[w:Poisson manifold#Symplectic leaves|symplectic leaves]], on which the Poisson bracket gives rise to a [[w:symplectic form|symplectic form]].{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=corollary 2.13}}

Each of these symplectic leaves has an iso-monodromy [[w:foliation|foliation]], namely, a decomposition into the common level sets of the remaining monodromy functions. By using again [[w:Sard's theorem|Sard's theorem]], they are generically [[w:Symplectic manifold#Lagrangian submanifolds|Lagrangian manifolds]].{{Sfn|Schwartz|2017|p=45}} Moreover, they are compact.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=§3.3. Compactness of the level sets}} Since the monodromy invariants Poisson-commute and there are enough of them, the discrete [[w:Liouville–Arnold theorem|Liouville–Arnold theorem]] can be applied to prove that the level sets are [[w:Torus#Flat torus|flat tori]] over which the dynamics is a translation.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=412}}
===Algebro-geometric integrability===

In {{Harvard citation|Soloviev|2013}}, it was shown that the pentagram map admits a [[w:Lax representation|Lax representation]] with a spectral parameter, which allows to prove its algebro-geometric integrability. This means that the space of polygons (either twisted or closed) is parametrized by its spectral data, consisting of [[w:Pentagram map#The spectral curve|its spectral curve]], with marked points and a [[w:Divisor (algebraic geometry)|divisor]] given by a [[w:Floquet theory|Floquet]]–[[w:Bloch's theorem|Bloch]] equation. This gives an embedding to the [[w:Jacobian variety|Jacobian variety]] through the [[w:Abel–Jacobi map|Abel–Jacobi map]], where the motion is expressed in terms of translation.{{sfn|Soloviev|2013|loc=theorems A, B and C}} The previously defined Poisson bracket is also retrieved.{{sfn|Soloviev|2013|loc=theorem D}} 

This integrability was generalized in {{Harvard citation|Weinreich|2022}} from the field of [[w:complex number|complex number]]s to any [[w:algebraically closed field|algebraically closed field]] of [[w:Characteristic (algebra)|characteristic]] different from 2. The translation on a torus is replaced by a translation on an [[w:Abelian variety|Abelian variety]] (in fact, a Jacobian variety again).{{sfn|Weinreich|2022|loc=theorem 1.4}}

=== Dimension of the invariant manifold ===

For twisted &lt;math&gt;n&lt;/math&gt;-gons, the [[w:dimension|dimension]] of the invariant tori (or Jacobian varieties) is{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=421}}

: &lt;math&gt;\begin{cases} 
n-1 &amp; \text{when }n \text{ is odd,}\\
n-2 &amp; \text{when }n \text{ is even.}
\end{cases}&lt;/math&gt;

Moreover, when &lt;math&gt;n&lt;/math&gt; is even, there are two isomorphic Jacobians on which the iterates of the pentagram map alternate. But on each of them, the second iterate is a translation.{{Sfn|Weinreich|2022|loc=theorem 1.4}}

=== For closed polygons ===
There is no Poisson structure on the space of closed polygons.{{Sfn|Soloviev|2013|loc=corollary 4.1}} Nevertheless, the one from twisted polygons can be used to prove integrability.{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=2153}}

Algebro-geometric integrability holds for closed polygons in a same manner as for the twisted ones.{{Sfn|Soloviev|2013|loc=theorem C}} However, Arnold-Liouville integrability is proved for real closed polygons only when they are convex. This is done by restricting the [[w:Hamiltonian vector field|Hamiltonian vector field]]s of monodromy functions to smaller dimensional tori, and showing that enough of them are still independent.{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|loc=corollary 1.1}}

In both situation, the dimension of the invariant manifolds decreases by &lt;math&gt;3&lt;/math&gt; for closed &lt;math&gt;n&lt;/math&gt;-gons (compared to the twisted case), and is equal to{{Sfn|Soloviev|2013|loc=theorem C}}{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|loc=theorem 1}}
:&lt;math&gt;\begin{cases} 
n-4 &amp; \text{when }n \text{ is odd,}\\
n-5 &amp; \text{when }n \text{ is even.}
\end{cases}&lt;/math&gt;

==Connections to other topics==

===The Boussinesq equation===

The continuous limit of a convex polygon is a parametrized convex curve in the plane. When the time parameter is suitably chosen, the [[w:Discretization|continuous limit]] of the pentagram map is the classical [[w:Boussinesq approximation (water waves)|Boussinesq equation]]. This equation is a classical example of an [[w:integrable|integrable]] [[w:partial differential equation|partial differential equation]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 5}}

Here is a description of the geometric action of the Boussinesq equation. Given a [[w:locally convex|locally convex]] curve &lt;math&gt; C:\mathbb R\to \mathbb R^2 &lt;/math&gt; and real numbers &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;t&lt;/math&gt;, consider the [[w:chord (geometry)|chord]] connecting &lt;math&gt; C(x-t) &lt;/math&gt; to &lt;math&gt; C(x+t) &lt;/math&gt;. The [[w:Envelope (mathematics)|envelope]] of all these chords is a new curve &lt;math&gt; C_t(x) &lt;/math&gt;. When &lt;math&gt;t&lt;/math&gt; is extremely small, the curve &lt;math&gt; C_t(x) &lt;/math&gt; is a good model for the time &lt;math&gt;t&lt;/math&gt; evolution of the original curve &lt;math&gt; C_0(x) &lt;/math&gt; under the Boussinesq equation. This construction is also similar to the pentagram map. Moreover, the pentagram invariant bracket is a discretization of a well known invariant Poisson bracket associated to the Boussinesq equation.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=§6.4 Discretization}}

===Cluster algebras===

The pentagram map{{Sfn|Glick|2011}} and some of its generalizations{{Sfn|Gekhtman|Shapiro|Tabachnikov|Vainshtein|2012}}{{Sfn|Glick|Pylyavskyy|2016}} are identified as special cases of discrete dynamical systems powered by [[w:cluster algebra|cluster algebra]]. Using the results from {{Harvard citation|Goncharov|Kenyon|2013}}, this provides a link with the [[w:Poisson–Lie group|Poisson–Lie group]]s, [[w:dimer model|dimer model]]s and other so-called cluster-integrable systems.{{Sfn|Fock|Marshakov|2016}} These methods allow to retrieve the Poisson-bracket and Hamiltonians used to prove complete integrability{{Sfn|Affolter|George|Ramassamy|2025|loc=§5 The pentagram map}} and provide [[w:Lax representation|Lax representation]]s.{{Sfn|Izosimov|2022b}}

=== Singularity theory ===
The pentagram map exhibit a property called singularity confinement, which is typical from [[w:integrable system|integrable system]]s.{{Sfn|Grammaticos|Ramani|Papageorgiou|1991}} It states that if a polygon &lt;math&gt;P&lt;/math&gt; is [[w:Singular point of an algebraic variety|singular]] for the pentagram map &lt;math&gt;T&lt;/math&gt;, then there exists an integer &lt;math&gt;m&lt;/math&gt; such that &lt;math&gt;P&lt;/math&gt; not singular for the iterate map &lt;math&gt;T^m&lt;/math&gt;.{{Sfn|Glick|2012}}

Moreover, the pentagram map (along with some of its generalizations and other discrete dynamical systems) exhibit the Devron property.{{Efn|The name comes from an episode of [[w:Star Trek|Star Trek]].{{Sfn|Glick|2015|loc=§1 Introduction}}}} This means that if a polygon &lt;math&gt;P&lt;/math&gt; is singular for some iterate of the pentagram map &lt;math&gt;T^m&lt;/math&gt;, then it will also be singular for some iterate of the inverse map &lt;math&gt;T^{-m'}&lt;/math&gt;.{{Sfn|Glick|2015}}

== Generalizations ==
The definition of twisted polygons still makes sense in any [[w:projective space|projective space]] &lt;math&gt;\mathbb P^d&lt;/math&gt;, under the action of the [[w:Projective linear group|projective group]] &lt;math&gt;\mathbb P \mathrm{GL}_{d+1}&lt;/math&gt;. The pentagram map can be generalized in many ways, and some of them are presented here. Not all of them are integrable.{{Sfn|Khesin|Soloviev|2015|}} Some are [[w:discretization|discretization]]s of [[w:PDEs|PDEs]] from the [[w:KdV hierarchy|KdV hierarchy]], seen as higher dimensional version of [[w:Boussinesq approximation (water waves)|Boussinesq]] or [[w:Kadomtsev–Petviashvili equation|KP]] equations.{{Sfn|Marí-Beffa|2012}}{{Sfn|Wang|2023}} The description of all generalized pentagram maps in terms of [[w:cluster algebra|cluster algebra]]s is still an open question.{{Sfn|Gekhtman|Izosimov|2025|p=14}}

=== Polygons in general positions ===
Let &lt;math&gt;d \geq 2&lt;/math&gt; and &lt;math&gt;P&lt;/math&gt; be a twisted polygon of &lt;math&gt;\mathbb P^d&lt;/math&gt; in [[w:general position|general position]].

==== Short diagonal pentagram maps ====
The &lt;math&gt;k&lt;/math&gt;-th ''short diagonal hyperplane'' &lt;math&gt;H_k^{sh}&lt;/math&gt; is uniquely defined by passing through the vertices &lt;math&gt;v_k,v_{k+2},\dots,v_{k+2d-2}&lt;/math&gt;. [[w:Generic property#In algebraic geometry|Generically]], the intersection of &lt;math&gt;d&lt;/math&gt; consecutive hyperplanes uniquely defines a new point

: &lt;math&gt;T_{sh}v_k:=H_k^{sh}\cap H_{k+1}^{sh}\cap \dots \cap H_{k+d-1}^{sh}.&lt;/math&gt;

Doing this for every vertex defines a new twisted polygon. This map, denoted by &lt;math&gt;T_{sh}&lt;/math&gt;, is again projectively equivariant.{{Sfn|Khesin|Soloviev|2013}}

==== Generalized pentagram maps ====
The previous procedure can be generalized. Let &lt;math&gt;I=(i_1,\dots,i_{d-1}),~J=(j_1,\dots,j_{d-1})&lt;/math&gt; be two sets of integers, respectively called the jump tuple and the intersection tuple. Define the &lt;math&gt;k&lt;/math&gt;-th hyperplane &lt;math&gt;H_k^I&lt;/math&gt; to be passing through the vertices &lt;math&gt;v_k,v_{k+i_1},\dots,v_{k+i_1+\dots+i_{d-1}}&lt;/math&gt;. A new point is given by the intersection

: &lt;math&gt;T_{I,J}v_k:=H_k^I \cap H_{k+j_1}^I \cap \dots \cap H_{k+j_1+\dots +j_{d-1}}^I.&lt;/math&gt;

The map &lt;math&gt;T_{I,J}&lt;/math&gt; is called a generalized pentagram map.{{Sfn|Khesin|Soloviev|2015a}} The original pentagram map is recovered by considering&lt;math&gt;d=2,~I=(2),~J=(1)&lt;/math&gt;.

Integrability can be numerically tested by picking a random polygon &lt;math&gt;P&lt;/math&gt; with [[wikipedia:Rational_point|rational coordinates]] and studying the growth rate of the [[wikipedia:Height_function|height]] of its iterates. This is called the [[wikipedia:Integrable_system#Diophantine_integrability|diophantine integrability]] test, and some generalized pentagram maps don't seem to pass it.{{Sfn|Khesin|Soloviev|2015a|loc=§5 and §6}} However, it is conjectured that the maps &lt;math&gt;T_{I,I}&lt;/math&gt; are integrable for any &lt;math&gt;I&lt;/math&gt;.{{Sfn|Bolsinov|Matveev|Miranda|Tabachnikov|2018|loc=conjecture 4.13 (B. Khesin, F. Soloviev)}}

Some of these maps are [[w:discretization|discretization]]s of higher dimensional counterpart of the [[w:Boussinesq approximation (water waves)|Boussinesq equation]] in the [[w:KdV hierarchy|KdV hierarchy]].{{Sfn|Khesin|Soloviev|2015b|loc=theorem 4.1}}{{Sfn|Izosimov|2022b|loc=theorem 4.1}}

==== Dented pentagram maps ====
Fix an integer &lt;math&gt;m\in \{1,\dots ,d-1\}&lt;/math&gt;. Consider the jump tuple &lt;math&gt;I_m:=(1,\dots,1,2,1,\dots,1)&lt;/math&gt;, where the &lt;math&gt;2&lt;/math&gt; is at the &lt;math&gt;m&lt;/math&gt;-th place, and the intersection tuple &lt;math&gt;J:=(1,\dots,1)&lt;/math&gt;. The dented pentagram map is &lt;math&gt;T_m :=T_{I_m,J}&lt;/math&gt;. They are proved to be integrable.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 2.14}}

For an integer &lt;math&gt;p \geq 2&lt;/math&gt;, the deep dented pentagram map (of depth &lt;math&gt;p&lt;/math&gt;) &lt;math&gt;T_m^p&lt;/math&gt; is the same map as before, but the number &lt;math&gt;2&lt;/math&gt; in the definition of &lt;math&gt;I_m&lt;/math&gt; is replaced by &lt;math&gt;p&lt;/math&gt;. This kind of pentagram maps are again integrable.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 6.2}}

=== Corrugated polygons ===
A twisted polygon &lt;math&gt;P&lt;/math&gt; lying in &lt;math&gt;\mathbb P^d&lt;/math&gt; is said to be corrugated if for any &lt;math&gt;k\in \mathbb Z&lt;/math&gt;, the vertices &lt;math&gt;v_k,v_{k+1},v_{k+d},v_{k+d+1}&lt;/math&gt; span a projective two-dimensional plane. Such polygons are not in [[w:general position|general position]]. A new point is defined by

: &lt;math&gt;T_\text{cor}v_k:=\overline{v_k v_{k+d}}\cap \overline{v_{k+1} v_{k+d+1}}.&lt;/math&gt;

The map &lt;math&gt;T_\text{cor}&lt;/math&gt; yields a new corrugated polygon. They are [[w:Integrable system#Hamiltonian systems and Liouville integrability|completely Liouville-integrable]].{{Sfn|Gekhtman|Shapiro|Tabachnikov|Vainshtein|2012|loc=theorem 4.4}}

In fact, they can be retrieved as some dented pentagram map applied on corrugated polygons.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 5.3}}

=== Grassmannian polygons ===
Let &lt;math&gt;d \geq 3, m \geq 1&lt;/math&gt; be integers. The pentagram map can also be generalized to the [[w:Grassmannian|Grassmannian]] space &lt;math&gt;\mathrm{Gr}(m,md)&lt;/math&gt;, which consists of &lt;math&gt;m&lt;/math&gt;-[[w:Dimension (vector space)|dimensional]] [[w:linear subspace|linear subspace]]s of an &lt;math&gt;md&lt;/math&gt;-dimensional [[w:vector space|vector space]]. When &lt;math&gt;m=1&lt;/math&gt;, the linear subspaces are [[w:Vector space#vector line|lines]], which retrieves the definition of [[w:projective space|projective space]]s &lt;math&gt;\mathbb P^d&lt;/math&gt;.{{Sfn|Felipe|Marí-Beffa|2019|loc=§2 definitions and notations}}

A point &lt;math&gt;v\in\operatorname{Gr}(m,md)&lt;/math&gt; is represented by an &lt;math&gt;md \times m&lt;/math&gt; matrix &lt;math&gt;X_v&lt;/math&gt; such that its columns form a [[w:Basis (linear algebra)|basis]] of &lt;math&gt;v&lt;/math&gt;. Consider the [[w:Group action|action]] of the [[w:general linear group|general linear group]] &lt;math&gt;\mathrm{GL}_{md}&lt;/math&gt; by multiplication on the left of &lt;math&gt;X_v&lt;/math&gt;. This defines an action on the Grassmannian, even though it is not [[w:Faithful action|faithful]].{{Efn|Because there can be many lifts for &lt;math&gt;v&lt;/math&gt;, and because some matrices act trivially.}} Hence, the polygons of &lt;math&gt;\mathrm{Gr}(m,md)&lt;/math&gt; and their moduli spaces are defined as before, after the change of underlying group.{{Sfn|Felipe|Marí-Beffa|2019|loc=§2 definitions and notations}}

Depending on the parity of &lt;math&gt;d&lt;/math&gt;, one can define linear subspaces spanned by some &lt;math&gt;X_{v_k}&lt;/math&gt;'s such that taking their intersection generically defines a new point &lt;math&gt;v\in\mathrm{Gr}(m,md)&lt;/math&gt;.{{Sfn|Felipe|Marí-Beffa|2019|loc=sections 4 and 5}} This generalization of the pentagram map is integrable in a [[w:noncommutative|noncommutative]] sense.{{Sfn|Ovenhouse|2020}}

=== Over rings ===
The pentagram map admits a generalization by considering [[w:Projective space#Generalizations|projective planes]] over [[w:stably finite ring|stably finite ring]]s, instead of [[w:Field (mathematics)|field]]s. In particular, this retrieves the pentagram map over Grassmannians. Again, it admits a [[w:Lax representation|Lax representation]].{{Sfn|Hand|Izosimov|2025}}

== References ==
{{reflist|25em}}

===Notes===
{{notelist}}

==Works cited==
*{{Cite journal |ref=harv |title=The Limit Point of the Pentagram Map and Infinitesimal Monodromy |url=https://academic.oup.com/imrn/article/2022/7/5383/5911460 |journal=International Mathematics Research Notices |date=2022-03-23 |issn=1073-7928 |pages=5383–5397 |volume=2022 |issue=7 |doi=10.1093/imrn/rnaa258 |language=en |first1=Quinton |last1=Aboud |first2=Anton |last2=Izosimov}}
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      <text bytes="59546" sha1="enq22nybon8qsobyyrsrnit58j5ss0j" xml:space="preserve">{{Article info
| last1 = Stiegler
| orcid1 = 0009-0001-5789-6923
| first1 = Jean-Baptiste
| affiliation1 = Université Paris-Saclay
| correspondence1 = jean-baptiste.stiegler@universite-paris-saclay.fr
| journal = WikiJournal of Science
| et_al = true
| w1 = Pentagram map
| from w1 = true
| keywords = Pentagram map, Dynamical system, Projective geometry, Moduli space, Integrable systems
| license = CC-BY-SA 4.0
| submitted = 2025-12-08
| abstract = In [[w:mathematics|mathematics]], the '''pentagram map''' is a [[w:Dynamical system#Discrete dynamical system|discrete dynamical system]] acting on [[w:polygons|polygons]] in the [[w:projective plane|projective plane]]. It defines a new polygon whose vertices are obtained as the intersection points of the shortest [[w:Diagonal|diagonals]] of the initial polygon. This is a [[w:Projective linear group|projectively]] [[w:Equivariant map|equivariant]] procedure, hence it [[w:Quotient space (topology)|descends]] to the [[w:moduli space|moduli space]] of polygons and defines another dynamical system (which is also referred to as the pentagram map). It was first introduced by [[w:Richard Schwartz (mathematician)|Richard Schwartz]] in 1992.{{Sfn|Schwartz|1992}}

The pentagram map on the moduli space is famous for its [[w:Completely integrable|complete integrability]] and its link with [[w:cluster algebra|cluster algebras]].{{sfn|Gekhtman|Izosimov|2025|p=14}}

It admits many generalizations in [[w:Projective space|projective spaces]] and other settings.
}}
== Introduction ==

=== Informal definition ===

==== On polygons ====
[[File:Pentagram pentagon nolabel big.svg|alt=|thumb|300x300px|The pentagram map applied on a [[w:Convex set|convex]] [[w:pentagon|pentagon]].]]
Initially, the pentagram map was defined for [[w:convex polygon|convex polygon]]s (with at least five sides) on the [[w:euclidean plane|Euclidean plane]]. Given such a polygon &lt;math&gt;P&lt;/math&gt; with &lt;math&gt;n&lt;/math&gt; sides, one can draw the &quot;shortest [[w:diagonal|diagonal]]s&quot;, meaning the [[w:Line segment|segments]] whose endpoints are a [[w:Vertex (geometry)|vertex]] and one of its second neighbors (as in Figure 1). The intersections of the shortest diagonals are then taken as the vertices of a new &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;T(P)&lt;/math&gt;; this new polygon is the output of the pentagram map.{{Sfn|Berger|2005}}

The same construction can be done on [[w:Concave polygon|non-convex polygons]], but there are several complications. First, some consecutive short diagonals may not intersect, so one must extend the segments to [[w:Line (geometry)|lines]]. Second, the image &lt;math&gt;T(P)&lt;/math&gt; can fail to be a new &lt;math&gt;n&lt;/math&gt;-gon because some consecutive vertices could coincide. However, this [[w:Generic property|generically]] doesn't happen.{{Sfn|Ovsienko|Schwartz|Tabachnikov|p=411|2009}} Finally, it is possible that two diagonals are [[w:Parallel (geometry)|parallel]] and don't intersect on the [[w:euclidean plane|Euclidean plane]]. This is resolved by extending the Euclidean plane to the [[w:real projective plane|real projective plane]] by the addition of a [[w:line at infinity|line at infinity]], where the [[w:Vanishing point|intersection point]] lies (see Figure 3). Hence, the pentagram map is defined for generic polygons in the real projective plane.{{Sfn|Berger|2005|p=25}}

More generally, the construction of the pentagram map is well defined whenever the concepts of lines and their intersections make sense. This is encompassed by the notion of a general [[w:projective plane|projective plane]], of which the real projective plane is one example; but the pentagram map can also be considered over other [[w:Field (mathematics)|fields]], for instance the [[w:complex number|complex number]]s, which give the [[w:complex projective plane|complex projective plane]].{{Sfn|Weinreich|2022|loc=§3.1.1}}

==== On the moduli space of polygons ====
Since the pentagram map is constructed by drawing lines and marking their intersections, it [[w:Commutative property|commutes]] with any transformation that sends lines to lines. Such maps are called [[w:projective transformations|projective transformations]]. This allows to identify polygons [[w:up to|up to]] [[w:Perspectivity#Projectivity|projective transformations]]. This identification gives a [[w:Quotient space (topology)|quotient space]] (technically called a [[w:moduli space|moduli space]]) of [[w:Equivalence class|classes]] of polygons. 

The pentagram map on polygons induces another dynamical system on the moduli space,{{Sfn|Schwartz|1992|loc=§1 Projective geometry}} whose behavior differs quite a lot from the initial one.{{Efn|Compare the paragraph about the [[w:Pentagram map#Collapsing of convex polygons|collapsing of convex polygons]] and the one about [[w:Pentagram map#Complete integrability|complete integrability]].}} The dynamic is trivial for the classes of pentagons and hexagons, but this stops to be the case for polygons with more vertices.{{Efn|See the paragraph about [[w:Pentagram_map#Pentagons_and_hexagons|pentagons and hexagons]].}}

=== Historical elements ===
The pentagram map for general polygons was introduced in {{Harvard citation|Schwartz|1992}}, but the simplest case is the one of [[w:pentagons|pentagons]], hence the name &quot;[[w:pentagram|pentagram]]&quot;.{{Sfn|Marí-Beffa|2014|p=1}} Their study goes back to {{Harvard citation|Clebsch|1871}},{{Sfn|Izosimov|2022a|p=1085}} {{Harvard citation|Kasner|1928}}{{Sfn|Tabachnikov|2019}} and {{Harvard citation|Motzkin|1945}}.{{Sfn|Schwartz|2013|p=1}}

The pentagram map interacts with some classical configuration theorems of [[w:projective geometry|projective geometry]]. It provides results analogous to the ones of [[w:Pascal's theorem|Pascal's theorem]] and [[wikipedia:Brianchon's_theorem|Brianchon's theorem]].{{Sfn|Schwartz|Tabachnikov|2010}} Some specific configurations make [[w:Desargues' theorem|Desargues's theorem]] and [[w:Poncelet's porism|Poncelet's porism]] appear.{{Sfn|Berger|2005|loc=§4 and §5}}{{Efn|See the paragraph about [[w:Pentagram map#Poncelet polygons|Poncelet polygons]].}}

==Definitions and first properties==

=== Definition of the map ===
[[File:Pentagram pentagon label big.svg|alt=|thumb|300x300px|The pentagram map on a convex pentagon, with vertices labeled.]]
[[File:Pentagram on nonconvex pentagon.svg|alt=|thumb|300x300px|The pentagram map applied on a [[w:Self-intersecting polygon|self-intersecting]] (in particular, non-convex) pentagon. The vertex &lt;math&gt;w_2&lt;/math&gt; is on the [[w:line at infinity|line at infinity]], because it is the [[w:Vanishing point|intersection of two parallel lines]].]]
Let &lt;math&gt;n\geq 5&lt;/math&gt; be an integer. A polygon &lt;math&gt;P&lt;/math&gt; with &lt;math&gt;n&lt;/math&gt; sides, or &lt;math&gt;n&lt;/math&gt;-gon, is a tuple of [[w:Vertex (geometry)|vertices]] &lt;math&gt;(v_1,\dots,v_n)&lt;/math&gt; lying in some [[w:projective plane|projective plane]] &lt;math&gt;\mathbb P ^2&lt;/math&gt;,{{Efn|In the following, the figures represent polygons on the real plane, where the intuition is easier to grasp.}} where the indices are understood [[w:Modular arithmetic|modulo]] &lt;math&gt;n&lt;/math&gt;. The [[w:Dimension of an algebraic variety|dimension]] of the space of &lt;math&gt;n&lt;/math&gt;-gons is &lt;math&gt;2n&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=definition 1.1}}

Suppose that the vertices are in sufficiently [[w:general position|general position]], meaning that no consecutive triple of points are [[w:Collinearity|collinear]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=1}} Taking the intersection of two consecutive &quot;shortest&quot; [[w:diagonal|diagonal]]s{{Efn|Meaning the line between a vertex &lt;math&gt;v_k&lt;/math&gt; and a &quot;second neighbour&quot; &lt;math&gt;v_{k\pm 2}&lt;/math&gt;.}} defines a new point&lt;math display=&quot;block&quot;&gt; w_k := \overline{v_{k-1} v_{k+1}} \cap \overline{v_{k} v_{k+2}}. &lt;/math&gt;This procedure defines a new &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;T(P)=(w_1,\dots,w_n)&lt;/math&gt;, as in Figure 2.{{Sfn|Schwartz|1992|p=71}}

The labeling of the indices of &lt;math&gt;T(P)&lt;/math&gt; is not [[w:canonical|canonical]]. In most papers, a choice is made at the beginning of the paper and the formulas are tuned accordingly.{{Sfn|Izosimov|2016|loc=remark 1.5}}

The pentagram map on polygons is a [[w:birational map|birational map]] &lt;math&gt;T:(\mathbb P^2)^n&lt;/math&gt;{{nowrap|{{font|size=145%|⇢}}}}&lt;math&gt;(\mathbb P^2)^n&lt;/math&gt;. Indeed, each [[w:Homogeneous coordinates|coordinate]] of &lt;math&gt;w_k&lt;/math&gt; is given as a [[w:rational function|rational function]] of the coordinates of &lt;math&gt;v_{k-1},\dots,v_{k+2}&lt;/math&gt;, since it is defined as the intersection of lines passing by them. Moreover, the [[w:inverse map|inverse map]] is given by taking the intersections &lt;math&gt;\overline{w_{k-2} w_{k-1}} \cap \overline{w_{k} w_{k+1}} &lt;/math&gt;, which is rational for the same reason.{{Sfnp|Weinreich|2022|loc=definition 1.2}}

=== Moduli space ===
The pentagram map is defined by taking [[w:Line (geometry)|lines]] and intersections of them. The biggest [[w:Group (mathematics)|group]] which maps lines to lines is the one of [[w:projective transformations|projective transformations]] &lt;math&gt;\mathbb P \mathrm{GL}_{3}&lt;/math&gt;. Such a transformation &lt;math&gt;M&lt;/math&gt; [[w:Group action|acts]] on a polygon &lt;math&gt;P&lt;/math&gt; by sending it to &lt;math&gt;M \cdot P:=(Mv_1,\dots,Mv_n)&lt;/math&gt;. The pentagram map [[w:Commutative property|commutes]] with this action, and thereby induces another [[w:dynamical system|dynamical system]] on the [[w:moduli space|moduli space]] of projective [[w:equivalence classes|equivalence classes]] of polygons. Its [[w:Dimension of an algebraic variety|dimension]] is &lt;math&gt;2n-8&lt;/math&gt;.{{Sfn|Schwartz|1992|loc=§1 Projective geometry}}

===Twisted polygons===
[[File:Twisted heptagon.svg|alt=|thumb|300x300px|An example of twisted [[w:heptagon|heptagon]] on the real plane.]]
The pentagram map naturally generalizes on the larger space of twisted polygons (see example in Figure 4). For any integer &lt;math&gt;n\geq5&lt;/math&gt;, a twisted &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;P&lt;/math&gt; is the data of:

* a [[w:Sequence#Indexing|bi-infinite sequence]] of points &lt;math&gt;(v_k)_{k\in\mathbb Z}&lt;/math&gt; in the projective plane (called the vertices),
* a [[w:projective transformation|projective transformation]] &lt;math&gt;M \in \mathbb P \mathrm{GL}_3&lt;/math&gt; (called the [[w:monodromy|monodromy]]),

such that for any &lt;math&gt;k \in \mathbb Z&lt;/math&gt;, the property &lt;math&gt;v_{k+n}=Mv_k&lt;/math&gt; is satisfied. The dimension of the space of twisted &lt;math&gt;n&lt;/math&gt;-gons is &lt;math&gt;2n+8&lt;/math&gt;.{{Sfn|Schwartz|2008}}

When &lt;math&gt;M=\mathrm{Id}&lt;/math&gt;, this gives back the initial definition of polygons (which are said to be closed). The space of closed &lt;math&gt;n&lt;/math&gt;-gons is of [[w:codimension|codimension]] &lt;math&gt;8&lt;/math&gt; in the space of twisted ones.{{Sfn|Soloviev|2013|p=2816}}

The action of projective transformations over the space of closed polygons generalizes to the space of twisted ones (the monodromy is changed by [[w:Matrix similarity|conjugation]]). This provides again a moduli space, of dimension &lt;math&gt;2n&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=definition 1.3}}

== Collapsing of convex polygons ==

=== Exponential shrinking ===
[[File:Pentagram map convex heptagon iterate.svg|alt=|thumb|300x300px|The pentagram map iterated on a convex [[w:heptagon|heptagon]], exhibiting the convergence.]]
Let &lt;math&gt;P&lt;/math&gt; be a closed [[w:Convex polygon#Strictly convex polygon|strictly convex polygon]] lying on the real plane. One of the first results proved by Richard Schwartz it that its iterates under the pentagram map shrink [[w:Exponential growth|exponentially fast]] to a point, as illustrated in Figure 5. This follows from two facts.
# The image of a strictly convex polygon is contained in its [[w:Interior (topology)|interior]], and is also strictly convex.{{Sfn|Glick|2020|p=2818}}
# There exists a constant &lt;math&gt;0&lt; \eta_P&lt;1&lt;/math&gt;, depending on &lt;math&gt;P&lt;/math&gt;, such that for any &lt;math&gt;N \in \mathbb N&lt;/math&gt;, the diameters of the iterates verify the inequality &lt;math display=&quot;inline&quot;&gt;\operatorname{diam}(T^N(P))\leq\eta_P^N \operatorname{diam}(P). &lt;/math&gt;{{Sfn|Schwartz|1992|loc=theorem 3.1}}

Hence, by [[w:Cantor's intersection theorem#Variant in complete metric spaces|Cantor's intersection theorem]], the sequence of polygons collapses toward a point.{{Sfn|Schwartz|1992|loc=§3 Convex polygons}}

The behavior on the moduli space is very different, since the dynamics is [[w:Recurrent point|recurrent]].{{Sfn|Schwartz|2001|loc=theorem 1.1}} It is even a [[w:quasiperiodic motion|quasiperiodic motion]],{{Sfn|Ovsienko|Schwartz|Tabachnikov|2009}} as discussed in [[w:Pentagram map#Complete integrability|the section about integrability]].

=== Coordinates of the limit point ===
The limit point coordinates are found in {{Harvard citation|Glick|2020}}. They satisfy some [[w:Degree of a polynomial|degree]] 3 [[w:polynomial equation|polynomial equations]], whose coefficients are [[w:rational function|rational function]]s in the coordinates of the vertices of the starting polygon. The proof relies on the fact that the limit point must be an [[w:eigenline|eigenline]] of a certain [[w:linear operator|linear operator]] of &lt;math&gt;\mathbb R^3&lt;/math&gt;.{{sfn|Glick|2020}}

This operator was reinterpreted in {{Harvard citation|Aboud|Izosimov|2022}} as the infinitesimal monodromy of the polygon. The [[w:Pentagram map#The scaling symmetry|scaling symmetry]] is used to [[w:Deformation (mathematics)|deform]] a closed polygon &lt;math&gt;P&lt;/math&gt; into a family of twisted ones '''&lt;math&gt;(P_z)_{z\in \mathbb C^*}&lt;/math&gt;''' with monodromy &lt;math&gt;M_z&lt;/math&gt;. The infinitesimal monodromy is defined to be:{{sfn|Aboud|Izosimov|2022}}
&lt;math display=&quot;block&quot;&gt;\left.\frac{dM_z}{dz}\right|_{z=1}.&lt;/math&gt;

=== Generalization ===
The collapsing of polygons may also happen in some [[w:Pentagram map#Generalizations|generalization of the pentagram map]], when considering some specific configurations of polygons in the real plane. The coordinates of the collapse point are given by a formula analogous to the one for the original pentagram map.{{Sfn|Schwartz|2026}}

== Periodic orbits on the moduli space ==
For some configurations of closed polygons, the iterate of the pentagram map will send &lt;math&gt;P&lt;/math&gt; to a projectively equivalent polygon (up to some shift of the indices). This means that, on the moduli space, the orbit of the class of &lt;math&gt;P&lt;/math&gt; is [[w:Periodic orbit|periodic]].

===Pentagons and hexagons===
[[File:penta hexagon.svg|300px|thumb|The outward hexagon is projectively equivalent to the inward one, with respect to their labeling.]]The following two facts are proved by checking [[w:cross-ratio|cross-ratio]] equalities, so they are true for polygons in any [[w:projective plane|projective plane]] (not just the [[w:Real projective plane|real one]]).{{Sfn|Schwartz|1992|loc=§2 Pentagons and hexagons}}

The pentagram map &lt;math&gt;T&lt;/math&gt; is the identity on the moduli space of [[w:pentagon|pentagon]]s.{{Sfn|Schwartz|1992|loc=theorem 2.1}}{{Sfn|Clebsch|1871}}{{Sfn|Motzkin|1945}} The second iterate &lt;math&gt;T^2&lt;/math&gt; is the identity on the space of labeled [[w:hexagon|hexagon]]s, up to a shift of labeling (see Figure 6).{{Sfn|Schwartz|1992|loc=theorem 2.3}} This phenomenon doesn't generalize to generic polygons with at least seven sides, for which the motion is [[w:Quasiperiodic motion|quasi-periodic]].{{Sfn|Tupan|2022}}

==== Generalization ====
The result about pentagons and hexagons generalizes to some [[w:Pentagram map#Generalizations|higher pentagram maps]] in &lt;math&gt;\mathbb P ^k&lt;/math&gt;, for polygons with &lt;math&gt;k+3&lt;/math&gt; or &lt;math&gt;2k+2&lt;/math&gt; sides. The proof uses a generalization of the [[w:Gale transform|Gale transform]].{{Sfn|Dirdak|2024}}

=== Poncelet polygons ===
A polygon is said to be Poncelet{{Efn|The name comes from [[w:Jean-Victor Poncelet|Jean-Victor Poncelet]] and [[w:Poncelet porism|his porism]].{{Sfn|Izosimov|2022a|p=1085}}}} if it is [[w:Inscribed figure|inscribed]] in a [[w:Conic section|conic]] and circumscribed about another one.{{Sfn|Schwartz|2015|loc=|p=433}}{{Efn|In particular, pentagons are Poncelet since [[w:five points determine a conic|five points determine a conic]].{{Sfn|Schwartz|2015|loc=|p=433}}}} For a convex Poncelet &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;P&lt;/math&gt; lying on the [[w:real projective plane|real projective plane]], the polygon &lt;math&gt;T^2(P)&lt;/math&gt; is projectively equivalent to &lt;math&gt;P&lt;/math&gt;.{{Sfn|Schwartz|2015|loc=theorem 1.1}} In fact, when &lt;math&gt;n&lt;/math&gt; is odd, the converse is also true.{{Sfn|Izosimov|2022a|loc=corollary 1.1}}

However, this converse statement is no longer true when the polygons are considered over the [[w:complex projective plane|complex projective plane]].{{Sfn|Izosimov|2022a|loc=remark 1.3}}

==Coordinates for the moduli space==
The moduli space can be described by different [[w:Coordinate_system|coordinate systems]]. The following ones are practical to explicit the dynamic, as presented in the next section.

=== Corner coordinates ===
[[File:Corner coordinates big.svg|thumb|300x300px|The geometric construction of the points defining the corner invariants.]]
Define the [[w:cross-ratio|cross-ratio]] of four [[w:Collinearity|collinear]] points to be

: &lt;math&gt; [a,b,c,d]=\frac{(a-b)(c-d)}{(a-c)(b-d)}. &lt;/math&gt;

The corner invariants are a system of coordinates on the space of twisted polygons, constructed by taking intersections as in Figure 7.{{Sfn|Schwartz|2001|loc=figure 2}} The left and right invariants are respectively defined{{Efn|The ordering of the vertices in the cross-ratios can differ from a paper to another one, which slightly changes the formulas in the following sections.}} as the following cross-ratios:

: &lt;math&gt;x_k:=[v_{k-2},v_{k-1},\overline{v_{k-2}v_{k-1}}\cap\overline{v_{k}v_{k+1}},\overline{v_{k-2}v_{k-1}}\cap\overline{v_{k+1}v_{k+2}}],&lt;/math&gt;

: &lt;math&gt;y_k:=[\overline{v_{k+1}v_{k+2}}\cap\overline{v_{k-2}v_{k-1}}, \overline{v_{k+1}v_{k+2}}\cap\overline{v_{k-1}v_{k}},v_{k+1},v_{k+2}].&lt;/math&gt;

Since the cross-ratio is [[w:Cross-ratio#Projective geometry|projective invariant]], the sequences &lt;math&gt;(x_k)_{k \in \mathbb Z}&lt;/math&gt; and &lt;math&gt;(y_k)_{k \in \mathbb Z}&lt;/math&gt; associated to a twisted &lt;math&gt;n&lt;/math&gt;-gon are &lt;math&gt;n&lt;/math&gt; periodic.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=415}}

The corner invariants are elements of &lt;math&gt;\mathbb{P}^1\smallsetminus\{0,1,\infty\}&lt;/math&gt;, and they realize an [[w:Isomorphism_of_varieties|isomorphism of variety]] between the moduli space of twisted &lt;math&gt;n&lt;/math&gt;-gons and &lt;math&gt;(\mathbb{P}^1\smallsetminus\{0,1,\infty\})^{2n}&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=theorem 3.6}}

===ab-coordinates===

There is a second set of coordinates for the moduli space of twisted &lt;math&gt;n&lt;/math&gt;-gons defined over a [[w:Field (mathematics)|field]] &lt;math&gt;F&lt;/math&gt; satisfying &lt;math&gt;\mathrm{SL}_3(F)\cong \mathbb P\mathrm{GL}_3(F)&lt;/math&gt;,{{Sfn|Weinreich|2022|loc=remark 3.8}} and such that &lt;math&gt;n&lt;/math&gt; is not divisible by &lt;math&gt;3&lt;/math&gt;.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=section 4.1}}

The vertices &lt;math&gt;v_k&lt;/math&gt; in the [[w:projective plane|projective plane]] &lt;math&gt;\mathbb P^2(F)&lt;/math&gt; can be [[w:Lift (mathematics)|lifted]] to [[w:Vector space|vectors]] &lt;math&gt;V_k&lt;/math&gt; in the [[w:affine space|affine space]] &lt;math&gt;F^3&lt;/math&gt; so that each consecutive triple of vectors spans a [[w:parallelepiped|parallelepiped]] having [[w:determinant|determinant]] equal to &lt;math&gt;1&lt;/math&gt;. This leads to the relation{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=equation 4.1}}

: &lt;math&gt;V_{k+3} = a_k V_{k+2} + b_k V_{k+1} + V_k.&lt;/math&gt;

This bring out an analogy between twisted polygons and solutions of third order linear [[w:ordinary differential equations|ordinary differential equations]], normalized to have unit [[w:Wronskian|Wronskian]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=remark 6.6}}

They are linked to the corner coordinates by:{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=lemma 4.5}}
: &lt;math&gt;x_k=\frac{a_{k-2}}{b_{k-2}b_{k-1}},&lt;/math&gt;
: &lt;math&gt;y_k=-\frac{b_{k-1}}{a_{k-2}a_{k-1}}.&lt;/math&gt;

==Formulas on the moduli space==

===As a birational map ===
The pentagram map is a [[w:birational map|birational map]] on the moduli space, because it can be decomposed as the [[w:Function composition|composition]] of two [[w:Birational geometry|birational]] [[w:Involution (mathematics)|involutions]].{{Sfn|Schwartz|2008|loc=§1.2 The Pentagram Map}} The corner invariants change in the following way:{{Sfn|Ovsienko|Schwartz|loc=lemma 2.4|Tabachnikov|2010}}

: &lt;math&gt;x_k'=x_k\frac{1-x_{k-1} y_{k-1}}{1-x_{k+1}y_{k+1}},&lt;/math&gt;

: &lt;math&gt;y_k'=y_{k+1}\frac{1-x_{k+2} y_{k+2}}{1-x_k y_k}.&lt;/math&gt;

=== The scaling symmetry ===
The [[w:multiplicative group|multiplicative group]] &lt;math&gt;F\smallsetminus\{0\}&lt;/math&gt; [[w:One-parameter group|acts]] on the moduli space in the following way:

: &lt;math&gt;R_s\cdot(x_1,\dots,x_n,y_1,\dots,y_n)=(sx_1,\dots,sx_n,s^{-1}y_1,\dots,s^{-1}y_n),&lt;/math&gt;

where &lt;math&gt;R&lt;/math&gt; is called the scaling action and &lt;math&gt;s&lt;/math&gt; is the scaling parameter. This action commutes with the pentagram map on the moduli space (as presented in the previous formulas). This property is called the scaling symmetry, and is instrumental in proving the [[w:Pentagram map#Complete integrability|complete integrability]] of the dynamics.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=corollary 2.5}}

==Invariant structures==

===Monodromy invariants===

The monodromy invariants, introduced in {{Harvard citation|Schwartz|2008}}, are a collection of [[w:Function (mathematics)|functions]] on the [[w:moduli space|moduli space]] that are invariant under the pentagram map.{{Sfn|Schwartz|2008|loc=theorem 1.2}} The simplest example of them are 
:&lt;math&gt; O_n= x_1x_2\cdots x_{n}, \quad E_n = y_1y_2\cdots y_n. &lt;/math&gt;
The other monodromy invariants can be retrieved through different points of view: through the [[w:Pentagram map#The scaling symmetry|scaling symmetry]], as [[w:Combinatorics|combinatorial]] objects, or as some [[w:determinant|determinant]]s.{{Sfn|Schwartz|Tabachnikov|2011|loc=§2 The Monodromy Invariants}} The one involving scaling symmetry is presented here.

Let &lt;math&gt;M\in \mathrm{GL}_3&lt;/math&gt; be a [[w:Lift (mathematics)|lift]] of the monodromy of a twisted &lt;math&gt;n&lt;/math&gt;-gon. The quantities

: &lt;math&gt;\Omega_1=\frac{\operatorname{trace}^3(M)}{\det(M)}, \quad \Omega_2=\frac{\operatorname{trace}^3(M^{-1})}{\det(M^{-1})},&lt;/math&gt;

are independent of the choice of lift and are invariant under [[w:Matrix similarity|conjugation]], so they are well defined for the projective class of the polygon. They are invariant under the pentagram map, since the monodromy matrix doesn't change.{{Sfn|Schwartz|Tabachnikov|2011|loc=|p=5}} Now, the quantities

: &lt;math&gt;\tilde{\Omega}_1=O_n^2E_n\Omega_1, \quad \tilde{\Omega}_2=O_nE_n^2\Omega_2,&lt;/math&gt;

have the same properties, but turn out to be polynomials in the corner invariants.{{Efn|Some papers consider the cube roots of this functions, but it doesn't change the following definitions of the monodromy invariants.}} They can be written as{{Sfn|Schwartz|Tabachnikov|2011|loc=|p=5}}

: &lt;math&gt;
\tilde{\Omega}_1=\biggl(\sum_{k=0}^{\lfloor n/2\rfloor}O_k\biggr)^3, \quad
\tilde{\Omega}_2=\biggl(\sum_{k=0}^{\lfloor n/2\rfloor}E_k\biggr)^3,
&lt;/math&gt;

where each &lt;math&gt;O_k&lt;/math&gt; and &lt;math&gt;E_k&lt;/math&gt; are [[w:homogeneous polynomial|homogeneous polynomial]]s respectively of weight &lt;math&gt;k&lt;/math&gt; and &lt;math&gt;-k&lt;/math&gt;,{{Sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=11}} meaning they change under the [[w:Pentagram map#The scaling symmetry|rescaling action]] on variables by{{Sfn|Schwartz|Tabachnikov|2011|p=5}}

: &lt;math&gt; R_s(O_k)= s^k O_k, \quad R_s(E_k)= s^{-k} E_k. &lt;/math&gt;

The quantities &lt;math&gt;O_1,\dots,O_{\lfloor n/2 \rfloor},O_n, E_1,\dots,E_{\lfloor n/2 \rfloor},E_n,&lt;/math&gt; are unchanged by the dynamics, and are called the monodromy invariants. Moreover, they are [[w:algebraically independent|algebraically independent]].{{Sfn|Schwartz|2008|loc=theorem 1.2}}

==== Polygons on conics ====
Whenever &lt;math&gt;P&lt;/math&gt; is [[w:Inscribed figure|inscribed]] on a [[w:conic section|conic section]], one has &lt;math&gt;O_k(P)=E_k(P)&lt;/math&gt; for all &lt;math&gt;k&lt;/math&gt;.{{Sfn|Schwartz|Tabachnikov|2011|loc=theorem 1.1}} Moreover, if &lt;math&gt;P&lt;/math&gt; is circumscribed about another conic,{{Efn|See the paragraph about [[w:Pentagram map#Poncelet polygons|Poncelet polygons]].}} then its monodromy invariants are characterized by the pair of conics.{{Sfn|Schwartz|2015|loc=theorem 1.2}} For such odd-gons, the translation on the [[w:Jacobian variety|Jacobian variety]]{{Efn|See the paragraph about [[w:Pentagram map#Algebro-geometric integrability|algebraic integrability]].}} is restricted to the [[w:Prym variety|Prym variety]] (which is a half-dimensional torus in the Jacobian).{{Sfn|Izosimov|2016|loc=theorem 1.3}}

===Poisson bracket===

An invariant [[w:Poisson bracket|Poisson bracket]] on the space of twisted polygons was found in {{Harvard citation|Ovsienko|Schwartz|Tabachnikov|2010}}. The monodromy invariants [[w:Poisson bracket#Constants of motion|commute]] with respect to it:
&lt;math display=&quot;block&quot;&gt; \{O_i,O_j\}=\{O_i,E_j\}=\{E_i,E_j\}=0 &lt;/math&gt;for all &lt;math&gt;i,j&lt;/math&gt;.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}}

The Poisson bracket is defined in terms of the corner coordinates by:
&lt;math display=&quot;block&quot;&gt; \begin{align}
 \{x_i,x_{i\pm1}\} &amp;= \mp x_i x_{i+1}, \\
 \{y_i,y_{i\pm 1}\} &amp;= \mp y_i y_{i+1}, \\
 \{x_i,x_j\} &amp;= \{y_i,y_j\} = \{x_i,y_j\} = 0
\end{align}&lt;/math&gt;for all other &lt;math&gt; i,j.&lt;/math&gt;{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=equation 2.16}}

=== The spectral curve ===

Let &lt;math&gt;\zeta&lt;/math&gt; be an element of the [[w:multiplicative group|multiplicative group]] and &lt;math&gt;P_\zeta&lt;/math&gt; be the polygon obtained by applying the [[w:Pentagram map#The scaling symmetry|rescaling action]] &lt;math&gt;R_\zeta&lt;/math&gt; on &lt;math&gt;P&lt;/math&gt;.  A [[w:Lax matrix|Lax matrix]] &lt;math&gt;\hat{T}(\zeta) \in \mathrm{GL}_3&lt;/math&gt; is a lift of the monodromy of &lt;math&gt;P_\zeta&lt;/math&gt; satisfying a [[w:Lax pair#Zero-curvature equation|zero-curvature equation]].{{Sfn|Weinreich|2022|loc=§5 The Lax representation}} Then, the spectral function is the [[w:Bivariate polynomial|bivariate]] [[w:characteristic polynomial|characteristic polynomial]]
&lt;math display=&quot;block&quot;&gt; Q(\lambda,\zeta) := \det(\lambda\operatorname{Id}-\hat{T}(\zeta)),&lt;/math&gt;or some renormalization of it. The [[w:spectral curve|spectral curve]] is the [[w:Projective variety#projective completion|projective completion]] of the [[w:Algebraic curve|affine curve]] defined by the equation &lt;math&gt;Q(\lambda,\zeta)=0&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=§6. The geometry of the spectral curve}} It is invariant under the pentagram map, and the monodromy invariants appear as the [[w:coefficient|coefficient]]s of &lt;math&gt;Q&lt;/math&gt;.{{Sfn|Soloviev|2013|loc=theorem 6.4}} Its [[w:geometric genus|geometric genus]] is &lt;math&gt;n-1&lt;/math&gt; if &lt;math&gt;n&lt;/math&gt; is odd, and &lt;math&gt;n-2&lt;/math&gt; if &lt;math&gt;n&lt;/math&gt; is even.{{Sfn|Weinreich|2022|p=|loc=theorem 6.4}}

It was first introduced in {{Harv|Soloviev|2013|ps=|p=}} for his proof of [[w:Pentagram map#Algebro-geometric integrability|algebro-geometric integrability]].{{sfn|Soloviev|2013}}

==Complete integrability==
The pentagram map on the moduli space has been proved to be a [[w:completely integrable|completely integrable]] [[w:discrete dynamical system|discrete dynamical system]], both in the [[w:Integrable system#Hamiltonian systems and Liouville integrability|Arnold-Liouville]]{{Efn|Over the [[w:real number|real number]]s.}} and the [[w:Integrable system#Complete integrability over the complex numbers|algebro-geometric]]{{Efn|Over [[w:algebraically closed field|algebraically closed field]]s of [[w:Characteristic (algebra)|characteristic]] different from 2.}} senses. In any case, this means that the moduli space is [[w:almost everywhere|almost everywhere]] [[w:Foliation|foliated]] by [[w:Torus#Flat torus|flat tori]] (or in the algebraic setting, [[w:Abelian variety|Abelian varieties]]), where the motion is a [[w:Translation (geometry)|translation]]. This [[w:Generic property|generically]] induces a [[w:quasiperiodic motion|quasiperiodic motion]] on the corresponding torus.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2009}}

===Arnold–Liouville integrability===

The proof of the integrability of the pentagram map on a real twisted polygon was achieved in {{Harvard citation|Ovsienko|Schwartz|Tabachnikov|2010}}. This is done by noticing that the monodromy invariants &lt;math&gt;O_n&lt;/math&gt; and &lt;math&gt;E_n&lt;/math&gt; are [[w:Casimir invariant|Casimir invariant]]s for the bracket, meaning (in this context) that&lt;math display=&quot;block&quot;&gt; \{O_n,f\}=\{E_n,f\} = 0 &lt;/math&gt;for all functions &lt;math&gt;f&lt;/math&gt;.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}} When &lt;math&gt;n&lt;/math&gt; is even, this is also true for the monodromy invariants &lt;math&gt;O_{\lfloor n/2 \rfloor }&lt;/math&gt; and &lt;math&gt;E_{\lfloor n/2 \rfloor }&lt;/math&gt;.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}}

This allows to consider the Casimir [[w:level set|level set]], where each Casimir has a specified value. Because of [[w:Sard's theorem|Sard's theorem]], any generic level set is a [[w:smooth manifold|smooth manifold]].{{Sfn|Schwartz|2017|p=44}} They form a [[w:foliation|foliation]] in [[w:Poisson manifold#Symplectic leaves|symplectic leaves]], on which the Poisson bracket gives rise to a [[w:symplectic form|symplectic form]].{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=corollary 2.13}}

Each of these symplectic leaves has an iso-monodromy [[w:foliation|foliation]], namely, a decomposition into the common level sets of the remaining monodromy functions. By using again [[w:Sard's theorem|Sard's theorem]], they are generically [[w:Symplectic manifold#Lagrangian submanifolds|Lagrangian manifolds]].{{Sfn|Schwartz|2017|p=45}} Moreover, they are compact.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=§3.3. Compactness of the level sets}} Since the monodromy invariants Poisson-commute and there are enough of them, the discrete [[w:Liouville–Arnold theorem|Liouville–Arnold theorem]] can be applied to prove that the level sets are [[w:Torus#Flat torus|flat tori]] over which the dynamics is a translation.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=412}}
===Algebro-geometric integrability===

In {{Harvard citation|Soloviev|2013}}, it was shown that the pentagram map admits a [[w:Lax representation|Lax representation]] with a spectral parameter, which allows to prove its algebro-geometric integrability. This means that the space of polygons (either twisted or closed) is parametrized by its spectral data, consisting of [[w:Pentagram map#The spectral curve|its spectral curve]], with marked points and a [[w:Divisor (algebraic geometry)|divisor]] given by a [[w:Floquet theory|Floquet]]–[[w:Bloch's theorem|Bloch]] equation. This gives an embedding to the [[w:Jacobian variety|Jacobian variety]] through the [[w:Abel–Jacobi map|Abel–Jacobi map]], where the motion is expressed in terms of translation.{{sfn|Soloviev|2013|loc=theorems A, B and C}} The previously defined Poisson bracket is also retrieved.{{sfn|Soloviev|2013|loc=theorem D}} 

This integrability was generalized in {{Harvard citation|Weinreich|2022}} from the field of [[w:complex number|complex number]]s to any [[w:algebraically closed field|algebraically closed field]] of [[w:Characteristic (algebra)|characteristic]] different from 2. The translation on a torus is replaced by a translation on an [[w:Abelian variety|Abelian variety]] (in fact, a Jacobian variety again).{{sfn|Weinreich|2022|loc=theorem 1.4}}

=== Dimension of the invariant manifold ===

For twisted &lt;math&gt;n&lt;/math&gt;-gons, the [[w:dimension|dimension]] of the invariant tori (or Jacobian varieties) is{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=421}}

: &lt;math&gt;\begin{cases} 
n-1 &amp; \text{when }n \text{ is odd,}\\
n-2 &amp; \text{when }n \text{ is even.}
\end{cases}&lt;/math&gt;

Moreover, when &lt;math&gt;n&lt;/math&gt; is even, there are two isomorphic Jacobians on which the iterates of the pentagram map alternate. But on each of them, the second iterate is a translation.{{Sfn|Weinreich|2022|loc=theorem 1.4}}

=== For closed polygons ===
There is no Poisson structure on the space of closed polygons.{{Sfn|Soloviev|2013|loc=corollary 4.1}} Nevertheless, the one from twisted polygons can be used to prove integrability.{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=2153}}

Algebro-geometric integrability holds for closed polygons in a same manner as for the twisted ones.{{Sfn|Soloviev|2013|loc=theorem C}} However, Arnold-Liouville integrability is proved for real closed polygons only when they are convex. This is done by restricting the [[w:Hamiltonian vector field|Hamiltonian vector field]]s of monodromy functions to smaller dimensional tori, and showing that enough of them are still independent.{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|loc=corollary 1.1}}

In both situation, the dimension of the invariant manifolds decreases by &lt;math&gt;3&lt;/math&gt; for closed &lt;math&gt;n&lt;/math&gt;-gons (compared to the twisted case), and is equal to{{Sfn|Soloviev|2013|loc=theorem C}}{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|loc=theorem 1}}
:&lt;math&gt;\begin{cases} 
n-4 &amp; \text{when }n \text{ is odd,}\\
n-5 &amp; \text{when }n \text{ is even.}
\end{cases}&lt;/math&gt;

==Connections to other topics==

===The Boussinesq equation===

The continuous limit of a convex polygon is a parametrized convex curve in the plane. When the time parameter is suitably chosen, the [[w:Discretization|continuous limit]] of the pentagram map is the classical [[w:Boussinesq approximation (water waves)|Boussinesq equation]]. This equation is a classical example of an [[w:integrable|integrable]] [[w:partial differential equation|partial differential equation]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 5}}

Here is a description of the geometric action of the Boussinesq equation. Given a [[w:locally convex|locally convex]] curve &lt;math&gt; C:\mathbb R\to \mathbb R^2 &lt;/math&gt; and real numbers &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;t&lt;/math&gt;, consider the [[w:chord (geometry)|chord]] connecting &lt;math&gt; C(x-t) &lt;/math&gt; to &lt;math&gt; C(x+t) &lt;/math&gt;. The [[w:Envelope (mathematics)|envelope]] of all these chords is a new curve &lt;math&gt; C_t(x) &lt;/math&gt;. When &lt;math&gt;t&lt;/math&gt; is extremely small, the curve &lt;math&gt; C_t(x) &lt;/math&gt; is a good model for the time &lt;math&gt;t&lt;/math&gt; evolution of the original curve &lt;math&gt; C_0(x) &lt;/math&gt; under the Boussinesq equation. This construction is also similar to the pentagram map. Moreover, the pentagram invariant bracket is a discretization of a well known invariant Poisson bracket associated to the Boussinesq equation.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=§6.4 Discretization}}

===Cluster algebras===

The pentagram map{{Sfn|Glick|2011}} and some of its generalizations{{Sfn|Gekhtman|Shapiro|Tabachnikov|Vainshtein|2012}}{{Sfn|Glick|Pylyavskyy|2016}} are identified as special cases of discrete dynamical systems powered by [[w:cluster algebra|cluster algebra]]. Using the results from {{Harvard citation|Goncharov|Kenyon|2013}}, this provides a link with the [[w:Poisson–Lie group|Poisson–Lie group]]s, [[w:dimer model|dimer model]]s and other so-called cluster-integrable systems.{{Sfn|Fock|Marshakov|2016}}{{Sfn|Izosimov|2022c}} These methods allow to retrieve the Poisson-bracket and Hamiltonians used to prove complete integrability{{Sfn|Affolter|George|Ramassamy|2025|loc=§5 The pentagram map}} and provide [[w:Lax representation|Lax representation]]s.{{Sfn|Izosimov|2022b}}

=== Singularity theory ===
The pentagram map exhibit a property called singularity confinement, which is typical from [[w:integrable system|integrable system]]s.{{Sfn|Grammaticos|Ramani|Papageorgiou|1991}} It states that if a polygon &lt;math&gt;P&lt;/math&gt; is [[w:Singular point of an algebraic variety|singular]] for the pentagram map &lt;math&gt;T&lt;/math&gt;, then there exists an integer &lt;math&gt;m&lt;/math&gt; such that &lt;math&gt;P&lt;/math&gt; not singular for the iterate map &lt;math&gt;T^m&lt;/math&gt;.{{Sfn|Glick|2012}}

Moreover, the pentagram map (along with some of its generalizations and other discrete dynamical systems) exhibit the Devron property.{{Efn|The name comes from an episode of [[w:Star Trek|Star Trek]].{{Sfn|Glick|2015|loc=§1 Introduction}}}} This means that if a polygon &lt;math&gt;P&lt;/math&gt; is singular for some iterate of the pentagram map &lt;math&gt;T^m&lt;/math&gt;, then it will also be singular for some iterate of the inverse map &lt;math&gt;T^{-m'}&lt;/math&gt;.{{Sfn|Glick|2015}}

== Generalizations ==
The definition of twisted polygons still makes sense in any [[w:projective space|projective space]] &lt;math&gt;\mathbb P^d&lt;/math&gt;, under the action of the [[w:Projective linear group|projective group]] &lt;math&gt;\mathbb P \mathrm{GL}_{d+1}&lt;/math&gt;. The pentagram map can be generalized in many ways, and some of them are presented here. Not all of them are integrable.{{Sfn|Khesin|Soloviev|2015|}} Some are [[w:discretization|discretization]]s of [[w:PDEs|PDEs]] from the [[w:KdV hierarchy|KdV hierarchy]], seen as higher dimensional version of [[w:Boussinesq approximation (water waves)|Boussinesq]] or [[w:Kadomtsev–Petviashvili equation|KP]] equations.{{Sfn|Marí-Beffa|2012}}{{Sfn|Wang|2023}} The description of all generalized pentagram maps in terms of [[w:cluster algebra|cluster algebra]]s is still an open question.{{Sfn|Gekhtman|Izosimov|2025|p=14}}

=== Polygons in general positions ===
Let &lt;math&gt;d \geq 2&lt;/math&gt; and &lt;math&gt;P&lt;/math&gt; be a twisted polygon of &lt;math&gt;\mathbb P^d&lt;/math&gt; in [[w:general position|general position]].

==== Short diagonal pentagram maps ====
The &lt;math&gt;k&lt;/math&gt;-th ''short diagonal hyperplane'' &lt;math&gt;H_k^{sh}&lt;/math&gt; is uniquely defined by passing through the vertices &lt;math&gt;v_k,v_{k+2},\dots,v_{k+2d-2}&lt;/math&gt;. [[w:Generic property#In algebraic geometry|Generically]], the intersection of &lt;math&gt;d&lt;/math&gt; consecutive hyperplanes uniquely defines a new point

: &lt;math&gt;T_{sh}v_k:=H_k^{sh}\cap H_{k+1}^{sh}\cap \dots \cap H_{k+d-1}^{sh}.&lt;/math&gt;

Doing this for every vertex defines a new twisted polygon. This map, denoted by &lt;math&gt;T_{sh}&lt;/math&gt;, is again projectively equivariant.{{Sfn|Khesin|Soloviev|2013}}

==== Generalized pentagram maps ====
The previous procedure can be generalized. Let &lt;math&gt;I=(i_1,\dots,i_{d-1}),~J=(j_1,\dots,j_{d-1})&lt;/math&gt; be two sets of integers, respectively called the jump tuple and the intersection tuple. Define the &lt;math&gt;k&lt;/math&gt;-th hyperplane &lt;math&gt;H_k^I&lt;/math&gt; to be passing through the vertices &lt;math&gt;v_k,v_{k+i_1},\dots,v_{k+i_1+\dots+i_{d-1}}&lt;/math&gt;. A new point is given by the intersection

: &lt;math&gt;T_{I,J}v_k:=H_k^I \cap H_{k+j_1}^I \cap \dots \cap H_{k+j_1+\dots +j_{d-1}}^I.&lt;/math&gt;

The map &lt;math&gt;T_{I,J}&lt;/math&gt; is called a generalized pentagram map.{{Sfn|Khesin|Soloviev|2015a}} The original pentagram map is recovered by considering&lt;math&gt;d=2,~I=(2),~J=(1)&lt;/math&gt;.

Integrability can be numerically tested by picking a random polygon &lt;math&gt;P&lt;/math&gt; with [[wikipedia:Rational_point|rational coordinates]] and studying the growth rate of the [[wikipedia:Height_function|height]] of its iterates. This is called the [[wikipedia:Integrable_system#Diophantine_integrability|diophantine integrability]] test, and some generalized pentagram maps don't seem to pass it.{{Sfn|Khesin|Soloviev|2015a|loc=§5 and §6}} However, it is conjectured that the maps &lt;math&gt;T_{I,I}&lt;/math&gt; are integrable for any &lt;math&gt;I&lt;/math&gt;.{{Sfn|Bolsinov|Matveev|Miranda|Tabachnikov|2018|loc=conjecture 4.13 (B. Khesin, F. Soloviev)}}

Some of these maps are [[w:discretization|discretization]]s of higher dimensional counterpart of the [[w:Boussinesq approximation (water waves)|Boussinesq equation]] in the [[w:KdV hierarchy|KdV hierarchy]].{{Sfn|Khesin|Soloviev|2015b|loc=theorem 4.1}}{{Sfn|Izosimov|2022b|loc=theorem 4.1}}

==== Dented pentagram maps ====
Fix an integer &lt;math&gt;m\in \{1,\dots ,d-1\}&lt;/math&gt;. Consider the jump tuple &lt;math&gt;I_m:=(1,\dots,1,2,1,\dots,1)&lt;/math&gt;, where the &lt;math&gt;2&lt;/math&gt; is at the &lt;math&gt;m&lt;/math&gt;-th place, and the intersection tuple &lt;math&gt;J:=(1,\dots,1)&lt;/math&gt;. The dented pentagram map is &lt;math&gt;T_m :=T_{I_m,J}&lt;/math&gt;. They are proved to be integrable.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 2.14}}

For an integer &lt;math&gt;p \geq 2&lt;/math&gt;, the deep dented pentagram map (of depth &lt;math&gt;p&lt;/math&gt;) &lt;math&gt;T_m^p&lt;/math&gt; is the same map as before, but the number &lt;math&gt;2&lt;/math&gt; in the definition of &lt;math&gt;I_m&lt;/math&gt; is replaced by &lt;math&gt;p&lt;/math&gt;. This kind of pentagram maps are again integrable.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 6.2}}

=== Corrugated polygons ===
A twisted polygon &lt;math&gt;P&lt;/math&gt; lying in &lt;math&gt;\mathbb P^d&lt;/math&gt; is said to be corrugated if for any &lt;math&gt;k\in \mathbb Z&lt;/math&gt;, the vertices &lt;math&gt;v_k,v_{k+1},v_{k+d},v_{k+d+1}&lt;/math&gt; span a projective two-dimensional plane. Such polygons are not in [[w:general position|general position]]. A new point is defined by

: &lt;math&gt;T_\text{cor}v_k:=\overline{v_k v_{k+d}}\cap \overline{v_{k+1} v_{k+d+1}}.&lt;/math&gt;

The map &lt;math&gt;T_\text{cor}&lt;/math&gt; yields a new corrugated polygon. They are [[w:Integrable system#Hamiltonian systems and Liouville integrability|completely Liouville-integrable]].{{Sfn|Gekhtman|Shapiro|Tabachnikov|Vainshtein|2012|loc=theorem 4.4}}

In fact, they can be retrieved as some dented pentagram map applied on corrugated polygons.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 5.3}}

=== Grassmannian polygons ===
Let &lt;math&gt;d \geq 3, m \geq 1&lt;/math&gt; be integers. The pentagram map can also be generalized to the [[w:Grassmannian|Grassmannian]] space &lt;math&gt;\mathrm{Gr}(m,md)&lt;/math&gt;, which consists of &lt;math&gt;m&lt;/math&gt;-[[w:Dimension (vector space)|dimensional]] [[w:linear subspace|linear subspace]]s of an &lt;math&gt;md&lt;/math&gt;-dimensional [[w:vector space|vector space]]. When &lt;math&gt;m=1&lt;/math&gt;, the linear subspaces are [[w:Vector space#vector line|lines]], which retrieves the definition of [[w:projective space|projective space]]s &lt;math&gt;\mathbb P^d&lt;/math&gt;.{{Sfn|Felipe|Marí-Beffa|2019|loc=§2 definitions and notations}}

A point &lt;math&gt;v\in\operatorname{Gr}(m,md)&lt;/math&gt; is represented by an &lt;math&gt;md \times m&lt;/math&gt; matrix &lt;math&gt;X_v&lt;/math&gt; such that its columns form a [[w:Basis (linear algebra)|basis]] of &lt;math&gt;v&lt;/math&gt;. Consider the [[w:Group action|action]] of the [[w:general linear group|general linear group]] &lt;math&gt;\mathrm{GL}_{md}&lt;/math&gt; by multiplication on the left of &lt;math&gt;X_v&lt;/math&gt;. This defines an action on the Grassmannian, even though it is not [[w:Faithful action|faithful]].{{Efn|Because there can be many lifts for &lt;math&gt;v&lt;/math&gt;, and because some matrices act trivially.}} Hence, the polygons of &lt;math&gt;\mathrm{Gr}(m,md)&lt;/math&gt; and their moduli spaces are defined as before, after the change of underlying group.{{Sfn|Felipe|Marí-Beffa|2019|loc=§2 definitions and notations}}

Depending on the parity of &lt;math&gt;d&lt;/math&gt;, one can define linear subspaces spanned by some &lt;math&gt;X_{v_k}&lt;/math&gt;'s such that taking their intersection generically defines a new point &lt;math&gt;v\in\mathrm{Gr}(m,md)&lt;/math&gt;.{{Sfn|Felipe|Marí-Beffa|2019|loc=sections 4 and 5}} This generalization of the pentagram map is integrable in a [[w:noncommutative|noncommutative]] sense.{{Sfn|Ovenhouse|2020}}

=== Over rings ===
The pentagram map admits a generalization by considering [[w:Projective space#Generalizations|projective planes]] over [[w:stably finite ring|stably finite ring]]s, instead of [[w:Field (mathematics)|field]]s. In particular, this retrieves the pentagram map over Grassmannians. Again, it admits a [[w:Lax representation|Lax representation]].{{Sfn|Hand|Izosimov|2025}}

== References ==
{{reflist|25em}}

===Notes===
{{notelist}}

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      <text bytes="59806" sha1="pf12af3nawdr9s9w5l066oq13egy26u" xml:space="preserve">{{Article info
| last1 = Stiegler
| orcid1 = 0009-0001-5789-6923
| first1 = Jean-Baptiste
| affiliation1 = Université Paris-Saclay
| correspondence1 = jean-baptiste.stiegler@universite-paris-saclay.fr
| journal = WikiJournal of Science
| et_al = true
| w1 = Pentagram map
| from w1 = true
| keywords = Pentagram map, Dynamical system, Projective geometry, Moduli space, Integrable systems
| license = CC-BY-SA 4.0
| submitted = 2025-12-08
| abstract = In [[w:mathematics|mathematics]], the '''pentagram map''' is a [[w:Dynamical system#Discrete dynamical system|discrete dynamical system]] acting on [[w:polygons|polygons]] in the [[w:projective plane|projective plane]]. It defines a new polygon whose vertices are obtained as the intersection points of the shortest [[w:Diagonal|diagonals]] of the initial polygon. This is a [[w:Projective linear group|projectively]] [[w:Equivariant map|equivariant]] procedure, hence it [[w:Quotient space (topology)|descends]] to the [[w:moduli space|moduli space]] of polygons and defines another dynamical system (which is also referred to as the pentagram map). It was first introduced by [[w:Richard Schwartz (mathematician)|Richard Schwartz]] in 1992.{{Sfn|Schwartz|1992}}

The pentagram map on the moduli space is famous for its [[w:Completely integrable|complete integrability]] and its link with [[w:cluster algebra|cluster algebras]].{{sfn|Gekhtman|Izosimov|2025|p=14}}

It admits many generalizations in [[w:Projective space|projective spaces]] and other settings.
}}
== Introduction ==

=== Informal definition ===

==== On polygons ====
[[File:Pentagram pentagon nolabel big.svg|alt=|thumb|300x300px|The pentagram map applied on a [[w:Convex set|convex]] [[w:pentagon|pentagon]].]]
Initially, the pentagram map was defined for [[w:convex polygon|convex polygon]]s (with at least five sides) on the [[w:euclidean plane|Euclidean plane]]. Given such a polygon &lt;math&gt;P&lt;/math&gt; with &lt;math&gt;n&lt;/math&gt; sides, one can draw the &quot;shortest [[w:diagonal|diagonal]]s&quot;, meaning the [[w:Line segment|segments]] whose endpoints are a [[w:Vertex (geometry)|vertex]] and one of its second neighbors (as in Figure 1). The intersections of the shortest diagonals are then taken as the vertices of a new &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;T(P)&lt;/math&gt;; this new polygon is the output of the pentagram map.{{Sfn|Berger|2005}}

The same construction can be done on [[w:Concave polygon|non-convex polygons]], but there are several complications. First, some consecutive short diagonals may not intersect, so one must extend the segments to [[w:Line (geometry)|lines]]. Second, the image &lt;math&gt;T(P)&lt;/math&gt; can fail to be a new &lt;math&gt;n&lt;/math&gt;-gon because some consecutive vertices could coincide. However, this [[w:Generic property|generically]] doesn't happen.{{Sfn|Ovsienko|Schwartz|Tabachnikov|p=411|2009}} Finally, it is possible that two diagonals are [[w:Parallel (geometry)|parallel]] and don't intersect on the [[w:euclidean plane|Euclidean plane]]. This is resolved by extending the Euclidean plane to the [[w:real projective plane|real projective plane]] by the addition of a [[w:line at infinity|line at infinity]], where the [[w:Vanishing point|intersection point]] lies (see Figure 3). Hence, the pentagram map is defined for generic polygons in the real projective plane.{{Sfn|Berger|2005|p=25}}

More generally, the construction of the pentagram map is well defined whenever the concepts of lines and their intersections make sense. This is encompassed by the notion of a general [[w:projective plane|projective plane]], of which the real projective plane is one example; but the pentagram map can also be considered over other [[w:Field (mathematics)|fields]], for instance the [[w:complex number|complex number]]s, which give the [[w:complex projective plane|complex projective plane]].{{Sfn|Weinreich|2022|loc=§3.1.1}}

==== On the moduli space of polygons ====
Since the pentagram map is constructed by drawing lines and marking their intersections, it [[w:Commutative property|commutes]] with any transformation that sends lines to lines. Such maps are called [[w:projective transformations|projective transformations]]. This allows to identify polygons [[w:up to|up to]] [[w:Perspectivity#Projectivity|projective transformations]]. This identification gives a [[w:Quotient space (topology)|quotient space]] (technically called a [[w:moduli space|moduli space]]) of [[w:Equivalence class|classes]] of polygons. 

The pentagram map on polygons induces another dynamical system on the moduli space,{{Sfn|Schwartz|1992|loc=§1 Projective geometry}} whose behavior differs quite a lot from the initial one.{{Efn|Compare the paragraph about the [[w:Pentagram map#Collapsing of convex polygons|collapsing of convex polygons]] and the one about [[w:Pentagram map#Complete integrability|complete integrability]].}} The dynamic is trivial for the classes of pentagons and hexagons, but this stops to be the case for polygons with more vertices.{{Efn|See the paragraph about [[w:Pentagram_map#Pentagons_and_hexagons|pentagons and hexagons]].}}

=== Historical elements ===
The pentagram map for general polygons was introduced in {{Harvard citation|Schwartz|1992}}, but the simplest case is the one of [[w:pentagons|pentagons]], hence the name &quot;[[w:pentagram|pentagram]]&quot;.{{Sfn|Marí-Beffa|2014|p=1}} Their study goes back to {{Harvard citation|Clebsch|1871}},{{Sfn|Izosimov|2022a|p=1085}} {{Harvard citation|Kasner|1928}}{{Sfn|Tabachnikov|2019}} and {{Harvard citation|Motzkin|1945}}.{{Sfn|Schwartz|2013|p=1}}

The pentagram map interacts with some classical configuration theorems of [[w:projective geometry|projective geometry]]. It provides results analogous to the ones of [[w:Pascal's theorem|Pascal's theorem]] and [[wikipedia:Brianchon's_theorem|Brianchon's theorem]].{{Sfn|Schwartz|Tabachnikov|2010}} Some specific configurations make [[w:Desargues' theorem|Desargues's theorem]] and [[w:Poncelet's porism|Poncelet's porism]] appear.{{Sfn|Berger|2005|loc=§4 and §5}}{{Efn|See the paragraph about [[w:Pentagram map#Poncelet polygons|Poncelet polygons]].}}

==Definitions and first properties==

=== Definition of the map ===
[[File:Pentagram pentagon label big.svg|alt=|thumb|300x300px|The pentagram map on a convex pentagon, with vertices labeled.]]
[[File:Pentagram on nonconvex pentagon.svg|alt=|thumb|300x300px|The pentagram map applied on a [[w:Self-intersecting polygon|self-intersecting]] (in particular, non-convex) pentagon. The vertex &lt;math&gt;w_2&lt;/math&gt; is on the [[w:line at infinity|line at infinity]], because it is the [[w:Vanishing point|intersection of two parallel lines]].]]
Let &lt;math&gt;n\geq 5&lt;/math&gt; be an integer. A polygon &lt;math&gt;P&lt;/math&gt; with &lt;math&gt;n&lt;/math&gt; sides, or &lt;math&gt;n&lt;/math&gt;-gon, is a tuple of [[w:Vertex (geometry)|vertices]] &lt;math&gt;(v_1,\dots,v_n)&lt;/math&gt; lying in some [[w:projective plane|projective plane]] &lt;math&gt;\mathbb P ^2&lt;/math&gt;,{{Efn|In the following, the figures represent polygons on the real plane, where the intuition is easier to grasp.}} where the indices are understood [[w:Modular arithmetic|modulo]] &lt;math&gt;n&lt;/math&gt;. The [[w:Dimension of an algebraic variety|dimension]] of the space of &lt;math&gt;n&lt;/math&gt;-gons is &lt;math&gt;2n&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=definition 1.1}}

Suppose that the vertices are in sufficiently [[w:general position|general position]], meaning that no consecutive triple of points are [[w:Collinearity|collinear]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=1}} Taking the intersection of two consecutive &quot;shortest&quot; [[w:diagonal|diagonal]]s{{Efn|Meaning the line between a vertex &lt;math&gt;v_k&lt;/math&gt; and a &quot;second neighbour&quot; &lt;math&gt;v_{k\pm 2}&lt;/math&gt;.}} defines a new point&lt;math display=&quot;block&quot;&gt; w_k := \overline{v_{k-1} v_{k+1}} \cap \overline{v_{k} v_{k+2}}. &lt;/math&gt;This procedure defines a new &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;T(P)=(w_1,\dots,w_n)&lt;/math&gt;, as in Figure 2.{{Sfn|Schwartz|1992|p=71}}

The labeling of the indices of &lt;math&gt;T(P)&lt;/math&gt; is not [[w:canonical|canonical]]. In most papers, a choice is made at the beginning of the paper and the formulas are tuned accordingly.{{Sfn|Izosimov|2016|loc=remark 1.5}}

The pentagram map on polygons is a [[w:birational map|birational map]] &lt;math&gt;T:(\mathbb P^2)^n&lt;/math&gt;{{nowrap|{{font|size=145%|⇢}}}}&lt;math&gt;(\mathbb P^2)^n&lt;/math&gt;. Indeed, each [[w:Homogeneous coordinates|coordinate]] of &lt;math&gt;w_k&lt;/math&gt; is given as a [[w:rational function|rational function]] of the coordinates of &lt;math&gt;v_{k-1},\dots,v_{k+2}&lt;/math&gt;, since it is defined as the intersection of lines passing by them. Moreover, the [[w:inverse map|inverse map]] is given by taking the intersections &lt;math&gt;\overline{w_{k-2} w_{k-1}} \cap \overline{w_{k} w_{k+1}} &lt;/math&gt;, which is rational for the same reason.{{Sfnp|Weinreich|2022|loc=definition 1.2}}

=== Moduli space ===
The pentagram map is defined by taking [[w:Line (geometry)|lines]] and intersections of them. The biggest [[w:Group (mathematics)|group]] which maps lines to lines is the one of [[w:projective transformations|projective transformations]] &lt;math&gt;\mathbb P \mathrm{GL}_{3}&lt;/math&gt;. Such a transformation &lt;math&gt;M&lt;/math&gt; [[w:Group action|acts]] on a polygon &lt;math&gt;P&lt;/math&gt; by sending it to &lt;math&gt;M \cdot P:=(Mv_1,\dots,Mv_n)&lt;/math&gt;. The pentagram map [[w:Commutative property|commutes]] with this action, and thereby induces another [[w:dynamical system|dynamical system]] on the [[w:moduli space|moduli space]] of projective [[w:equivalence classes|equivalence classes]] of polygons. Its [[w:Dimension of an algebraic variety|dimension]] is &lt;math&gt;2n-8&lt;/math&gt;.{{Sfn|Schwartz|1992|loc=§1 Projective geometry}}

===Twisted polygons===
[[File:Twisted heptagon.svg|alt=|thumb|300x300px|An example of twisted [[w:heptagon|heptagon]] on the real plane.]]
The pentagram map naturally generalizes on the larger space of twisted polygons (see example in Figure 4). For any integer &lt;math&gt;n\geq5&lt;/math&gt;, a twisted &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;P&lt;/math&gt; is the data of:

* a [[w:Sequence#Indexing|bi-infinite sequence]] of points &lt;math&gt;(v_k)_{k\in\mathbb Z}&lt;/math&gt; in the projective plane (called the vertices),
* a [[w:projective transformation|projective transformation]] &lt;math&gt;M \in \mathbb P \mathrm{GL}_3&lt;/math&gt; (called the [[w:monodromy|monodromy]]),

such that for any &lt;math&gt;k \in \mathbb Z&lt;/math&gt;, the property &lt;math&gt;v_{k+n}=Mv_k&lt;/math&gt; is satisfied. The dimension of the space of twisted &lt;math&gt;n&lt;/math&gt;-gons is &lt;math&gt;2n+8&lt;/math&gt;.{{Sfn|Schwartz|2008}}

When &lt;math&gt;M=\mathrm{Id}&lt;/math&gt;, this gives back the initial definition of polygons (which are said to be closed). The space of closed &lt;math&gt;n&lt;/math&gt;-gons is of [[w:codimension|codimension]] &lt;math&gt;8&lt;/math&gt; in the space of twisted ones.{{Sfn|Soloviev|2013|p=2816}}

The action of projective transformations over the space of closed polygons generalizes to the space of twisted ones (the monodromy is changed by [[w:Matrix similarity|conjugation]]). This provides again a moduli space, of dimension &lt;math&gt;2n&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=definition 1.3}}

== Collapsing of convex polygons ==

=== Exponential shrinking ===
[[File:Pentagram map convex heptagon iterate.svg|alt=|thumb|300x300px|The pentagram map iterated on a convex [[w:heptagon|heptagon]], exhibiting the convergence.]]
Let &lt;math&gt;P&lt;/math&gt; be a closed [[w:Convex polygon#Strictly convex polygon|strictly convex polygon]] lying on the real plane. One of the first results proved by Richard Schwartz it that its iterates under the pentagram map shrink [[w:Exponential growth|exponentially fast]] to a point, as illustrated in Figure 5. This follows from two facts.
# The image of a strictly convex polygon is contained in its [[w:Interior (topology)|interior]], and is also strictly convex.{{Sfn|Glick|2020|p=2818}}
# There exists a constant &lt;math&gt;0&lt; \eta_P&lt;1&lt;/math&gt;, depending on &lt;math&gt;P&lt;/math&gt;, such that for any &lt;math&gt;N \in \mathbb N&lt;/math&gt;, the diameters of the iterates verify the inequality &lt;math display=&quot;inline&quot;&gt;\operatorname{diam}(T^N(P))\leq\eta_P^N \operatorname{diam}(P). &lt;/math&gt;{{Sfn|Schwartz|1992|loc=theorem 3.1}}

Hence, by [[w:Cantor's intersection theorem#Variant in complete metric spaces|Cantor's intersection theorem]], the sequence of polygons collapses toward a point.{{Sfn|Schwartz|1992|loc=§3 Convex polygons}}

The behavior on the moduli space is very different, since the dynamics is [[w:Recurrent point|recurrent]].{{Sfn|Schwartz|2001|loc=theorem 1.1}} It is even a [[w:quasiperiodic motion|quasiperiodic motion]],{{Sfn|Ovsienko|Schwartz|Tabachnikov|2009}} as discussed in [[w:Pentagram map#Complete integrability|the section about integrability]].

=== Coordinates of the limit point ===
The limit point coordinates are found in {{Harvard citation|Glick|2020}}. They satisfy some [[w:Degree of a polynomial|degree]] 3 [[w:polynomial equation|polynomial equations]], whose coefficients are [[w:rational function|rational function]]s in the coordinates of the vertices of the starting polygon. The proof relies on the fact that the limit point must be an [[w:eigenline|eigenline]] of a certain [[w:linear operator|linear operator]] of &lt;math&gt;\mathbb R^3&lt;/math&gt;.{{sfn|Glick|2020}}

This operator was reinterpreted in {{Harvard citation|Aboud|Izosimov|2022}} as the infinitesimal monodromy of the polygon. The [[w:Pentagram map#The scaling symmetry|scaling symmetry]] is used to [[w:Deformation (mathematics)|deform]] a closed polygon &lt;math&gt;P&lt;/math&gt; into a family of twisted ones '''&lt;math&gt;(P_z)_{z\in \mathbb C^*}&lt;/math&gt;''' with monodromy &lt;math&gt;M_z&lt;/math&gt;. The infinitesimal monodromy is defined to be:{{sfn|Aboud|Izosimov|2022}}
&lt;math display=&quot;block&quot;&gt;\left.\frac{dM_z}{dz}\right|_{z=1}.&lt;/math&gt;

=== Generalization ===
The collapsing of polygons may also happen in some [[w:Pentagram map#Generalizations|generalization of the pentagram map]], when considering some specific configurations of polygons in the real plane. The coordinates of the collapse point are given by a formula analogous to the one for the original pentagram map.{{Sfn|Schwartz|2026}}

== Periodic orbits on the moduli space ==
For some configurations of closed polygons, the iterate of the pentagram map will send &lt;math&gt;P&lt;/math&gt; to a projectively equivalent polygon (up to some shift of the indices). This means that, on the moduli space, the orbit of the class of &lt;math&gt;P&lt;/math&gt; is [[w:Periodic orbit|periodic]].

===Pentagons and hexagons===
[[File:penta hexagon.svg|300px|thumb|The outward hexagon is projectively equivalent to the inward one, with respect to their labeling.]]The following two facts are proved by checking [[w:cross-ratio|cross-ratio]] equalities, so they are true for polygons in any [[w:projective plane|projective plane]] (not just the [[w:Real projective plane|real one]]).{{Sfn|Schwartz|1992|loc=§2 Pentagons and hexagons}}

The pentagram map &lt;math&gt;T&lt;/math&gt; is the identity on the moduli space of [[w:pentagon|pentagon]]s.{{Sfn|Schwartz|1992|loc=theorem 2.1}}{{Sfn|Clebsch|1871}}{{Sfn|Motzkin|1945}} The second iterate &lt;math&gt;T^2&lt;/math&gt; is the identity on the space of labeled [[w:hexagon|hexagon]]s, up to a shift of labeling (see Figure 6).{{Sfn|Schwartz|1992|loc=theorem 2.3}} This phenomenon doesn't generalize to generic polygons with at least seven sides, for which the motion is [[w:Quasiperiodic motion|quasi-periodic]].{{Sfn|Tupan|2022}}

==== Generalization ====
The result about pentagons and hexagons generalizes to some [[w:Pentagram map#Generalizations|higher pentagram maps]] in &lt;math&gt;\mathbb P ^k&lt;/math&gt;, for polygons with &lt;math&gt;k+3&lt;/math&gt; or &lt;math&gt;2k+2&lt;/math&gt; sides. The proof uses a generalization of the [[w:Gale transform|Gale transform]].{{Sfn|Dirdak|2024}}

=== Poncelet polygons ===
A polygon is said to be Poncelet{{Efn|The name comes from [[w:Jean-Victor Poncelet|Jean-Victor Poncelet]] and [[w:Poncelet porism|his porism]].{{Sfn|Izosimov|2022a|p=1085}}}} if it is [[w:Inscribed figure|inscribed]] in a [[w:Conic section|conic]] and circumscribed about another one.{{Sfn|Schwartz|2015|loc=|p=433}}{{Efn|In particular, pentagons are Poncelet since [[w:five points determine a conic|five points determine a conic]].{{Sfn|Schwartz|2015|loc=|p=433}}}} For a convex Poncelet &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;P&lt;/math&gt; lying on the [[w:real projective plane|real projective plane]], the polygon &lt;math&gt;T^2(P)&lt;/math&gt; is projectively equivalent to &lt;math&gt;P&lt;/math&gt;.{{Sfn|Schwartz|2015|loc=theorem 1.1}} In fact, when &lt;math&gt;n&lt;/math&gt; is odd, the converse is also true.{{Sfn|Izosimov|2022a|loc=corollary 1.1}}

However, this converse statement is no longer true when the polygons are considered over the [[w:complex projective plane|complex projective plane]].{{Sfn|Izosimov|2022a|loc=remark 1.3}}

==Coordinates for the moduli space==
The moduli space can be described by different [[w:Coordinate_system|coordinate systems]]. The following ones are practical to explicit the dynamic, as presented in the next section.

=== Corner coordinates ===
[[File:Corner coordinates big.svg|thumb|300x300px|The geometric construction of the points defining the corner invariants.]]
Define the [[w:cross-ratio|cross-ratio]] of four [[w:Collinearity|collinear]] points to be

: &lt;math&gt; [a,b,c,d]=\frac{(a-b)(c-d)}{(a-c)(b-d)}. &lt;/math&gt;

The corner invariants are a system of coordinates on the space of twisted polygons, constructed by taking intersections as in Figure 7.{{Sfn|Schwartz|2001|loc=figure 2}} The left and right invariants are respectively defined{{Efn|The ordering of the vertices in the cross-ratios can differ from a paper to another one, which slightly changes the formulas in the following sections.}} as the following cross-ratios:

: &lt;math&gt;x_k:=[v_{k-2},v_{k-1},\overline{v_{k-2}v_{k-1}}\cap\overline{v_{k}v_{k+1}},\overline{v_{k-2}v_{k-1}}\cap\overline{v_{k+1}v_{k+2}}],&lt;/math&gt;

: &lt;math&gt;y_k:=[\overline{v_{k+1}v_{k+2}}\cap\overline{v_{k-2}v_{k-1}}, \overline{v_{k+1}v_{k+2}}\cap\overline{v_{k-1}v_{k}},v_{k+1},v_{k+2}].&lt;/math&gt;

Since the cross-ratio is [[w:Cross-ratio#Projective geometry|projective invariant]], the sequences &lt;math&gt;(x_k)_{k \in \mathbb Z}&lt;/math&gt; and &lt;math&gt;(y_k)_{k \in \mathbb Z}&lt;/math&gt; associated to a twisted &lt;math&gt;n&lt;/math&gt;-gon are &lt;math&gt;n&lt;/math&gt; periodic.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=415}}

The corner invariants are elements of &lt;math&gt;\mathbb{P}^1\smallsetminus\{0,1,\infty\}&lt;/math&gt;, and they realize an [[w:Isomorphism_of_varieties|isomorphism of variety]] between the moduli space of twisted &lt;math&gt;n&lt;/math&gt;-gons and &lt;math&gt;(\mathbb{P}^1\smallsetminus\{0,1,\infty\})^{2n}&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=theorem 3.6}}

===ab-coordinates===

There is a second set of coordinates for the moduli space of twisted &lt;math&gt;n&lt;/math&gt;-gons defined over a [[w:Field (mathematics)|field]] &lt;math&gt;F&lt;/math&gt; satisfying &lt;math&gt;\mathrm{SL}_3(F)\cong \mathbb P\mathrm{GL}_3(F)&lt;/math&gt;,{{Sfn|Weinreich|2022|loc=remark 3.8}} and such that &lt;math&gt;n&lt;/math&gt; is not divisible by &lt;math&gt;3&lt;/math&gt;.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=section 4.1}}

The vertices &lt;math&gt;v_k&lt;/math&gt; in the [[w:projective plane|projective plane]] &lt;math&gt;\mathbb P^2(F)&lt;/math&gt; can be [[w:Lift (mathematics)|lifted]] to [[w:Vector space|vectors]] &lt;math&gt;V_k&lt;/math&gt; in the [[w:affine space|affine space]] &lt;math&gt;F^3&lt;/math&gt; so that each consecutive triple of vectors spans a [[w:parallelepiped|parallelepiped]] having [[w:determinant|determinant]] equal to &lt;math&gt;1&lt;/math&gt;. This leads to the relation{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=equation 4.1}}

: &lt;math&gt;V_{k+3} = a_k V_{k+2} + b_k V_{k+1} + V_k.&lt;/math&gt;

This bring out an analogy between twisted polygons and solutions of third order linear [[w:ordinary differential equations|ordinary differential equations]], normalized to have unit [[w:Wronskian|Wronskian]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=remark 6.6}}

They are linked to the corner coordinates by:{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=lemma 4.5}}
: &lt;math&gt;x_k=\frac{a_{k-2}}{b_{k-2}b_{k-1}},&lt;/math&gt;
: &lt;math&gt;y_k=-\frac{b_{k-1}}{a_{k-2}a_{k-1}}.&lt;/math&gt;

==Formulas on the moduli space==

===As a birational map ===
The pentagram map is a [[w:birational map|birational map]] on the moduli space, because it can be decomposed as the [[w:Function composition|composition]] of two [[w:Birational geometry|birational]] [[w:Involution (mathematics)|involutions]].{{Sfn|Schwartz|2008|loc=§1.2 The Pentagram Map}} The corner invariants change in the following way:{{Sfn|Ovsienko|Schwartz|loc=lemma 2.4|Tabachnikov|2010}}

: &lt;math&gt;x_k'=x_k\frac{1-x_{k-1} y_{k-1}}{1-x_{k+1}y_{k+1}},&lt;/math&gt;

: &lt;math&gt;y_k'=y_{k+1}\frac{1-x_{k+2} y_{k+2}}{1-x_k y_k}.&lt;/math&gt;

=== The scaling symmetry ===
The [[w:multiplicative group|multiplicative group]] &lt;math&gt;F\smallsetminus\{0\}&lt;/math&gt; [[w:One-parameter group|acts]] on the moduli space in the following way:

: &lt;math&gt;R_s\cdot(x_1,\dots,x_n,y_1,\dots,y_n)=(sx_1,\dots,sx_n,s^{-1}y_1,\dots,s^{-1}y_n),&lt;/math&gt;

where &lt;math&gt;R&lt;/math&gt; is called the scaling action and &lt;math&gt;s&lt;/math&gt; is the scaling parameter. This action commutes with the pentagram map on the moduli space (as presented in the previous formulas). This property is called the scaling symmetry, and is instrumental in proving the [[w:Pentagram map#Complete integrability|complete integrability]] of the dynamics.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=corollary 2.5}}

==Invariant structures==

===Monodromy invariants===

The monodromy invariants, introduced in {{Harvard citation|Schwartz|2008}}, are a collection of [[w:Function (mathematics)|functions]] on the [[w:moduli space|moduli space]] that are invariant under the pentagram map.{{Sfn|Schwartz|2008|loc=theorem 1.2}} The simplest example of them are 
:&lt;math&gt; O_n= x_1x_2\cdots x_{n}, \quad E_n = y_1y_2\cdots y_n. &lt;/math&gt;
The other monodromy invariants can be retrieved through different points of view: through the [[w:Pentagram map#The scaling symmetry|scaling symmetry]], as [[w:Combinatorics|combinatorial]] objects, or as some [[w:determinant|determinant]]s.{{Sfn|Schwartz|Tabachnikov|2011|loc=§2 The Monodromy Invariants}} The one involving scaling symmetry is presented here.

Let &lt;math&gt;M\in \mathrm{GL}_3&lt;/math&gt; be a [[w:Lift (mathematics)|lift]] of the monodromy of a twisted &lt;math&gt;n&lt;/math&gt;-gon. The quantities

: &lt;math&gt;\Omega_1=\frac{\operatorname{trace}^3(M)}{\det(M)}, \quad \Omega_2=\frac{\operatorname{trace}^3(M^{-1})}{\det(M^{-1})},&lt;/math&gt;

are independent of the choice of lift and are invariant under [[w:Matrix similarity|conjugation]], so they are well defined for the projective class of the polygon. They are invariant under the pentagram map, since the monodromy matrix doesn't change.{{Sfn|Schwartz|Tabachnikov|2011|loc=|p=5}} Now, the quantities

: &lt;math&gt;\tilde{\Omega}_1=O_n^2E_n\Omega_1, \quad \tilde{\Omega}_2=O_nE_n^2\Omega_2,&lt;/math&gt;

have the same properties, but turn out to be polynomials in the corner invariants.{{Efn|Some papers consider the cube roots of this functions, but it doesn't change the following definitions of the monodromy invariants.}} They can be written as{{Sfn|Schwartz|Tabachnikov|2011|loc=|p=5}}

: &lt;math&gt;
\tilde{\Omega}_1=\biggl(\sum_{k=0}^{\lfloor n/2\rfloor}O_k\biggr)^3, \quad
\tilde{\Omega}_2=\biggl(\sum_{k=0}^{\lfloor n/2\rfloor}E_k\biggr)^3,
&lt;/math&gt;

where each &lt;math&gt;O_k&lt;/math&gt; and &lt;math&gt;E_k&lt;/math&gt; are [[w:homogeneous polynomial|homogeneous polynomial]]s respectively of weight &lt;math&gt;k&lt;/math&gt; and &lt;math&gt;-k&lt;/math&gt;,{{Sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=11}} meaning they change under the [[w:Pentagram map#The scaling symmetry|rescaling action]] on variables by{{Sfn|Schwartz|Tabachnikov|2011|p=5}}

: &lt;math&gt; R_s(O_k)= s^k O_k, \quad R_s(E_k)= s^{-k} E_k. &lt;/math&gt;

The quantities &lt;math&gt;O_1,\dots,O_{\lfloor n/2 \rfloor},O_n, E_1,\dots,E_{\lfloor n/2 \rfloor},E_n,&lt;/math&gt; are unchanged by the dynamics, and are called the monodromy invariants. Moreover, they are [[w:algebraically independent|algebraically independent]].{{Sfn|Schwartz|2008|loc=theorem 1.2}}

==== Polygons on conics ====
Whenever &lt;math&gt;P&lt;/math&gt; is [[w:Inscribed figure|inscribed]] on a [[w:conic section|conic section]], one has &lt;math&gt;O_k(P)=E_k(P)&lt;/math&gt; for all &lt;math&gt;k&lt;/math&gt;.{{Sfn|Schwartz|Tabachnikov|2011|loc=theorem 1.1}} Moreover, if &lt;math&gt;P&lt;/math&gt; is circumscribed about another conic,{{Efn|See the paragraph about [[w:Pentagram map#Poncelet polygons|Poncelet polygons]].}} then its monodromy invariants are characterized by the pair of conics.{{Sfn|Schwartz|2015|loc=theorem 1.2}} For such odd-gons, the translation on the [[w:Jacobian variety|Jacobian variety]]{{Efn|See the paragraph about [[w:Pentagram map#Algebro-geometric integrability|algebraic integrability]].}} is restricted to the [[w:Prym variety|Prym variety]] (which is a half-dimensional torus in the Jacobian).{{Sfn|Izosimov|2016|loc=theorem 1.3}}

===Poisson bracket===

An invariant [[w:Poisson bracket|Poisson bracket]] on the space of twisted polygons was found in {{Harvard citation|Ovsienko|Schwartz|Tabachnikov|2010}}. The monodromy invariants [[w:Poisson bracket#Constants of motion|commute]] with respect to it:
&lt;math display=&quot;block&quot;&gt; \{O_i,O_j\}=\{O_i,E_j\}=\{E_i,E_j\}=0 &lt;/math&gt;for all &lt;math&gt;i,j&lt;/math&gt;.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}}

The Poisson bracket is defined in terms of the corner coordinates by:
&lt;math display=&quot;block&quot;&gt; \begin{align}
 \{x_i,x_{i\pm1}\} &amp;= \mp x_i x_{i+1}, \\
 \{y_i,y_{i\pm 1}\} &amp;= \mp y_i y_{i+1}, \\
 \{x_i,x_j\} &amp;= \{y_i,y_j\} = \{x_i,y_j\} = 0
\end{align}&lt;/math&gt;for all other &lt;math&gt; i,j.&lt;/math&gt;{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=equation 2.16}}

=== The spectral curve ===

Let &lt;math&gt;\zeta&lt;/math&gt; be an element of the [[w:multiplicative group|multiplicative group]] and &lt;math&gt;P_\zeta&lt;/math&gt; be the polygon obtained by applying the [[w:Pentagram map#The scaling symmetry|rescaling action]] &lt;math&gt;R_\zeta&lt;/math&gt; on &lt;math&gt;P&lt;/math&gt;.  A [[w:Lax matrix|Lax matrix]] &lt;math&gt;\hat{T}(\zeta) \in \mathrm{GL}_3&lt;/math&gt; is a lift of the monodromy of &lt;math&gt;P_\zeta&lt;/math&gt; satisfying a [[w:Lax pair#Zero-curvature equation|zero-curvature equation]].{{Sfn|Weinreich|2022|loc=§5 The Lax representation}} Then, the spectral function is the [[w:Bivariate polynomial|bivariate]] [[w:characteristic polynomial|characteristic polynomial]]
&lt;math display=&quot;block&quot;&gt; Q(\lambda,\zeta) := \det(\lambda\operatorname{Id}-\hat{T}(\zeta)),&lt;/math&gt;or some renormalization of it. The [[w:spectral curve|spectral curve]] is the [[w:Projective variety#projective completion|projective completion]] of the [[w:Algebraic curve|affine curve]] defined by the equation &lt;math&gt;Q(\lambda,\zeta)=0&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=§6. The geometry of the spectral curve}} It is invariant under the pentagram map, and the monodromy invariants appear as the [[w:coefficient|coefficient]]s of &lt;math&gt;Q&lt;/math&gt;.{{Sfn|Soloviev|2013|loc=theorem 6.4}} Its [[w:geometric genus|geometric genus]] is &lt;math&gt;n-1&lt;/math&gt; if &lt;math&gt;n&lt;/math&gt; is odd, and &lt;math&gt;n-2&lt;/math&gt; if &lt;math&gt;n&lt;/math&gt; is even.{{Sfn|Weinreich|2022|p=|loc=theorem 6.4}}

It was first introduced in {{Harv|Soloviev|2013|ps=|p=}} for his proof of [[w:Pentagram map#Algebro-geometric integrability|algebro-geometric integrability]].{{sfn|Soloviev|2013}}

==Complete integrability==
The pentagram map on the moduli space has been proved to be a [[w:completely integrable|completely integrable]] [[w:discrete dynamical system|discrete dynamical system]], both in the [[w:Integrable system#Hamiltonian systems and Liouville integrability|Arnold-Liouville]]{{Efn|Over the [[w:real number|real number]]s.}} and the [[w:Integrable system#Complete integrability over the complex numbers|algebro-geometric]]{{Efn|Over [[w:algebraically closed field|algebraically closed field]]s of [[w:Characteristic (algebra)|characteristic]] different from 2.}} senses. In any case, this means that the moduli space is [[w:almost everywhere|almost everywhere]] [[w:Foliation|foliated]] by [[w:Torus#Flat torus|flat tori]] (or in the algebraic setting, [[w:Abelian variety|Abelian varieties]]), where the motion is a [[w:Translation (geometry)|translation]]. This [[w:Generic property|generically]] induces a [[w:quasiperiodic motion|quasiperiodic motion]] on the corresponding torus.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2009}}

===Arnold–Liouville integrability===

The proof of the integrability of the pentagram map on a real twisted polygon was achieved in {{Harvard citation|Ovsienko|Schwartz|Tabachnikov|2010}}. This is done by noticing that the monodromy invariants &lt;math&gt;O_n&lt;/math&gt; and &lt;math&gt;E_n&lt;/math&gt; are [[w:Casimir invariant|Casimir invariant]]s for the bracket, meaning (in this context) that&lt;math display=&quot;block&quot;&gt; \{O_n,f\}=\{E_n,f\} = 0 &lt;/math&gt;for all functions &lt;math&gt;f&lt;/math&gt;.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}} When &lt;math&gt;n&lt;/math&gt; is even, this is also true for the monodromy invariants &lt;math&gt;O_{\lfloor n/2 \rfloor }&lt;/math&gt; and &lt;math&gt;E_{\lfloor n/2 \rfloor }&lt;/math&gt;.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}}

This allows to consider the Casimir [[w:level set|level set]], where each Casimir has a specified value. Because of [[w:Sard's theorem|Sard's theorem]], any generic level set is a [[w:smooth manifold|smooth manifold]].{{Sfn|Schwartz|2017|p=44}} They form a [[w:foliation|foliation]] in [[w:Poisson manifold#Symplectic leaves|symplectic leaves]], on which the Poisson bracket gives rise to a [[w:symplectic form|symplectic form]].{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=corollary 2.13}}

Each of these symplectic leaves has an iso-monodromy [[w:foliation|foliation]], namely, a decomposition into the common level sets of the remaining monodromy functions. By using again [[w:Sard's theorem|Sard's theorem]], they are generically [[w:Symplectic manifold#Lagrangian submanifolds|Lagrangian manifolds]].{{Sfn|Schwartz|2017|p=45}} Moreover, they are compact.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=§3.3. Compactness of the level sets}} Since the monodromy invariants Poisson-commute and there are enough of them, the discrete [[w:Liouville–Arnold theorem|Liouville–Arnold theorem]] can be applied to prove that the level sets are [[w:Torus#Flat torus|flat tori]] over which the dynamics is a translation.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=412}}
===Algebro-geometric integrability===

In {{Harvard citation|Soloviev|2013}}, it was shown that the pentagram map admits a [[w:Lax representation|Lax representation]] with a spectral parameter, which allows to prove its algebro-geometric integrability. This means that the space of polygons (either twisted or closed) is parametrized by its spectral data, consisting of [[w:Pentagram map#The spectral curve|its spectral curve]], with marked points and a [[w:Divisor (algebraic geometry)|divisor]] given by a [[w:Floquet theory|Floquet]]–[[w:Bloch's theorem|Bloch]] equation. This gives an embedding to the [[w:Jacobian variety|Jacobian variety]] through the [[w:Abel–Jacobi map|Abel–Jacobi map]], where the motion is expressed in terms of translation.{{sfn|Soloviev|2013|loc=theorems A, B and C}} The previously defined Poisson bracket is also retrieved.{{sfn|Soloviev|2013|loc=theorem D}} 

This integrability was generalized in {{Harvard citation|Weinreich|2022}} from the field of [[w:complex number|complex number]]s to any [[w:algebraically closed field|algebraically closed field]] of [[w:Characteristic (algebra)|characteristic]] different from 2. The translation on a torus is replaced by a translation on an [[w:Abelian variety|Abelian variety]] (in fact, a Jacobian variety again).{{sfn|Weinreich|2022|loc=theorem 1.4}}

=== Dimension of the invariant manifold ===

For twisted &lt;math&gt;n&lt;/math&gt;-gons, the [[w:dimension|dimension]] of the invariant tori (or Jacobian varieties) is{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=421}}

: &lt;math&gt;\begin{cases} 
n-1 &amp; \text{when }n \text{ is odd,}\\
n-2 &amp; \text{when }n \text{ is even.}
\end{cases}&lt;/math&gt;

Moreover, when &lt;math&gt;n&lt;/math&gt; is even, there are two isomorphic Jacobians on which the iterates of the pentagram map alternate. But on each of them, the second iterate is a translation.{{Sfn|Weinreich|2022|loc=theorem 1.4}}

=== For closed polygons ===
There is no Poisson structure on the space of closed polygons.{{Sfn|Soloviev|2013|loc=corollary 4.1}} Nevertheless, the one from twisted polygons can be used to prove integrability.{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=2153}}

Algebro-geometric integrability holds for closed polygons in a same manner as for the twisted ones.{{Sfn|Soloviev|2013|loc=theorem C}} However, Arnold-Liouville integrability is proved for real closed polygons only when they are convex. This is done by restricting the [[w:Hamiltonian vector field|Hamiltonian vector field]]s of monodromy functions to smaller dimensional tori, and showing that enough of them are still independent.{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|loc=corollary 1.1}}

In both situation, the dimension of the invariant manifolds decreases by &lt;math&gt;3&lt;/math&gt; for closed &lt;math&gt;n&lt;/math&gt;-gons (compared to the twisted case), and is equal to{{Sfn|Soloviev|2013|loc=theorem C}}{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|loc=theorem 1}}
:&lt;math&gt;\begin{cases} 
n-4 &amp; \text{when }n \text{ is odd,}\\
n-5 &amp; \text{when }n \text{ is even.}
\end{cases}&lt;/math&gt;

==Connections to other topics==

===The Boussinesq equation===

The continuous limit of a convex polygon is a parametrized convex curve in the plane. When the time parameter is suitably chosen, the [[w:Discretization|continuous limit]] of the pentagram map is the classical [[w:Boussinesq approximation (water waves)|Boussinesq equation]]. This equation is a classical example of an [[w:integrable|integrable]] [[w:partial differential equation|partial differential equation]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 5}}

Here is a description of the geometric action of the Boussinesq equation. Given a [[w:locally convex|locally convex]] curve &lt;math&gt; C:\mathbb R\to \mathbb R^2 &lt;/math&gt; and real numbers &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;t&lt;/math&gt;, consider the [[w:chord (geometry)|chord]] connecting &lt;math&gt; C(x-t) &lt;/math&gt; to &lt;math&gt; C(x+t) &lt;/math&gt;. The [[w:Envelope (mathematics)|envelope]] of all these chords is a new curve &lt;math&gt; C_t(x) &lt;/math&gt;. When &lt;math&gt;t&lt;/math&gt; is extremely small, the curve &lt;math&gt; C_t(x) &lt;/math&gt; is a good model for the time &lt;math&gt;t&lt;/math&gt; evolution of the original curve &lt;math&gt; C_0(x) &lt;/math&gt; under the Boussinesq equation. This construction is also similar to the pentagram map. Moreover, the pentagram invariant bracket is a discretization of a well known invariant Poisson bracket associated to the Boussinesq equation.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=§6.4 Discretization}}

===Cluster algebras===

The pentagram map{{Sfn|Glick|2011}} and some of its generalizations{{Sfn|Gekhtman|Shapiro|Tabachnikov|Vainshtein|2012}}{{Sfn|Glick|Pylyavskyy|2016}} are identified as special cases of discrete dynamical systems powered by [[w:cluster algebra|cluster algebra]]. Using the results from {{Harvard citation|Goncharov|Kenyon|2013}}, this provides a link with the [[w:Poisson–Lie group|Poisson–Lie group]]s, [[w:dimer model|dimer model]]s and other so-called cluster-integrable systems.{{Sfn|Fock|Marshakov|2016}}{{Sfn|Izosimov|2022c}} These methods allow to retrieve the Poisson-bracket and Hamiltonians used to prove complete integrability{{Sfn|Affolter|George|Ramassamy|2025|loc=§5 The pentagram map}} and provide [[w:Lax representation|Lax representation]]s.{{Sfn|Izosimov|2022b}}

=== Singularity theory ===
The pentagram map exhibit a property called singularity confinement, which is typical from [[w:integrable system|integrable system]]s.{{Sfn|Grammaticos|Ramani|Papageorgiou|1991}} It states that if a polygon &lt;math&gt;P&lt;/math&gt; is [[w:Singular point of an algebraic variety|singular]] for the pentagram map &lt;math&gt;T&lt;/math&gt;, then there exists an integer &lt;math&gt;m&lt;/math&gt; such that &lt;math&gt;P&lt;/math&gt; not singular for the iterate map &lt;math&gt;T^m&lt;/math&gt;.{{Sfn|Glick|2012}}

Moreover, the pentagram map (along with some of its generalizations and other discrete dynamical systems) exhibit the Devron property.{{Efn|The name comes from an episode of [[w:Star Trek|Star Trek]].{{Sfn|Glick|2015|loc=§1 Introduction}}}} This means that if a polygon &lt;math&gt;P&lt;/math&gt; is singular for some iterate of the pentagram map &lt;math&gt;T^m&lt;/math&gt;, then it will also be singular for some iterate of the inverse map &lt;math&gt;T^{-m'}&lt;/math&gt;.{{Sfn|Glick|2015}}

== Generalizations ==
The definition of twisted polygons still makes sense in any [[w:projective space|projective space]] &lt;math&gt;\mathbb P^d&lt;/math&gt;, under the action of the [[w:Projective linear group|projective group]] &lt;math&gt;\mathbb P \mathrm{GL}_{d+1}&lt;/math&gt;. The pentagram map can be generalized in many ways, and some of them are presented here. Not all of them are integrable.{{Sfn|Khesin|Soloviev|2015|}} Some are [[w:discretization|discretization]]s of [[w:PDEs|PDEs]] from the [[w:KdV hierarchy|KdV hierarchy]], seen as higher dimensional version of [[w:Boussinesq approximation (water waves)|Boussinesq]] or [[w:Kadomtsev–Petviashvili equation|KP]] equations.{{Sfn|Marí-Beffa|2012}}{{Sfn|Wang|2023}} The description of all generalized pentagram maps in terms of [[w:cluster algebra|cluster algebra]]s is still an open question.{{Sfn|Gekhtman|Izosimov|2025|p=14}}

=== Polygons in general positions ===
Let &lt;math&gt;d \geq 2&lt;/math&gt; and &lt;math&gt;P&lt;/math&gt; be a twisted polygon of &lt;math&gt;\mathbb P^d&lt;/math&gt; in [[w:general position|general position]].

==== Short diagonal pentagram maps ====
The &lt;math&gt;k&lt;/math&gt;-th ''short diagonal hyperplane'' &lt;math&gt;H_k^{sh}&lt;/math&gt; is uniquely defined by passing through the vertices &lt;math&gt;v_k,v_{k+2},\dots,v_{k+2d-2}&lt;/math&gt;. [[w:Generic property#In algebraic geometry|Generically]], the intersection of &lt;math&gt;d&lt;/math&gt; consecutive hyperplanes uniquely defines a new point

: &lt;math&gt;T_{sh}v_k:=H_k^{sh}\cap H_{k+1}^{sh}\cap \dots \cap H_{k+d-1}^{sh}.&lt;/math&gt;

Doing this for every vertex defines a new twisted polygon. This map, denoted by &lt;math&gt;T_{sh}&lt;/math&gt;, is again projectively equivariant.{{Sfn|Khesin|Soloviev|2013}}

==== Generalized pentagram maps ====
The previous procedure can be generalized. Let &lt;math&gt;I=(i_1,\dots,i_{d-1}),~J=(j_1,\dots,j_{d-1})&lt;/math&gt; be two sets of integers, respectively called the jump tuple and the intersection tuple. Define the &lt;math&gt;k&lt;/math&gt;-th hyperplane &lt;math&gt;H_k^I&lt;/math&gt; to be passing through the vertices &lt;math&gt;v_k,v_{k+i_1},\dots,v_{k+i_1+\dots+i_{d-1}}&lt;/math&gt;. A new point is given by the intersection

: &lt;math&gt;T_{I,J}v_k:=H_k^I \cap H_{k+j_1}^I \cap \dots \cap H_{k+j_1+\dots +j_{d-1}}^I.&lt;/math&gt;

The map &lt;math&gt;T_{I,J}&lt;/math&gt; is called a generalized pentagram map.{{Sfn|Khesin|Soloviev|2015a}} The original pentagram map is recovered by considering&lt;math&gt;d=2,~I=(2),~J=(1)&lt;/math&gt;.

Integrability can be numerically tested by picking a random polygon &lt;math&gt;P&lt;/math&gt; with [[wikipedia:Rational_point|rational coordinates]] and studying the growth rate of the [[wikipedia:Height_function|height]] of its iterates. This is called the [[wikipedia:Integrable_system#Diophantine_integrability|diophantine integrability]] test, and some generalized pentagram maps don't seem to pass it.{{Sfn|Khesin|Soloviev|2015a|loc=§5 and §6}} However, it is conjectured that the maps &lt;math&gt;T_{I,I}&lt;/math&gt; are integrable for any &lt;math&gt;I&lt;/math&gt;.{{Sfn|Bolsinov|Matveev|Miranda|Tabachnikov|2018|loc=conjecture 4.13 (B. Khesin, F. Soloviev)}}

Some of these maps are [[w:discretization|discretization]]s of higher dimensional counterpart of the [[w:Boussinesq approximation (water waves)|Boussinesq equation]] in the [[w:KdV hierarchy|KdV hierarchy]].{{Sfn|Khesin|Soloviev|2015b|loc=theorem 4.1}}{{Sfn|Izosimov|2022b|loc=theorem 4.1}}

==== Dented pentagram maps ====
Fix an integer &lt;math&gt;m\in \{1,\dots ,d-1\}&lt;/math&gt;. Consider the jump tuple &lt;math&gt;I_m:=(1,\dots,1,2,1,\dots,1)&lt;/math&gt;, where the &lt;math&gt;2&lt;/math&gt; is at the &lt;math&gt;m&lt;/math&gt;-th place, and the intersection tuple &lt;math&gt;J:=(1,\dots,1)&lt;/math&gt;. The dented pentagram map is &lt;math&gt;T_m :=T_{I_m,J}&lt;/math&gt;. They are proved to be integrable.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 2.14}}

For an integer &lt;math&gt;p \geq 2&lt;/math&gt;, the deep dented pentagram map (of depth &lt;math&gt;p&lt;/math&gt;) &lt;math&gt;T_m^p&lt;/math&gt; is the same map as before, but the number &lt;math&gt;2&lt;/math&gt; in the definition of &lt;math&gt;I_m&lt;/math&gt; is replaced by &lt;math&gt;p&lt;/math&gt;. This kind of pentagram maps are again integrable.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 6.2}}

=== Corrugated polygons ===
A twisted polygon &lt;math&gt;P&lt;/math&gt; lying in &lt;math&gt;\mathbb P^d&lt;/math&gt; is said to be corrugated if for any &lt;math&gt;k\in \mathbb Z&lt;/math&gt;, the vertices &lt;math&gt;v_k,v_{k+1},v_{k+d},v_{k+d+1}&lt;/math&gt; span a projective two-dimensional plane. Such polygons are not in [[w:general position|general position]]. A new point is defined by

: &lt;math&gt;T_\text{cor}v_k:=\overline{v_k v_{k+d}}\cap \overline{v_{k+1} v_{k+d+1}}.&lt;/math&gt;

The map &lt;math&gt;T_\text{cor}&lt;/math&gt; yields a new corrugated polygon. They are [[w:Integrable system#Hamiltonian systems and Liouville integrability|completely Liouville-integrable]].{{Sfn|Gekhtman|Shapiro|Tabachnikov|Vainshtein|2012|loc=theorem 4.4}}

In fact, they can be retrieved as some dented pentagram map applied on corrugated polygons.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 5.3}}

=== Grassmannian polygons ===
Let &lt;math&gt;d \geq 3, m \geq 1&lt;/math&gt; be integers. The pentagram map can also be generalized to the [[w:Grassmannian|Grassmannian]] space &lt;math&gt;\mathrm{Gr}(m,md)&lt;/math&gt;, which consists of &lt;math&gt;m&lt;/math&gt;-[[w:Dimension (vector space)|dimensional]] [[w:linear subspace|linear subspace]]s of an &lt;math&gt;md&lt;/math&gt;-dimensional [[w:vector space|vector space]]. When &lt;math&gt;m=1&lt;/math&gt;, the linear subspaces are [[w:Vector space#vector line|lines]], which retrieves the definition of [[w:projective space|projective space]]s &lt;math&gt;\mathbb P^d&lt;/math&gt;.{{Sfn|Felipe|Marí-Beffa|2019|loc=§2 definitions and notations}}

A point &lt;math&gt;v\in\operatorname{Gr}(m,md)&lt;/math&gt; is represented by an &lt;math&gt;md \times m&lt;/math&gt; matrix &lt;math&gt;X_v&lt;/math&gt; such that its columns form a [[w:Basis (linear algebra)|basis]] of &lt;math&gt;v&lt;/math&gt;. Consider the [[w:Group action|action]] of the [[w:general linear group|general linear group]] &lt;math&gt;\mathrm{GL}_{md}&lt;/math&gt; by multiplication on the left of &lt;math&gt;X_v&lt;/math&gt;. This defines an action on the Grassmannian, even though it is not [[w:Faithful action|faithful]].{{Efn|Because there can be many lifts for &lt;math&gt;v&lt;/math&gt;, and because some matrices act trivially.}} Hence, the polygons of &lt;math&gt;\mathrm{Gr}(m,md)&lt;/math&gt; and their moduli spaces are defined as before, after the change of underlying group.{{Sfn|Felipe|Marí-Beffa|2019|loc=§2 definitions and notations}}

Depending on the parity of &lt;math&gt;d&lt;/math&gt;, one can define linear subspaces spanned by some &lt;math&gt;X_{v_k}&lt;/math&gt;'s such that taking their intersection generically defines a new point &lt;math&gt;v\in\mathrm{Gr}(m,md)&lt;/math&gt;.{{Sfn|Felipe|Marí-Beffa|2019|loc=sections 4 and 5}} This generalization of the pentagram map is integrable in a [[w:noncommutative|noncommutative]] sense.{{Sfn|Ovenhouse|2020}}

=== Over rings ===
The pentagram map admits a generalization by considering [[w:Projective space#Generalizations|projective planes]] over [[w:stably finite ring|stably finite ring]]s, instead of [[w:Field (mathematics)|field]]s. In particular, this retrieves the pentagram map over Grassmannians. Again, it admits a [[w:Lax representation|Lax representation]].{{Sfn|Hand|Izosimov|2025}}

== References ==
{{reflist|25em}}

===Notes===
{{notelist}}

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*{{Cite journal|ref=harv |last=Schwartz|first=Richard|date=2026-02-14|title=The Flapping Birds in the Pentagram Zoo|url=https://armj.math.stonybrook.edu/Articles/241224-Schwartz/index.html|journal=Arnold Mathematical Journal|volume=011|issue=004|pages=10|doi=10.56994/ARMJ.011.004.002|issn=2199-6792}}
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*{{Cite journal |ref=harv |last=Tupan |first=Alexandru |date=2022-07-03 |title=Pentagram Configurations for Pentagons and Hexagons |url=https://www.tandfonline.com/doi/full/10.1080/00029890.2022.2060695 |journal=The American Mathematical Monthly |language=en |volume=129 |issue=6 |pages=554–565 |doi=10.1080/00029890.2022.2060695 |issn=0002-9890}}
*{{Cite journal|ref=harv |title=Pentagram-Type Maps and the Discrete KP Equation|url=https://link.springer.com/10.1007/s00332-023-09961-7|journal=Journal of Nonlinear Science|date=2023|issn=0938-8974|volume=33|issue=6|doi=10.1007/s00332-023-09961-7|language=en|first=Bao|last=Wang |article-number=101 |bibcode=2023JNS....33..101W }}
*{{Cite journal |ref=harv |title=The algebraic dynamics of the pentagram map |journal=Ergodic Theory and Dynamical Systems |date=2022-11-25 |issn=0143-3857 |pages=3460–3505 |volume=43 |issue=10 |doi=10.1017/etds.2022.82 |first=Max H. |last=Weinreich}}</text>
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      <comment>/* Connections to other topics */ added citation for the octahedron reccurrence</comment>
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      <text bytes="60216" sha1="l1o42dyd7a0z3s8rxucgta3gtgkt7ca" xml:space="preserve">{{Article info
| last1 = Stiegler
| orcid1 = 0009-0001-5789-6923
| first1 = Jean-Baptiste
| affiliation1 = Université Paris-Saclay
| correspondence1 = jean-baptiste.stiegler@universite-paris-saclay.fr
| journal = WikiJournal of Science
| et_al = true
| w1 = Pentagram map
| from w1 = true
| keywords = Pentagram map, Dynamical system, Projective geometry, Moduli space, Integrable systems
| license = CC-BY-SA 4.0
| submitted = 2025-12-08
| abstract = In [[w:mathematics|mathematics]], the '''pentagram map''' is a [[w:Dynamical system#Discrete dynamical system|discrete dynamical system]] acting on [[w:polygons|polygons]] in the [[w:projective plane|projective plane]]. It defines a new polygon whose vertices are obtained as the intersection points of the shortest [[w:Diagonal|diagonals]] of the initial polygon. This is a [[w:Projective linear group|projectively]] [[w:Equivariant map|equivariant]] procedure, hence it [[w:Quotient space (topology)|descends]] to the [[w:moduli space|moduli space]] of polygons and defines another dynamical system (which is also referred to as the pentagram map). It was first introduced by [[w:Richard Schwartz (mathematician)|Richard Schwartz]] in 1992.{{Sfn|Schwartz|1992}}

The pentagram map on the moduli space is famous for its [[w:Completely integrable|complete integrability]] and its link with [[w:cluster algebra|cluster algebras]].{{sfn|Gekhtman|Izosimov|2025|p=14}}

It admits many generalizations in [[w:Projective space|projective spaces]] and other settings.
}}
== Introduction ==

=== Informal definition ===

==== On polygons ====
[[File:Pentagram pentagon nolabel big.svg|alt=|thumb|300x300px|The pentagram map applied on a [[w:Convex set|convex]] [[w:pentagon|pentagon]].]]
Initially, the pentagram map was defined for [[w:convex polygon|convex polygon]]s (with at least five sides) on the [[w:euclidean plane|Euclidean plane]]. Given such a polygon &lt;math&gt;P&lt;/math&gt; with &lt;math&gt;n&lt;/math&gt; sides, one can draw the &quot;shortest [[w:diagonal|diagonal]]s&quot;, meaning the [[w:Line segment|segments]] whose endpoints are a [[w:Vertex (geometry)|vertex]] and one of its second neighbors (as in Figure 1). The intersections of the shortest diagonals are then taken as the vertices of a new &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;T(P)&lt;/math&gt;; this new polygon is the output of the pentagram map.{{Sfn|Berger|2005}}

The same construction can be done on [[w:Concave polygon|non-convex polygons]], but there are several complications. First, some consecutive short diagonals may not intersect, so one must extend the segments to [[w:Line (geometry)|lines]]. Second, the image &lt;math&gt;T(P)&lt;/math&gt; can fail to be a new &lt;math&gt;n&lt;/math&gt;-gon because some consecutive vertices could coincide. However, this [[w:Generic property|generically]] doesn't happen.{{Sfn|Ovsienko|Schwartz|Tabachnikov|p=411|2009}} Finally, it is possible that two diagonals are [[w:Parallel (geometry)|parallel]] and don't intersect on the [[w:euclidean plane|Euclidean plane]]. This is resolved by extending the Euclidean plane to the [[w:real projective plane|real projective plane]] by the addition of a [[w:line at infinity|line at infinity]], where the [[w:Vanishing point|intersection point]] lies (see Figure 3). Hence, the pentagram map is defined for generic polygons in the real projective plane.{{Sfn|Berger|2005|p=25}}

More generally, the construction of the pentagram map is well defined whenever the concepts of lines and their intersections make sense. This is encompassed by the notion of a general [[w:projective plane|projective plane]], of which the real projective plane is one example; but the pentagram map can also be considered over other [[w:Field (mathematics)|fields]], for instance the [[w:complex number|complex number]]s, which give the [[w:complex projective plane|complex projective plane]].{{Sfn|Weinreich|2022|loc=§3.1.1}}

==== On the moduli space of polygons ====
Since the pentagram map is constructed by drawing lines and marking their intersections, it [[w:Commutative property|commutes]] with any transformation that sends lines to lines. Such maps are called [[w:projective transformations|projective transformations]]. This allows to identify polygons [[w:up to|up to]] [[w:Perspectivity#Projectivity|projective transformations]]. This identification gives a [[w:Quotient space (topology)|quotient space]] (technically called a [[w:moduli space|moduli space]]) of [[w:Equivalence class|classes]] of polygons. 

The pentagram map on polygons induces another dynamical system on the moduli space,{{Sfn|Schwartz|1992|loc=§1 Projective geometry}} whose behavior differs quite a lot from the initial one.{{Efn|Compare the paragraph about the [[w:Pentagram map#Collapsing of convex polygons|collapsing of convex polygons]] and the one about [[w:Pentagram map#Complete integrability|complete integrability]].}} The dynamic is trivial for the classes of pentagons and hexagons, but this stops to be the case for polygons with more vertices.{{Efn|See the paragraph about [[w:Pentagram_map#Pentagons_and_hexagons|pentagons and hexagons]].}}

=== Historical elements ===
The pentagram map for general polygons was introduced in {{Harvard citation|Schwartz|1992}}, but the simplest case is the one of [[w:pentagons|pentagons]], hence the name &quot;[[w:pentagram|pentagram]]&quot;.{{Sfn|Marí-Beffa|2014|p=1}} Their study goes back to {{Harvard citation|Clebsch|1871}},{{Sfn|Izosimov|2022a|p=1085}} {{Harvard citation|Kasner|1928}}{{Sfn|Tabachnikov|2019}} and {{Harvard citation|Motzkin|1945}}.{{Sfn|Schwartz|2013|p=1}}

The pentagram map interacts with some classical configuration theorems of [[w:projective geometry|projective geometry]]. It provides results analogous to the ones of [[w:Pascal's theorem|Pascal's theorem]] and [[wikipedia:Brianchon's_theorem|Brianchon's theorem]].{{Sfn|Schwartz|Tabachnikov|2010}} Some specific configurations make [[w:Desargues' theorem|Desargues's theorem]] and [[w:Poncelet's porism|Poncelet's porism]] appear.{{Sfn|Berger|2005|loc=§4 and §5}}{{Efn|See the paragraph about [[w:Pentagram map#Poncelet polygons|Poncelet polygons]].}}

==Definitions and first properties==

=== Definition of the map ===
[[File:Pentagram pentagon label big.svg|alt=|thumb|300x300px|The pentagram map on a convex pentagon, with vertices labeled.]]
[[File:Pentagram on nonconvex pentagon.svg|alt=|thumb|300x300px|The pentagram map applied on a [[w:Self-intersecting polygon|self-intersecting]] (in particular, non-convex) pentagon. The vertex &lt;math&gt;w_2&lt;/math&gt; is on the [[w:line at infinity|line at infinity]], because it is the [[w:Vanishing point|intersection of two parallel lines]].]]
Let &lt;math&gt;n\geq 5&lt;/math&gt; be an integer. A polygon &lt;math&gt;P&lt;/math&gt; with &lt;math&gt;n&lt;/math&gt; sides, or &lt;math&gt;n&lt;/math&gt;-gon, is a tuple of [[w:Vertex (geometry)|vertices]] &lt;math&gt;(v_1,\dots,v_n)&lt;/math&gt; lying in some [[w:projective plane|projective plane]] &lt;math&gt;\mathbb P ^2&lt;/math&gt;,{{Efn|In the following, the figures represent polygons on the real plane, where the intuition is easier to grasp.}} where the indices are understood [[w:Modular arithmetic|modulo]] &lt;math&gt;n&lt;/math&gt;. The [[w:Dimension of an algebraic variety|dimension]] of the space of &lt;math&gt;n&lt;/math&gt;-gons is &lt;math&gt;2n&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=definition 1.1}}

Suppose that the vertices are in sufficiently [[w:general position|general position]], meaning that no consecutive triple of points are [[w:Collinearity|collinear]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=1}} Taking the intersection of two consecutive &quot;shortest&quot; [[w:diagonal|diagonal]]s{{Efn|Meaning the line between a vertex &lt;math&gt;v_k&lt;/math&gt; and a &quot;second neighbour&quot; &lt;math&gt;v_{k\pm 2}&lt;/math&gt;.}} defines a new point&lt;math display=&quot;block&quot;&gt; w_k := \overline{v_{k-1} v_{k+1}} \cap \overline{v_{k} v_{k+2}}. &lt;/math&gt;This procedure defines a new &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;T(P)=(w_1,\dots,w_n)&lt;/math&gt;, as in Figure 2.{{Sfn|Schwartz|1992|p=71}}

The labeling of the indices of &lt;math&gt;T(P)&lt;/math&gt; is not [[w:canonical|canonical]]. In most papers, a choice is made at the beginning of the paper and the formulas are tuned accordingly.{{Sfn|Izosimov|2016|loc=remark 1.5}}

The pentagram map on polygons is a [[w:birational map|birational map]] &lt;math&gt;T:(\mathbb P^2)^n&lt;/math&gt;{{nowrap|{{font|size=145%|⇢}}}}&lt;math&gt;(\mathbb P^2)^n&lt;/math&gt;. Indeed, each [[w:Homogeneous coordinates|coordinate]] of &lt;math&gt;w_k&lt;/math&gt; is given as a [[w:rational function|rational function]] of the coordinates of &lt;math&gt;v_{k-1},\dots,v_{k+2}&lt;/math&gt;, since it is defined as the intersection of lines passing by them. Moreover, the [[w:inverse map|inverse map]] is given by taking the intersections &lt;math&gt;\overline{w_{k-2} w_{k-1}} \cap \overline{w_{k} w_{k+1}} &lt;/math&gt;, which is rational for the same reason.{{Sfnp|Weinreich|2022|loc=definition 1.2}}

=== Moduli space ===
The pentagram map is defined by taking [[w:Line (geometry)|lines]] and intersections of them. The biggest [[w:Group (mathematics)|group]] which maps lines to lines is the one of [[w:projective transformations|projective transformations]] &lt;math&gt;\mathbb P \mathrm{GL}_{3}&lt;/math&gt;. Such a transformation &lt;math&gt;M&lt;/math&gt; [[w:Group action|acts]] on a polygon &lt;math&gt;P&lt;/math&gt; by sending it to &lt;math&gt;M \cdot P:=(Mv_1,\dots,Mv_n)&lt;/math&gt;. The pentagram map [[w:Commutative property|commutes]] with this action, and thereby induces another [[w:dynamical system|dynamical system]] on the [[w:moduli space|moduli space]] of projective [[w:equivalence classes|equivalence classes]] of polygons. Its [[w:Dimension of an algebraic variety|dimension]] is &lt;math&gt;2n-8&lt;/math&gt;.{{Sfn|Schwartz|1992|loc=§1 Projective geometry}}

===Twisted polygons===
[[File:Twisted heptagon.svg|alt=|thumb|300x300px|An example of twisted [[w:heptagon|heptagon]] on the real plane.]]
The pentagram map naturally generalizes on the larger space of twisted polygons (see example in Figure 4). For any integer &lt;math&gt;n\geq5&lt;/math&gt;, a twisted &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;P&lt;/math&gt; is the data of:

* a [[w:Sequence#Indexing|bi-infinite sequence]] of points &lt;math&gt;(v_k)_{k\in\mathbb Z}&lt;/math&gt; in the projective plane (called the vertices),
* a [[w:projective transformation|projective transformation]] &lt;math&gt;M \in \mathbb P \mathrm{GL}_3&lt;/math&gt; (called the [[w:monodromy|monodromy]]),

such that for any &lt;math&gt;k \in \mathbb Z&lt;/math&gt;, the property &lt;math&gt;v_{k+n}=Mv_k&lt;/math&gt; is satisfied. The dimension of the space of twisted &lt;math&gt;n&lt;/math&gt;-gons is &lt;math&gt;2n+8&lt;/math&gt;.{{Sfn|Schwartz|2008}}

When &lt;math&gt;M=\mathrm{Id}&lt;/math&gt;, this gives back the initial definition of polygons (which are said to be closed). The space of closed &lt;math&gt;n&lt;/math&gt;-gons is of [[w:codimension|codimension]] &lt;math&gt;8&lt;/math&gt; in the space of twisted ones.{{Sfn|Soloviev|2013|p=2816}}

The action of projective transformations over the space of closed polygons generalizes to the space of twisted ones (the monodromy is changed by [[w:Matrix similarity|conjugation]]). This provides again a moduli space, of dimension &lt;math&gt;2n&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=definition 1.3}}

== Collapsing of convex polygons ==

=== Exponential shrinking ===
[[File:Pentagram map convex heptagon iterate.svg|alt=|thumb|300x300px|The pentagram map iterated on a convex [[w:heptagon|heptagon]], exhibiting the convergence.]]
Let &lt;math&gt;P&lt;/math&gt; be a closed [[w:Convex polygon#Strictly convex polygon|strictly convex polygon]] lying on the real plane. One of the first results proved by Richard Schwartz it that its iterates under the pentagram map shrink [[w:Exponential growth|exponentially fast]] to a point, as illustrated in Figure 5. This follows from two facts.
# The image of a strictly convex polygon is contained in its [[w:Interior (topology)|interior]], and is also strictly convex.{{Sfn|Glick|2020|p=2818}}
# There exists a constant &lt;math&gt;0&lt; \eta_P&lt;1&lt;/math&gt;, depending on &lt;math&gt;P&lt;/math&gt;, such that for any &lt;math&gt;N \in \mathbb N&lt;/math&gt;, the diameters of the iterates verify the inequality &lt;math display=&quot;inline&quot;&gt;\operatorname{diam}(T^N(P))\leq\eta_P^N \operatorname{diam}(P). &lt;/math&gt;{{Sfn|Schwartz|1992|loc=theorem 3.1}}

Hence, by [[w:Cantor's intersection theorem#Variant in complete metric spaces|Cantor's intersection theorem]], the sequence of polygons collapses toward a point.{{Sfn|Schwartz|1992|loc=§3 Convex polygons}}

The behavior on the moduli space is very different, since the dynamics is [[w:Recurrent point|recurrent]].{{Sfn|Schwartz|2001|loc=theorem 1.1}} It is even a [[w:quasiperiodic motion|quasiperiodic motion]],{{Sfn|Ovsienko|Schwartz|Tabachnikov|2009}} as discussed in [[w:Pentagram map#Complete integrability|the section about integrability]].

=== Coordinates of the limit point ===
The limit point coordinates are found in {{Harvard citation|Glick|2020}}. They satisfy some [[w:Degree of a polynomial|degree]] 3 [[w:polynomial equation|polynomial equations]], whose coefficients are [[w:rational function|rational function]]s in the coordinates of the vertices of the starting polygon. The proof relies on the fact that the limit point must be an [[w:eigenline|eigenline]] of a certain [[w:linear operator|linear operator]] of &lt;math&gt;\mathbb R^3&lt;/math&gt;.{{sfn|Glick|2020}}

This operator was reinterpreted in {{Harvard citation|Aboud|Izosimov|2022}} as the infinitesimal monodromy of the polygon. The [[w:Pentagram map#The scaling symmetry|scaling symmetry]] is used to [[w:Deformation (mathematics)|deform]] a closed polygon &lt;math&gt;P&lt;/math&gt; into a family of twisted ones '''&lt;math&gt;(P_z)_{z\in \mathbb C^*}&lt;/math&gt;''' with monodromy &lt;math&gt;M_z&lt;/math&gt;. The infinitesimal monodromy is defined to be:{{sfn|Aboud|Izosimov|2022}}
&lt;math display=&quot;block&quot;&gt;\left.\frac{dM_z}{dz}\right|_{z=1}.&lt;/math&gt;

=== Generalization ===
The collapsing of polygons may also happen in some [[w:Pentagram map#Generalizations|generalization of the pentagram map]], when considering some specific configurations of polygons in the real plane. The coordinates of the collapse point are given by a formula analogous to the one for the original pentagram map.{{Sfn|Schwartz|2026}}

== Periodic orbits on the moduli space ==
For some configurations of closed polygons, the iterate of the pentagram map will send &lt;math&gt;P&lt;/math&gt; to a projectively equivalent polygon (up to some shift of the indices). This means that, on the moduli space, the orbit of the class of &lt;math&gt;P&lt;/math&gt; is [[w:Periodic orbit|periodic]].

===Pentagons and hexagons===
[[File:penta hexagon.svg|300px|thumb|The outward hexagon is projectively equivalent to the inward one, with respect to their labeling.]]The following two facts are proved by checking [[w:cross-ratio|cross-ratio]] equalities, so they are true for polygons in any [[w:projective plane|projective plane]] (not just the [[w:Real projective plane|real one]]).{{Sfn|Schwartz|1992|loc=§2 Pentagons and hexagons}}

The pentagram map &lt;math&gt;T&lt;/math&gt; is the identity on the moduli space of [[w:pentagon|pentagon]]s.{{Sfn|Schwartz|1992|loc=theorem 2.1}}{{Sfn|Clebsch|1871}}{{Sfn|Motzkin|1945}} The second iterate &lt;math&gt;T^2&lt;/math&gt; is the identity on the space of labeled [[w:hexagon|hexagon]]s, up to a shift of labeling (see Figure 6).{{Sfn|Schwartz|1992|loc=theorem 2.3}} This phenomenon doesn't generalize to generic polygons with at least seven sides, for which the motion is [[w:Quasiperiodic motion|quasi-periodic]].{{Sfn|Tupan|2022}}

==== Generalization ====
The result about pentagons and hexagons generalizes to some [[w:Pentagram map#Generalizations|higher pentagram maps]] in &lt;math&gt;\mathbb P ^k&lt;/math&gt;, for polygons with &lt;math&gt;k+3&lt;/math&gt; or &lt;math&gt;2k+2&lt;/math&gt; sides. The proof uses a generalization of the [[w:Gale transform|Gale transform]].{{Sfn|Dirdak|2024}}

=== Poncelet polygons ===
A polygon is said to be Poncelet{{Efn|The name comes from [[w:Jean-Victor Poncelet|Jean-Victor Poncelet]] and [[w:Poncelet porism|his porism]].{{Sfn|Izosimov|2022a|p=1085}}}} if it is [[w:Inscribed figure|inscribed]] in a [[w:Conic section|conic]] and circumscribed about another one.{{Sfn|Schwartz|2015|loc=|p=433}}{{Efn|In particular, pentagons are Poncelet since [[w:five points determine a conic|five points determine a conic]].{{Sfn|Schwartz|2015|loc=|p=433}}}} For a convex Poncelet &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;P&lt;/math&gt; lying on the [[w:real projective plane|real projective plane]], the polygon &lt;math&gt;T^2(P)&lt;/math&gt; is projectively equivalent to &lt;math&gt;P&lt;/math&gt;.{{Sfn|Schwartz|2015|loc=theorem 1.1}} In fact, when &lt;math&gt;n&lt;/math&gt; is odd, the converse is also true.{{Sfn|Izosimov|2022a|loc=corollary 1.1}}

However, this converse statement is no longer true when the polygons are considered over the [[w:complex projective plane|complex projective plane]].{{Sfn|Izosimov|2022a|loc=remark 1.3}}

==Coordinates for the moduli space==
The moduli space can be described by different [[w:Coordinate_system|coordinate systems]]. The following ones are practical to explicit the dynamic, as presented in the next section.

=== Corner coordinates ===
[[File:Corner coordinates big.svg|thumb|300x300px|The geometric construction of the points defining the corner invariants.]]
Define the [[w:cross-ratio|cross-ratio]] of four [[w:Collinearity|collinear]] points to be

: &lt;math&gt; [a,b,c,d]=\frac{(a-b)(c-d)}{(a-c)(b-d)}. &lt;/math&gt;

The corner invariants are a system of coordinates on the space of twisted polygons, constructed by taking intersections as in Figure 7.{{Sfn|Schwartz|2001|loc=figure 2}} The left and right invariants are respectively defined{{Efn|The ordering of the vertices in the cross-ratios can differ from a paper to another one, which slightly changes the formulas in the following sections.}} as the following cross-ratios:

: &lt;math&gt;x_k:=[v_{k-2},v_{k-1},\overline{v_{k-2}v_{k-1}}\cap\overline{v_{k}v_{k+1}},\overline{v_{k-2}v_{k-1}}\cap\overline{v_{k+1}v_{k+2}}],&lt;/math&gt;

: &lt;math&gt;y_k:=[\overline{v_{k+1}v_{k+2}}\cap\overline{v_{k-2}v_{k-1}}, \overline{v_{k+1}v_{k+2}}\cap\overline{v_{k-1}v_{k}},v_{k+1},v_{k+2}].&lt;/math&gt;

Since the cross-ratio is [[w:Cross-ratio#Projective geometry|projective invariant]], the sequences &lt;math&gt;(x_k)_{k \in \mathbb Z}&lt;/math&gt; and &lt;math&gt;(y_k)_{k \in \mathbb Z}&lt;/math&gt; associated to a twisted &lt;math&gt;n&lt;/math&gt;-gon are &lt;math&gt;n&lt;/math&gt; periodic.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=415}}

The corner invariants are elements of &lt;math&gt;\mathbb{P}^1\smallsetminus\{0,1,\infty\}&lt;/math&gt;, and they realize an [[w:Isomorphism_of_varieties|isomorphism of variety]] between the moduli space of twisted &lt;math&gt;n&lt;/math&gt;-gons and &lt;math&gt;(\mathbb{P}^1\smallsetminus\{0,1,\infty\})^{2n}&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=theorem 3.6}}

===ab-coordinates===

There is a second set of coordinates for the moduli space of twisted &lt;math&gt;n&lt;/math&gt;-gons defined over a [[w:Field (mathematics)|field]] &lt;math&gt;F&lt;/math&gt; satisfying &lt;math&gt;\mathrm{SL}_3(F)\cong \mathbb P\mathrm{GL}_3(F)&lt;/math&gt;,{{Sfn|Weinreich|2022|loc=remark 3.8}} and such that &lt;math&gt;n&lt;/math&gt; is not divisible by &lt;math&gt;3&lt;/math&gt;.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=section 4.1}}

The vertices &lt;math&gt;v_k&lt;/math&gt; in the [[w:projective plane|projective plane]] &lt;math&gt;\mathbb P^2(F)&lt;/math&gt; can be [[w:Lift (mathematics)|lifted]] to [[w:Vector space|vectors]] &lt;math&gt;V_k&lt;/math&gt; in the [[w:affine space|affine space]] &lt;math&gt;F^3&lt;/math&gt; so that each consecutive triple of vectors spans a [[w:parallelepiped|parallelepiped]] having [[w:determinant|determinant]] equal to &lt;math&gt;1&lt;/math&gt;. This leads to the relation{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=equation 4.1}}

: &lt;math&gt;V_{k+3} = a_k V_{k+2} + b_k V_{k+1} + V_k.&lt;/math&gt;

This bring out an analogy between twisted polygons and solutions of third order linear [[w:ordinary differential equations|ordinary differential equations]], normalized to have unit [[w:Wronskian|Wronskian]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=remark 6.6}}

They are linked to the corner coordinates by:{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=lemma 4.5}}
: &lt;math&gt;x_k=\frac{a_{k-2}}{b_{k-2}b_{k-1}},&lt;/math&gt;
: &lt;math&gt;y_k=-\frac{b_{k-1}}{a_{k-2}a_{k-1}}.&lt;/math&gt;

==Formulas on the moduli space==

===As a birational map ===
The pentagram map is a [[w:birational map|birational map]] on the moduli space, because it can be decomposed as the [[w:Function composition|composition]] of two [[w:Birational geometry|birational]] [[w:Involution (mathematics)|involutions]].{{Sfn|Schwartz|2008|loc=§1.2 The Pentagram Map}} The corner invariants change in the following way:{{Sfn|Ovsienko|Schwartz|loc=lemma 2.4|Tabachnikov|2010}}

: &lt;math&gt;x_k'=x_k\frac{1-x_{k-1} y_{k-1}}{1-x_{k+1}y_{k+1}},&lt;/math&gt;

: &lt;math&gt;y_k'=y_{k+1}\frac{1-x_{k+2} y_{k+2}}{1-x_k y_k}.&lt;/math&gt;

=== The scaling symmetry ===
The [[w:multiplicative group|multiplicative group]] &lt;math&gt;F\smallsetminus\{0\}&lt;/math&gt; [[w:One-parameter group|acts]] on the moduli space in the following way:

: &lt;math&gt;R_s\cdot(x_1,\dots,x_n,y_1,\dots,y_n)=(sx_1,\dots,sx_n,s^{-1}y_1,\dots,s^{-1}y_n),&lt;/math&gt;

where &lt;math&gt;R&lt;/math&gt; is called the scaling action and &lt;math&gt;s&lt;/math&gt; is the scaling parameter. This action commutes with the pentagram map on the moduli space (as presented in the previous formulas). This property is called the scaling symmetry, and is instrumental in proving the [[w:Pentagram map#Complete integrability|complete integrability]] of the dynamics.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=corollary 2.5}}

==Invariant structures==

===Monodromy invariants===

The monodromy invariants, introduced in {{Harvard citation|Schwartz|2008}}, are a collection of [[w:Function (mathematics)|functions]] on the [[w:moduli space|moduli space]] that are invariant under the pentagram map.{{Sfn|Schwartz|2008|loc=theorem 1.2}} The simplest example of them are 
:&lt;math&gt; O_n= x_1x_2\cdots x_{n}, \quad E_n = y_1y_2\cdots y_n. &lt;/math&gt;
The other monodromy invariants can be retrieved through different points of view: through the [[w:Pentagram map#The scaling symmetry|scaling symmetry]], as [[w:Combinatorics|combinatorial]] objects, or as some [[w:determinant|determinant]]s.{{Sfn|Schwartz|Tabachnikov|2011|loc=§2 The Monodromy Invariants}} The one involving scaling symmetry is presented here.

Let &lt;math&gt;M\in \mathrm{GL}_3&lt;/math&gt; be a [[w:Lift (mathematics)|lift]] of the monodromy of a twisted &lt;math&gt;n&lt;/math&gt;-gon. The quantities

: &lt;math&gt;\Omega_1=\frac{\operatorname{trace}^3(M)}{\det(M)}, \quad \Omega_2=\frac{\operatorname{trace}^3(M^{-1})}{\det(M^{-1})},&lt;/math&gt;

are independent of the choice of lift and are invariant under [[w:Matrix similarity|conjugation]], so they are well defined for the projective class of the polygon. They are invariant under the pentagram map, since the monodromy matrix doesn't change.{{Sfn|Schwartz|Tabachnikov|2011|loc=|p=5}} Now, the quantities

: &lt;math&gt;\tilde{\Omega}_1=O_n^2E_n\Omega_1, \quad \tilde{\Omega}_2=O_nE_n^2\Omega_2,&lt;/math&gt;

have the same properties, but turn out to be polynomials in the corner invariants.{{Efn|Some papers consider the cube roots of this functions, but it doesn't change the following definitions of the monodromy invariants.}} They can be written as{{Sfn|Schwartz|Tabachnikov|2011|loc=|p=5}}

: &lt;math&gt;
\tilde{\Omega}_1=\biggl(\sum_{k=0}^{\lfloor n/2\rfloor}O_k\biggr)^3, \quad
\tilde{\Omega}_2=\biggl(\sum_{k=0}^{\lfloor n/2\rfloor}E_k\biggr)^3,
&lt;/math&gt;

where each &lt;math&gt;O_k&lt;/math&gt; and &lt;math&gt;E_k&lt;/math&gt; are [[w:homogeneous polynomial|homogeneous polynomial]]s respectively of weight &lt;math&gt;k&lt;/math&gt; and &lt;math&gt;-k&lt;/math&gt;,{{Sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=11}} meaning they change under the [[w:Pentagram map#The scaling symmetry|rescaling action]] on variables by{{Sfn|Schwartz|Tabachnikov|2011|p=5}}

: &lt;math&gt; R_s(O_k)= s^k O_k, \quad R_s(E_k)= s^{-k} E_k. &lt;/math&gt;

The quantities &lt;math&gt;O_1,\dots,O_{\lfloor n/2 \rfloor},O_n, E_1,\dots,E_{\lfloor n/2 \rfloor},E_n,&lt;/math&gt; are unchanged by the dynamics, and are called the monodromy invariants. Moreover, they are [[w:algebraically independent|algebraically independent]].{{Sfn|Schwartz|2008|loc=theorem 1.2}}

==== Polygons on conics ====
Whenever &lt;math&gt;P&lt;/math&gt; is [[w:Inscribed figure|inscribed]] on a [[w:conic section|conic section]], one has &lt;math&gt;O_k(P)=E_k(P)&lt;/math&gt; for all &lt;math&gt;k&lt;/math&gt;.{{Sfn|Schwartz|Tabachnikov|2011|loc=theorem 1.1}} Moreover, if &lt;math&gt;P&lt;/math&gt; is circumscribed about another conic,{{Efn|See the paragraph about [[w:Pentagram map#Poncelet polygons|Poncelet polygons]].}} then its monodromy invariants are characterized by the pair of conics.{{Sfn|Schwartz|2015|loc=theorem 1.2}} For such odd-gons, the translation on the [[w:Jacobian variety|Jacobian variety]]{{Efn|See the paragraph about [[w:Pentagram map#Algebro-geometric integrability|algebraic integrability]].}} is restricted to the [[w:Prym variety|Prym variety]] (which is a half-dimensional torus in the Jacobian).{{Sfn|Izosimov|2016|loc=theorem 1.3}}

===Poisson bracket===

An invariant [[w:Poisson bracket|Poisson bracket]] on the space of twisted polygons was found in {{Harvard citation|Ovsienko|Schwartz|Tabachnikov|2010}}. The monodromy invariants [[w:Poisson bracket#Constants of motion|commute]] with respect to it:
&lt;math display=&quot;block&quot;&gt; \{O_i,O_j\}=\{O_i,E_j\}=\{E_i,E_j\}=0 &lt;/math&gt;for all &lt;math&gt;i,j&lt;/math&gt;.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}}

The Poisson bracket is defined in terms of the corner coordinates by:
&lt;math display=&quot;block&quot;&gt; \begin{align}
 \{x_i,x_{i\pm1}\} &amp;= \mp x_i x_{i+1}, \\
 \{y_i,y_{i\pm 1}\} &amp;= \mp y_i y_{i+1}, \\
 \{x_i,x_j\} &amp;= \{y_i,y_j\} = \{x_i,y_j\} = 0
\end{align}&lt;/math&gt;for all other &lt;math&gt; i,j.&lt;/math&gt;{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=equation 2.16}}

=== The spectral curve ===

Let &lt;math&gt;\zeta&lt;/math&gt; be an element of the [[w:multiplicative group|multiplicative group]] and &lt;math&gt;P_\zeta&lt;/math&gt; be the polygon obtained by applying the [[w:Pentagram map#The scaling symmetry|rescaling action]] &lt;math&gt;R_\zeta&lt;/math&gt; on &lt;math&gt;P&lt;/math&gt;.  A [[w:Lax matrix|Lax matrix]] &lt;math&gt;\hat{T}(\zeta) \in \mathrm{GL}_3&lt;/math&gt; is a lift of the monodromy of &lt;math&gt;P_\zeta&lt;/math&gt; satisfying a [[w:Lax pair#Zero-curvature equation|zero-curvature equation]].{{Sfn|Weinreich|2022|loc=§5 The Lax representation}} Then, the spectral function is the [[w:Bivariate polynomial|bivariate]] [[w:characteristic polynomial|characteristic polynomial]]
&lt;math display=&quot;block&quot;&gt; Q(\lambda,\zeta) := \det(\lambda\operatorname{Id}-\hat{T}(\zeta)),&lt;/math&gt;or some renormalization of it. The [[w:spectral curve|spectral curve]] is the [[w:Projective variety#projective completion|projective completion]] of the [[w:Algebraic curve|affine curve]] defined by the equation &lt;math&gt;Q(\lambda,\zeta)=0&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=§6. The geometry of the spectral curve}} It is invariant under the pentagram map, and the monodromy invariants appear as the [[w:coefficient|coefficient]]s of &lt;math&gt;Q&lt;/math&gt;.{{Sfn|Soloviev|2013|loc=theorem 6.4}} Its [[w:geometric genus|geometric genus]] is &lt;math&gt;n-1&lt;/math&gt; if &lt;math&gt;n&lt;/math&gt; is odd, and &lt;math&gt;n-2&lt;/math&gt; if &lt;math&gt;n&lt;/math&gt; is even.{{Sfn|Weinreich|2022|p=|loc=theorem 6.4}}

It was first introduced in {{Harv|Soloviev|2013|ps=|p=}} for his proof of [[w:Pentagram map#Algebro-geometric integrability|algebro-geometric integrability]].{{sfn|Soloviev|2013}}

==Complete integrability==
The pentagram map on the moduli space has been proved to be a [[w:completely integrable|completely integrable]] [[w:discrete dynamical system|discrete dynamical system]], both in the [[w:Integrable system#Hamiltonian systems and Liouville integrability|Arnold-Liouville]]{{Efn|Over the [[w:real number|real number]]s.}} and the [[w:Integrable system#Complete integrability over the complex numbers|algebro-geometric]]{{Efn|Over [[w:algebraically closed field|algebraically closed field]]s of [[w:Characteristic (algebra)|characteristic]] different from 2.}} senses. In any case, this means that the moduli space is [[w:almost everywhere|almost everywhere]] [[w:Foliation|foliated]] by [[w:Torus#Flat torus|flat tori]] (or in the algebraic setting, [[w:Abelian variety|Abelian varieties]]), where the motion is a [[w:Translation (geometry)|translation]]. This [[w:Generic property|generically]] induces a [[w:quasiperiodic motion|quasiperiodic motion]] on the corresponding torus.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2009}}

===Arnold–Liouville integrability===

The proof of the integrability of the pentagram map on a real twisted polygon was achieved in {{Harvard citation|Ovsienko|Schwartz|Tabachnikov|2010}}. This is done by noticing that the monodromy invariants &lt;math&gt;O_n&lt;/math&gt; and &lt;math&gt;E_n&lt;/math&gt; are [[w:Casimir invariant|Casimir invariant]]s for the bracket, meaning (in this context) that&lt;math display=&quot;block&quot;&gt; \{O_n,f\}=\{E_n,f\} = 0 &lt;/math&gt;for all functions &lt;math&gt;f&lt;/math&gt;.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}} When &lt;math&gt;n&lt;/math&gt; is even, this is also true for the monodromy invariants &lt;math&gt;O_{\lfloor n/2 \rfloor }&lt;/math&gt; and &lt;math&gt;E_{\lfloor n/2 \rfloor }&lt;/math&gt;.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}}

This allows to consider the Casimir [[w:level set|level set]], where each Casimir has a specified value. Because of [[w:Sard's theorem|Sard's theorem]], any generic level set is a [[w:smooth manifold|smooth manifold]].{{Sfn|Schwartz|2017|p=44}} They form a [[w:foliation|foliation]] in [[w:Poisson manifold#Symplectic leaves|symplectic leaves]], on which the Poisson bracket gives rise to a [[w:symplectic form|symplectic form]].{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=corollary 2.13}}

Each of these symplectic leaves has an iso-monodromy [[w:foliation|foliation]], namely, a decomposition into the common level sets of the remaining monodromy functions. By using again [[w:Sard's theorem|Sard's theorem]], they are generically [[w:Symplectic manifold#Lagrangian submanifolds|Lagrangian manifolds]].{{Sfn|Schwartz|2017|p=45}} Moreover, they are compact.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=§3.3. Compactness of the level sets}} Since the monodromy invariants Poisson-commute and there are enough of them, the discrete [[w:Liouville–Arnold theorem|Liouville–Arnold theorem]] can be applied to prove that the level sets are [[w:Torus#Flat torus|flat tori]] over which the dynamics is a translation.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=412}}
===Algebro-geometric integrability===

In {{Harvard citation|Soloviev|2013}}, it was shown that the pentagram map admits a [[w:Lax representation|Lax representation]] with a spectral parameter, which allows to prove its algebro-geometric integrability. This means that the space of polygons (either twisted or closed) is parametrized by its spectral data, consisting of [[w:Pentagram map#The spectral curve|its spectral curve]], with marked points and a [[w:Divisor (algebraic geometry)|divisor]] given by a [[w:Floquet theory|Floquet]]–[[w:Bloch's theorem|Bloch]] equation. This gives an embedding to the [[w:Jacobian variety|Jacobian variety]] through the [[w:Abel–Jacobi map|Abel–Jacobi map]], where the motion is expressed in terms of translation.{{sfn|Soloviev|2013|loc=theorems A, B and C}} The previously defined Poisson bracket is also retrieved.{{sfn|Soloviev|2013|loc=theorem D}} 

This integrability was generalized in {{Harvard citation|Weinreich|2022}} from the field of [[w:complex number|complex number]]s to any [[w:algebraically closed field|algebraically closed field]] of [[w:Characteristic (algebra)|characteristic]] different from 2. The translation on a torus is replaced by a translation on an [[w:Abelian variety|Abelian variety]] (in fact, a Jacobian variety again).{{sfn|Weinreich|2022|loc=theorem 1.4}}

=== Dimension of the invariant manifold ===

For twisted &lt;math&gt;n&lt;/math&gt;-gons, the [[w:dimension|dimension]] of the invariant tori (or Jacobian varieties) is{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=421}}

: &lt;math&gt;\begin{cases} 
n-1 &amp; \text{when }n \text{ is odd,}\\
n-2 &amp; \text{when }n \text{ is even.}
\end{cases}&lt;/math&gt;

Moreover, when &lt;math&gt;n&lt;/math&gt; is even, there are two isomorphic Jacobians on which the iterates of the pentagram map alternate. But on each of them, the second iterate is a translation.{{Sfn|Weinreich|2022|loc=theorem 1.4}}

=== For closed polygons ===
There is no Poisson structure on the space of closed polygons.{{Sfn|Soloviev|2013|loc=corollary 4.1}} Nevertheless, the one from twisted polygons can be used to prove integrability.{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=2153}}

Algebro-geometric integrability holds for closed polygons in a same manner as for the twisted ones.{{Sfn|Soloviev|2013|loc=theorem C}} However, Arnold-Liouville integrability is proved for real closed polygons only when they are convex. This is done by restricting the [[w:Hamiltonian vector field|Hamiltonian vector field]]s of monodromy functions to smaller dimensional tori, and showing that enough of them are still independent.{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|loc=corollary 1.1}}

In both situation, the dimension of the invariant manifolds decreases by &lt;math&gt;3&lt;/math&gt; for closed &lt;math&gt;n&lt;/math&gt;-gons (compared to the twisted case), and is equal to{{Sfn|Soloviev|2013|loc=theorem C}}{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|loc=theorem 1}}
:&lt;math&gt;\begin{cases} 
n-4 &amp; \text{when }n \text{ is odd,}\\
n-5 &amp; \text{when }n \text{ is even.}
\end{cases}&lt;/math&gt;

==Connections to other topics==

===The Boussinesq equation===

The continuous limit of a convex polygon is a parametrized convex curve in the plane. When the time parameter is suitably chosen, the [[w:Discretization|continuous limit]] of the pentagram map is the classical [[w:Boussinesq approximation (water waves)|Boussinesq equation]]. This equation is a classical example of an [[w:integrable|integrable]] [[w:partial differential equation|partial differential equation]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 5}}

Here is a description of the geometric action of the Boussinesq equation. Given a [[w:locally convex|locally convex]] curve &lt;math&gt; C:\mathbb R\to \mathbb R^2 &lt;/math&gt; and real numbers &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;t&lt;/math&gt;, consider the [[w:chord (geometry)|chord]] connecting &lt;math&gt; C(x-t) &lt;/math&gt; to &lt;math&gt; C(x+t) &lt;/math&gt;. The [[w:Envelope (mathematics)|envelope]] of all these chords is a new curve &lt;math&gt; C_t(x) &lt;/math&gt;. When &lt;math&gt;t&lt;/math&gt; is extremely small, the curve &lt;math&gt; C_t(x) &lt;/math&gt; is a good model for the time &lt;math&gt;t&lt;/math&gt; evolution of the original curve &lt;math&gt; C_0(x) &lt;/math&gt; under the Boussinesq equation. This construction is also similar to the pentagram map. Moreover, the pentagram invariant bracket is a discretization of a well known invariant Poisson bracket associated to the Boussinesq equation.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=§6.4 Discretization}}

===Cluster algebras===

The pentagram map{{Sfn|Glick|2011}} and some of its generalizations{{Sfn|Gekhtman|Shapiro|Tabachnikov|Vainshtein|2012}}{{Sfn|Glick|Pylyavskyy|2016}} are identified as special cases of discrete dynamical systems powered by [[w:cluster algebra|cluster algebra]]. Using the results from {{Harvard citation|Goncharov|Kenyon|2013}}, this provides a link with the [[w:Poisson–Lie group|Poisson–Lie group]]s, [[w:dimer model|dimer model]]s and other so-called cluster-integrable systems.{{Sfn|Fock|Marshakov|2016}}{{Sfn|Izosimov|2022c}} These methods allow to retrieve the Poisson-bracket and Hamiltonians used to prove complete integrability{{Sfn|Affolter|George|Ramassamy|2025|loc=§5 The pentagram map}} and provide [[w:Lax representation|Lax representation]]s.{{Sfn|Izosimov|2022b}}

=== Singularity theory ===
The pentagram map exhibit a property called singularity confinement, which is typical from [[w:integrable system|integrable system]]s.{{Sfn|Grammaticos|Ramani|Papageorgiou|1991}} It states that if a polygon &lt;math&gt;P&lt;/math&gt; is [[w:Singular point of an algebraic variety|singular]] for the pentagram map &lt;math&gt;T&lt;/math&gt;, then there exists an integer &lt;math&gt;m&lt;/math&gt; such that &lt;math&gt;P&lt;/math&gt; not singular for the iterate map &lt;math&gt;T^m&lt;/math&gt;.{{Sfn|Glick|2012}}

Moreover, the pentagram map (along with some of its generalizations and other discrete dynamical systems) exhibit the Devron property.{{Efn|The name comes from an episode of [[w:Star Trek|Star Trek]].{{Sfn|Glick|2015|loc=§1 Introduction}}}} This means that if a polygon &lt;math&gt;P&lt;/math&gt; is singular for some iterate of the pentagram map &lt;math&gt;T^m&lt;/math&gt;, then it will also be singular for some iterate of the inverse map &lt;math&gt;T^{-m'}&lt;/math&gt;.{{Sfn|Glick|2015}}

== Generalizations ==
The definition of twisted polygons still makes sense in any [[w:projective space|projective space]] &lt;math&gt;\mathbb P^d&lt;/math&gt;, under the action of the [[w:Projective linear group|projective group]] &lt;math&gt;\mathbb P \mathrm{GL}_{d+1}&lt;/math&gt;. The pentagram map can be generalized in many ways, and some of them are presented here. Not all of them are integrable.{{Sfn|Khesin|Soloviev|2015|}} Some are [[w:discretization|discretization]]s of [[w:PDEs|PDEs]] from the [[w:KdV hierarchy|KdV hierarchy]], seen as higher dimensional version of [[w:Boussinesq approximation (water waves)|Boussinesq]] or [[w:Kadomtsev–Petviashvili equation|KP]] equations.{{Sfn|Marí-Beffa|2012}}{{Sfn|Wang|2023}} The description of all generalized pentagram maps in terms of [[w:cluster algebra|cluster algebra]]s is still an open question.{{Sfn|Gekhtman|Izosimov|2025|p=14}}

=== Polygons in general positions ===
Let &lt;math&gt;d \geq 2&lt;/math&gt; and &lt;math&gt;P&lt;/math&gt; be a twisted polygon of &lt;math&gt;\mathbb P^d&lt;/math&gt; in [[w:general position|general position]].

==== Short diagonal pentagram maps ====
The &lt;math&gt;k&lt;/math&gt;-th ''short diagonal hyperplane'' &lt;math&gt;H_k^{sh}&lt;/math&gt; is uniquely defined by passing through the vertices &lt;math&gt;v_k,v_{k+2},\dots,v_{k+2d-2}&lt;/math&gt;. [[w:Generic property#In algebraic geometry|Generically]], the intersection of &lt;math&gt;d&lt;/math&gt; consecutive hyperplanes uniquely defines a new point

: &lt;math&gt;T_{sh}v_k:=H_k^{sh}\cap H_{k+1}^{sh}\cap \dots \cap H_{k+d-1}^{sh}.&lt;/math&gt;

Doing this for every vertex defines a new twisted polygon. This map, denoted by &lt;math&gt;T_{sh}&lt;/math&gt;, is again projectively equivariant.{{Sfn|Khesin|Soloviev|2013}}

==== Generalized pentagram maps ====
The previous procedure can be generalized. Let &lt;math&gt;I=(i_1,\dots,i_{d-1}),~J=(j_1,\dots,j_{d-1})&lt;/math&gt; be two sets of integers, respectively called the jump tuple and the intersection tuple. Define the &lt;math&gt;k&lt;/math&gt;-th hyperplane &lt;math&gt;H_k^I&lt;/math&gt; to be passing through the vertices &lt;math&gt;v_k,v_{k+i_1},\dots,v_{k+i_1+\dots+i_{d-1}}&lt;/math&gt;. A new point is given by the intersection

: &lt;math&gt;T_{I,J}v_k:=H_k^I \cap H_{k+j_1}^I \cap \dots \cap H_{k+j_1+\dots +j_{d-1}}^I.&lt;/math&gt;

The map &lt;math&gt;T_{I,J}&lt;/math&gt; is called a generalized pentagram map.{{Sfn|Khesin|Soloviev|2015a}} The original pentagram map is recovered by considering&lt;math&gt;d=2,~I=(2),~J=(1)&lt;/math&gt;.

Integrability can be numerically tested by picking a random polygon &lt;math&gt;P&lt;/math&gt; with [[wikipedia:Rational_point|rational coordinates]] and studying the growth rate of the [[wikipedia:Height_function|height]] of its iterates. This is called the [[wikipedia:Integrable_system#Diophantine_integrability|diophantine integrability]] test, and some generalized pentagram maps don't seem to pass it.{{Sfn|Khesin|Soloviev|2015a|loc=§5 and §6}} However, it is conjectured that the maps &lt;math&gt;T_{I,I}&lt;/math&gt; are integrable for any &lt;math&gt;I&lt;/math&gt;.{{Sfn|Bolsinov|Matveev|Miranda|Tabachnikov|2018|loc=conjecture 4.13 (B. Khesin, F. Soloviev)}}

Some of these maps are [[w:discretization|discretization]]s of higher dimensional counterpart of the [[w:Boussinesq approximation (water waves)|Boussinesq equation]] in the [[w:KdV hierarchy|KdV hierarchy]].{{Sfn|Khesin|Soloviev|2015b|loc=theorem 4.1}}{{Sfn|Izosimov|2022b|loc=theorem 4.1}}

==== Dented pentagram maps ====
Fix an integer &lt;math&gt;m\in \{1,\dots ,d-1\}&lt;/math&gt;. Consider the jump tuple &lt;math&gt;I_m:=(1,\dots,1,2,1,\dots,1)&lt;/math&gt;, where the &lt;math&gt;2&lt;/math&gt; is at the &lt;math&gt;m&lt;/math&gt;-th place, and the intersection tuple &lt;math&gt;J:=(1,\dots,1)&lt;/math&gt;. The dented pentagram map is &lt;math&gt;T_m :=T_{I_m,J}&lt;/math&gt;. They are proved to be integrable.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 2.14}}

For an integer &lt;math&gt;p \geq 2&lt;/math&gt;, the deep dented pentagram map (of depth &lt;math&gt;p&lt;/math&gt;) &lt;math&gt;T_m^p&lt;/math&gt; is the same map as before, but the number &lt;math&gt;2&lt;/math&gt; in the definition of &lt;math&gt;I_m&lt;/math&gt; is replaced by &lt;math&gt;p&lt;/math&gt;. This kind of pentagram maps are again integrable.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 6.2}}

=== Corrugated polygons ===
A twisted polygon &lt;math&gt;P&lt;/math&gt; lying in &lt;math&gt;\mathbb P^d&lt;/math&gt; is said to be corrugated if for any &lt;math&gt;k\in \mathbb Z&lt;/math&gt;, the vertices &lt;math&gt;v_k,v_{k+1},v_{k+d},v_{k+d+1}&lt;/math&gt; span a projective two-dimensional plane. Such polygons are not in [[w:general position|general position]]. A new point is defined by

: &lt;math&gt;T_\text{cor}v_k:=\overline{v_k v_{k+d}}\cap \overline{v_{k+1} v_{k+d+1}}.&lt;/math&gt;

The map &lt;math&gt;T_\text{cor}&lt;/math&gt; yields a new corrugated polygon. They are [[w:Integrable system#Hamiltonian systems and Liouville integrability|completely Liouville-integrable]].{{Sfn|Gekhtman|Shapiro|Tabachnikov|Vainshtein|2012|loc=theorem 4.4}}

In fact, they can be retrieved as some dented pentagram map applied on corrugated polygons.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 5.3}}

=== Grassmannian polygons ===
Let &lt;math&gt;d \geq 3, m \geq 1&lt;/math&gt; be integers. The pentagram map can also be generalized to the [[w:Grassmannian|Grassmannian]] space &lt;math&gt;\mathrm{Gr}(m,md)&lt;/math&gt;, which consists of &lt;math&gt;m&lt;/math&gt;-[[w:Dimension (vector space)|dimensional]] [[w:linear subspace|linear subspace]]s of an &lt;math&gt;md&lt;/math&gt;-dimensional [[w:vector space|vector space]]. When &lt;math&gt;m=1&lt;/math&gt;, the linear subspaces are [[w:Vector space#vector line|lines]], which retrieves the definition of [[w:projective space|projective space]]s &lt;math&gt;\mathbb P^d&lt;/math&gt;.{{Sfn|Felipe|Marí-Beffa|2019|loc=§2 definitions and notations}}

A point &lt;math&gt;v\in\operatorname{Gr}(m,md)&lt;/math&gt; is represented by an &lt;math&gt;md \times m&lt;/math&gt; matrix &lt;math&gt;X_v&lt;/math&gt; such that its columns form a [[w:Basis (linear algebra)|basis]] of &lt;math&gt;v&lt;/math&gt;. Consider the [[w:Group action|action]] of the [[w:general linear group|general linear group]] &lt;math&gt;\mathrm{GL}_{md}&lt;/math&gt; by multiplication on the left of &lt;math&gt;X_v&lt;/math&gt;. This defines an action on the Grassmannian, even though it is not [[w:Faithful action|faithful]].{{Efn|Because there can be many lifts for &lt;math&gt;v&lt;/math&gt;, and because some matrices act trivially.}} Hence, the polygons of &lt;math&gt;\mathrm{Gr}(m,md)&lt;/math&gt; and their moduli spaces are defined as before, after the change of underlying group.{{Sfn|Felipe|Marí-Beffa|2019|loc=§2 definitions and notations}}

Depending on the parity of &lt;math&gt;d&lt;/math&gt;, one can define linear subspaces spanned by some &lt;math&gt;X_{v_k}&lt;/math&gt;'s such that taking their intersection generically defines a new point &lt;math&gt;v\in\mathrm{Gr}(m,md)&lt;/math&gt;.{{Sfn|Felipe|Marí-Beffa|2019|loc=sections 4 and 5}} This generalization of the pentagram map is integrable in a [[w:noncommutative|noncommutative]] sense.{{Sfn|Ovenhouse|2020}}

=== Over rings ===
The pentagram map admits a generalization by considering [[w:Projective space#Generalizations|projective planes]] over [[w:stably finite ring|stably finite ring]]s, instead of [[w:Field (mathematics)|field]]s. In particular, this retrieves the pentagram map over Grassmannians. Again, it admits a [[w:Lax representation|Lax representation]].{{Sfn|Hand|Izosimov|2025}}

== References ==
{{reflist|25em}}

===Notes===
{{notelist}}

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*{{Cite journal|ref=harv |title=Pentagram-Type Maps and the Discrete KP Equation|url=https://link.springer.com/10.1007/s00332-023-09961-7|journal=Journal of Nonlinear Science|date=2023|issn=0938-8974|volume=33|issue=6|doi=10.1007/s00332-023-09961-7|language=en|first=Bao|last=Wang |article-number=101 |bibcode=2023JNS....33..101W }}
*{{Cite journal |ref=harv |title=The algebraic dynamics of the pentagram map |journal=Ergodic Theory and Dynamical Systems |date=2022-11-25 |issn=0143-3857 |pages=3460–3505 |volume=43 |issue=10 |doi=10.1017/etds.2022.82 |first=Max H. |last=Weinreich}}</text>
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      <comment>section about octahedron reccurrence</comment>
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      <text bytes="60915" sha1="oy4f9ot0g7a2hrec6ecymc96qewsh5o" xml:space="preserve">{{Article info
| last1 = Stiegler
| orcid1 = 0009-0001-5789-6923
| first1 = Jean-Baptiste
| affiliation1 = Université Paris-Saclay
| correspondence1 = jean-baptiste.stiegler@universite-paris-saclay.fr
| journal = WikiJournal of Science
| et_al = true
| w1 = Pentagram map
| from w1 = true
| keywords = Pentagram map, Dynamical system, Projective geometry, Moduli space, Integrable systems
| license = CC-BY-SA 4.0
| submitted = 2025-12-08
| abstract = In [[w:mathematics|mathematics]], the '''pentagram map''' is a [[w:Dynamical system#Discrete dynamical system|discrete dynamical system]] acting on [[w:polygons|polygons]] in the [[w:projective plane|projective plane]]. It defines a new polygon whose vertices are obtained as the intersection points of the shortest [[w:Diagonal|diagonals]] of the initial polygon. This is a [[w:Projective linear group|projectively]] [[w:Equivariant map|equivariant]] procedure, hence it [[w:Quotient space (topology)|descends]] to the [[w:moduli space|moduli space]] of polygons and defines another dynamical system (which is also referred to as the pentagram map). It was first introduced by [[w:Richard Schwartz (mathematician)|Richard Schwartz]] in 1992.{{Sfn|Schwartz|1992}}

The pentagram map on the moduli space is famous for its [[w:Completely integrable|complete integrability]] and its link with [[w:cluster algebra|cluster algebras]].{{sfn|Gekhtman|Izosimov|2025|p=14}}

It admits many generalizations in [[w:Projective space|projective spaces]] and other settings.
}}
== Introduction ==

=== Informal definition ===

==== On polygons ====
[[File:Pentagram pentagon nolabel big.svg|alt=|thumb|300x300px|The pentagram map applied on a [[w:Convex set|convex]] [[w:pentagon|pentagon]].]]
Initially, the pentagram map was defined for [[w:convex polygon|convex polygon]]s (with at least five sides) on the [[w:euclidean plane|Euclidean plane]]. Given such a polygon &lt;math&gt;P&lt;/math&gt; with &lt;math&gt;n&lt;/math&gt; sides, one can draw the &quot;shortest [[w:diagonal|diagonal]]s&quot;, meaning the [[w:Line segment|segments]] whose endpoints are a [[w:Vertex (geometry)|vertex]] and one of its second neighbors (as in Figure 1). The intersections of the shortest diagonals are then taken as the vertices of a new &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;T(P)&lt;/math&gt;; this new polygon is the output of the pentagram map.{{Sfn|Berger|2005}}

The same construction can be done on [[w:Concave polygon|non-convex polygons]], but there are several complications. First, some consecutive short diagonals may not intersect, so one must extend the segments to [[w:Line (geometry)|lines]]. Second, the image &lt;math&gt;T(P)&lt;/math&gt; can fail to be a new &lt;math&gt;n&lt;/math&gt;-gon because some consecutive vertices could coincide. However, this [[w:Generic property|generically]] doesn't happen.{{Sfn|Ovsienko|Schwartz|Tabachnikov|p=411|2009}} Finally, it is possible that two diagonals are [[w:Parallel (geometry)|parallel]] and don't intersect on the [[w:euclidean plane|Euclidean plane]]. This is resolved by extending the Euclidean plane to the [[w:real projective plane|real projective plane]] by the addition of a [[w:line at infinity|line at infinity]], where the [[w:Vanishing point|intersection point]] lies (see Figure 3). Hence, the pentagram map is defined for generic polygons in the real projective plane.{{Sfn|Berger|2005|p=25}}

More generally, the construction of the pentagram map is well defined whenever the concepts of lines and their intersections make sense. This is encompassed by the notion of a general [[w:projective plane|projective plane]], of which the real projective plane is one example; but the pentagram map can also be considered over other [[w:Field (mathematics)|fields]], for instance the [[w:complex number|complex number]]s, which give the [[w:complex projective plane|complex projective plane]].{{Sfn|Weinreich|2022|loc=§3.1.1}}

==== On the moduli space of polygons ====
Since the pentagram map is constructed by drawing lines and marking their intersections, it [[w:Commutative property|commutes]] with any transformation that sends lines to lines. Such maps are called [[w:projective transformations|projective transformations]]. This allows to identify polygons [[w:up to|up to]] [[w:Perspectivity#Projectivity|projective transformations]]. This identification gives a [[w:Quotient space (topology)|quotient space]] (technically called a [[w:moduli space|moduli space]]) of [[w:Equivalence class|classes]] of polygons. 

The pentagram map on polygons induces another dynamical system on the moduli space,{{Sfn|Schwartz|1992|loc=§1 Projective geometry}} whose behavior differs quite a lot from the initial one.{{Efn|Compare the paragraph about the [[w:Pentagram map#Collapsing of convex polygons|collapsing of convex polygons]] and the one about [[w:Pentagram map#Complete integrability|complete integrability]].}} The dynamic is trivial for the classes of pentagons and hexagons, but this stops to be the case for polygons with more vertices.{{Efn|See the paragraph about [[w:Pentagram_map#Pentagons_and_hexagons|pentagons and hexagons]].}}

=== Historical elements ===
The pentagram map for general polygons was introduced in {{Harvard citation|Schwartz|1992}}, but the simplest case is the one of [[w:pentagons|pentagons]], hence the name &quot;[[w:pentagram|pentagram]]&quot;.{{Sfn|Marí-Beffa|2014|p=1}} Their study goes back to {{Harvard citation|Clebsch|1871}},{{Sfn|Izosimov|2022a|p=1085}} {{Harvard citation|Kasner|1928}}{{Sfn|Tabachnikov|2019}} and {{Harvard citation|Motzkin|1945}}.{{Sfn|Schwartz|2013|p=1}}

The pentagram map interacts with some classical configuration theorems of [[w:projective geometry|projective geometry]]. It provides results analogous to the ones of [[w:Pascal's theorem|Pascal's theorem]] and [[wikipedia:Brianchon's_theorem|Brianchon's theorem]].{{Sfn|Schwartz|Tabachnikov|2010}} Some specific configurations make [[w:Desargues' theorem|Desargues's theorem]] and [[w:Poncelet's porism|Poncelet's porism]] appear.{{Sfn|Berger|2005|loc=§4 and §5}}{{Efn|See the paragraph about [[w:Pentagram map#Poncelet polygons|Poncelet polygons]].}}

==Definitions and first properties==

=== Definition of the map ===
[[File:Pentagram pentagon label big.svg|alt=|thumb|300x300px|The pentagram map on a convex pentagon, with vertices labeled.]]
[[File:Pentagram on nonconvex pentagon.svg|alt=|thumb|300x300px|The pentagram map applied on a [[w:Self-intersecting polygon|self-intersecting]] (in particular, non-convex) pentagon. The vertex &lt;math&gt;w_2&lt;/math&gt; is on the [[w:line at infinity|line at infinity]], because it is the [[w:Vanishing point|intersection of two parallel lines]].]]
Let &lt;math&gt;n\geq 5&lt;/math&gt; be an integer. A polygon &lt;math&gt;P&lt;/math&gt; with &lt;math&gt;n&lt;/math&gt; sides, or &lt;math&gt;n&lt;/math&gt;-gon, is a tuple of [[w:Vertex (geometry)|vertices]] &lt;math&gt;(v_1,\dots,v_n)&lt;/math&gt; lying in some [[w:projective plane|projective plane]] &lt;math&gt;\mathbb P ^2&lt;/math&gt;,{{Efn|In the following, the figures represent polygons on the real plane, where the intuition is easier to grasp.}} where the indices are understood [[w:Modular arithmetic|modulo]] &lt;math&gt;n&lt;/math&gt;. The [[w:Dimension of an algebraic variety|dimension]] of the space of &lt;math&gt;n&lt;/math&gt;-gons is &lt;math&gt;2n&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=definition 1.1}}

Suppose that the vertices are in sufficiently [[w:general position|general position]], meaning that no consecutive triple of points are [[w:Collinearity|collinear]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=1}} Taking the intersection of two consecutive &quot;shortest&quot; [[w:diagonal|diagonal]]s{{Efn|Meaning the line between a vertex &lt;math&gt;v_k&lt;/math&gt; and a &quot;second neighbour&quot; &lt;math&gt;v_{k\pm 2}&lt;/math&gt;.}} defines a new point&lt;math display=&quot;block&quot;&gt; w_k := \overline{v_{k-1} v_{k+1}} \cap \overline{v_{k} v_{k+2}}. &lt;/math&gt;This procedure defines a new &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;T(P)=(w_1,\dots,w_n)&lt;/math&gt;, as in Figure 2.{{Sfn|Schwartz|1992|p=71}}

The labeling of the indices of &lt;math&gt;T(P)&lt;/math&gt; is not [[w:canonical|canonical]]. In most papers, a choice is made at the beginning of the paper and the formulas are tuned accordingly.{{Sfn|Izosimov|2016|loc=remark 1.5}}

The pentagram map on polygons is a [[w:birational map|birational map]] &lt;math&gt;T:(\mathbb P^2)^n&lt;/math&gt;{{nowrap|{{font|size=145%|⇢}}}}&lt;math&gt;(\mathbb P^2)^n&lt;/math&gt;. Indeed, each [[w:Homogeneous coordinates|coordinate]] of &lt;math&gt;w_k&lt;/math&gt; is given as a [[w:rational function|rational function]] of the coordinates of &lt;math&gt;v_{k-1},\dots,v_{k+2}&lt;/math&gt;, since it is defined as the intersection of lines passing by them. Moreover, the [[w:inverse map|inverse map]] is given by taking the intersections &lt;math&gt;\overline{w_{k-2} w_{k-1}} \cap \overline{w_{k} w_{k+1}} &lt;/math&gt;, which is rational for the same reason.{{Sfnp|Weinreich|2022|loc=definition 1.2}}

=== Moduli space ===
The pentagram map is defined by taking [[w:Line (geometry)|lines]] and intersections of them. The biggest [[w:Group (mathematics)|group]] which maps lines to lines is the one of [[w:projective transformations|projective transformations]] &lt;math&gt;\mathbb P \mathrm{GL}_{3}&lt;/math&gt;. Such a transformation &lt;math&gt;M&lt;/math&gt; [[w:Group action|acts]] on a polygon &lt;math&gt;P&lt;/math&gt; by sending it to &lt;math&gt;M \cdot P:=(Mv_1,\dots,Mv_n)&lt;/math&gt;. The pentagram map [[w:Commutative property|commutes]] with this action, and thereby induces another [[w:dynamical system|dynamical system]] on the [[w:moduli space|moduli space]] of projective [[w:equivalence classes|equivalence classes]] of polygons. Its [[w:Dimension of an algebraic variety|dimension]] is &lt;math&gt;2n-8&lt;/math&gt;.{{Sfn|Schwartz|1992|loc=§1 Projective geometry}}

===Twisted polygons===
[[File:Twisted heptagon.svg|alt=|thumb|300x300px|An example of twisted [[w:heptagon|heptagon]] on the real plane.]]
The pentagram map naturally generalizes on the larger space of twisted polygons (see example in Figure 4). For any integer &lt;math&gt;n\geq5&lt;/math&gt;, a twisted &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;P&lt;/math&gt; is the data of:

* a [[w:Sequence#Indexing|bi-infinite sequence]] of points &lt;math&gt;(v_k)_{k\in\mathbb Z}&lt;/math&gt; in the projective plane (called the vertices),
* a [[w:projective transformation|projective transformation]] &lt;math&gt;M \in \mathbb P \mathrm{GL}_3&lt;/math&gt; (called the [[w:monodromy|monodromy]]),

such that for any &lt;math&gt;k \in \mathbb Z&lt;/math&gt;, the property &lt;math&gt;v_{k+n}=Mv_k&lt;/math&gt; is satisfied. The dimension of the space of twisted &lt;math&gt;n&lt;/math&gt;-gons is &lt;math&gt;2n+8&lt;/math&gt;.{{Sfn|Schwartz|2008}}

When &lt;math&gt;M=\mathrm{Id}&lt;/math&gt;, this gives back the initial definition of polygons (which are said to be closed). The space of closed &lt;math&gt;n&lt;/math&gt;-gons is of [[w:codimension|codimension]] &lt;math&gt;8&lt;/math&gt; in the space of twisted ones.{{Sfn|Soloviev|2013|p=2816}}

The action of projective transformations over the space of closed polygons generalizes to the space of twisted ones (the monodromy is changed by [[w:Matrix similarity|conjugation]]). This provides again a moduli space, of dimension &lt;math&gt;2n&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=definition 1.3}}

== Collapsing of convex polygons ==

=== Exponential shrinking ===
[[File:Pentagram map convex heptagon iterate.svg|alt=|thumb|300x300px|The pentagram map iterated on a convex [[w:heptagon|heptagon]], exhibiting the convergence.]]
Let &lt;math&gt;P&lt;/math&gt; be a closed [[w:Convex polygon#Strictly convex polygon|strictly convex polygon]] lying on the real plane. One of the first results proved by Richard Schwartz it that its iterates under the pentagram map shrink [[w:Exponential growth|exponentially fast]] to a point, as illustrated in Figure 5. This follows from two facts.
# The image of a strictly convex polygon is contained in its [[w:Interior (topology)|interior]], and is also strictly convex.{{Sfn|Glick|2020|p=2818}}
# There exists a constant &lt;math&gt;0&lt; \eta_P&lt;1&lt;/math&gt;, depending on &lt;math&gt;P&lt;/math&gt;, such that for any &lt;math&gt;N \in \mathbb N&lt;/math&gt;, the diameters of the iterates verify the inequality &lt;math display=&quot;inline&quot;&gt;\operatorname{diam}(T^N(P))\leq\eta_P^N \operatorname{diam}(P). &lt;/math&gt;{{Sfn|Schwartz|1992|loc=theorem 3.1}}

Hence, by [[w:Cantor's intersection theorem#Variant in complete metric spaces|Cantor's intersection theorem]], the sequence of polygons collapses toward a point.{{Sfn|Schwartz|1992|loc=§3 Convex polygons}}

The behavior on the moduli space is very different, since the dynamics is [[w:Recurrent point|recurrent]].{{Sfn|Schwartz|2001|loc=theorem 1.1}} It is even a [[w:quasiperiodic motion|quasiperiodic motion]],{{Sfn|Ovsienko|Schwartz|Tabachnikov|2009}} as discussed in [[w:Pentagram map#Complete integrability|the section about integrability]].

=== Coordinates of the limit point ===
The limit point coordinates are found in {{Harvard citation|Glick|2020}}. They satisfy some [[w:Degree of a polynomial|degree]] 3 [[w:polynomial equation|polynomial equations]], whose coefficients are [[w:rational function|rational function]]s in the coordinates of the vertices of the starting polygon. The proof relies on the fact that the limit point must be an [[w:eigenline|eigenline]] of a certain [[w:linear operator|linear operator]] of &lt;math&gt;\mathbb R^3&lt;/math&gt;.{{sfn|Glick|2020}}

This operator was reinterpreted in {{Harvard citation|Aboud|Izosimov|2022}} as the infinitesimal monodromy of the polygon. The [[w:Pentagram map#The scaling symmetry|scaling symmetry]] is used to [[w:Deformation (mathematics)|deform]] a closed polygon &lt;math&gt;P&lt;/math&gt; into a family of twisted ones '''&lt;math&gt;(P_z)_{z\in \mathbb C^*}&lt;/math&gt;''' with monodromy &lt;math&gt;M_z&lt;/math&gt;. The infinitesimal monodromy is defined to be:{{sfn|Aboud|Izosimov|2022}}
&lt;math display=&quot;block&quot;&gt;\left.\frac{dM_z}{dz}\right|_{z=1}.&lt;/math&gt;

=== Generalization ===
The collapsing of polygons may also happen in some [[w:Pentagram map#Generalizations|generalization of the pentagram map]], when considering some specific configurations of polygons in the real plane. The coordinates of the collapse point are given by a formula analogous to the one for the original pentagram map.{{Sfn|Schwartz|2026}}

== Periodic orbits on the moduli space ==
For some configurations of closed polygons, the iterate of the pentagram map will send &lt;math&gt;P&lt;/math&gt; to a projectively equivalent polygon (up to some shift of the indices). This means that, on the moduli space, the orbit of the class of &lt;math&gt;P&lt;/math&gt; is [[w:Periodic orbit|periodic]].

===Pentagons and hexagons===
[[File:penta hexagon.svg|300px|thumb|The outward hexagon is projectively equivalent to the inward one, with respect to their labeling.]]The following two facts are proved by checking [[w:cross-ratio|cross-ratio]] equalities, so they are true for polygons in any [[w:projective plane|projective plane]] (not just the [[w:Real projective plane|real one]]).{{Sfn|Schwartz|1992|loc=§2 Pentagons and hexagons}}

The pentagram map &lt;math&gt;T&lt;/math&gt; is the identity on the moduli space of [[w:pentagon|pentagon]]s.{{Sfn|Schwartz|1992|loc=theorem 2.1}}{{Sfn|Clebsch|1871}}{{Sfn|Motzkin|1945}} The second iterate &lt;math&gt;T^2&lt;/math&gt; is the identity on the space of labeled [[w:hexagon|hexagon]]s, up to a shift of labeling (see Figure 6).{{Sfn|Schwartz|1992|loc=theorem 2.3}} This phenomenon doesn't generalize to generic polygons with at least seven sides, for which the motion is [[w:Quasiperiodic motion|quasi-periodic]].{{Sfn|Tupan|2022}}

==== Generalization ====
The result about pentagons and hexagons generalizes to some [[w:Pentagram map#Generalizations|higher pentagram maps]] in &lt;math&gt;\mathbb P ^k&lt;/math&gt;, for polygons with &lt;math&gt;k+3&lt;/math&gt; or &lt;math&gt;2k+2&lt;/math&gt; sides. The proof uses a generalization of the [[w:Gale transform|Gale transform]].{{Sfn|Dirdak|2024}}

=== Poncelet polygons ===
A polygon is said to be Poncelet{{Efn|The name comes from [[w:Jean-Victor Poncelet|Jean-Victor Poncelet]] and [[w:Poncelet porism|his porism]].{{Sfn|Izosimov|2022a|p=1085}}}} if it is [[w:Inscribed figure|inscribed]] in a [[w:Conic section|conic]] and circumscribed about another one.{{Sfn|Schwartz|2015|loc=|p=433}}{{Efn|In particular, pentagons are Poncelet since [[w:five points determine a conic|five points determine a conic]].{{Sfn|Schwartz|2015|loc=|p=433}}}} For a convex Poncelet &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;P&lt;/math&gt; lying on the [[w:real projective plane|real projective plane]], the polygon &lt;math&gt;T^2(P)&lt;/math&gt; is projectively equivalent to &lt;math&gt;P&lt;/math&gt;.{{Sfn|Schwartz|2015|loc=theorem 1.1}} In fact, when &lt;math&gt;n&lt;/math&gt; is odd, the converse is also true.{{Sfn|Izosimov|2022a|loc=corollary 1.1}}

However, this converse statement is no longer true when the polygons are considered over the [[w:complex projective plane|complex projective plane]].{{Sfn|Izosimov|2022a|loc=remark 1.3}}

==Coordinates for the moduli space==
The moduli space can be described by different [[w:Coordinate_system|coordinate systems]]. The following ones are practical to explicit the dynamic, as presented in the next section.

=== Corner coordinates ===
[[File:Corner coordinates big.svg|thumb|300x300px|The geometric construction of the points defining the corner invariants.]]
Define the [[w:cross-ratio|cross-ratio]] of four [[w:Collinearity|collinear]] points to be

: &lt;math&gt; [a,b,c,d]=\frac{(a-b)(c-d)}{(a-c)(b-d)}. &lt;/math&gt;

The corner invariants are a system of coordinates on the space of twisted polygons, constructed by taking intersections as in Figure 7.{{Sfn|Schwartz|2001|loc=figure 2}} The left and right invariants are respectively defined{{Efn|The ordering of the vertices in the cross-ratios can differ from a paper to another one, which slightly changes the formulas in the following sections.}} as the following cross-ratios:

: &lt;math&gt;x_k:=[v_{k-2},v_{k-1},\overline{v_{k-2}v_{k-1}}\cap\overline{v_{k}v_{k+1}},\overline{v_{k-2}v_{k-1}}\cap\overline{v_{k+1}v_{k+2}}],&lt;/math&gt;

: &lt;math&gt;y_k:=[\overline{v_{k+1}v_{k+2}}\cap\overline{v_{k-2}v_{k-1}}, \overline{v_{k+1}v_{k+2}}\cap\overline{v_{k-1}v_{k}},v_{k+1},v_{k+2}].&lt;/math&gt;

Since the cross-ratio is [[w:Cross-ratio#Projective geometry|projective invariant]], the sequences &lt;math&gt;(x_k)_{k \in \mathbb Z}&lt;/math&gt; and &lt;math&gt;(y_k)_{k \in \mathbb Z}&lt;/math&gt; associated to a twisted &lt;math&gt;n&lt;/math&gt;-gon are &lt;math&gt;n&lt;/math&gt; periodic.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=415}}

The corner invariants are elements of &lt;math&gt;\mathbb{P}^1\smallsetminus\{0,1,\infty\}&lt;/math&gt;, and they realize an [[w:Isomorphism_of_varieties|isomorphism of variety]] between the moduli space of twisted &lt;math&gt;n&lt;/math&gt;-gons and &lt;math&gt;(\mathbb{P}^1\smallsetminus\{0,1,\infty\})^{2n}&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=theorem 3.6}}

===ab-coordinates===

There is a second set of coordinates for the moduli space of twisted &lt;math&gt;n&lt;/math&gt;-gons defined over a [[w:Field (mathematics)|field]] &lt;math&gt;F&lt;/math&gt; satisfying &lt;math&gt;\mathrm{SL}_3(F)\cong \mathbb P\mathrm{GL}_3(F)&lt;/math&gt;,{{Sfn|Weinreich|2022|loc=remark 3.8}} and such that &lt;math&gt;n&lt;/math&gt; is not divisible by &lt;math&gt;3&lt;/math&gt;.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=section 4.1}}

The vertices &lt;math&gt;v_k&lt;/math&gt; in the [[w:projective plane|projective plane]] &lt;math&gt;\mathbb P^2(F)&lt;/math&gt; can be [[w:Lift (mathematics)|lifted]] to [[w:Vector space|vectors]] &lt;math&gt;V_k&lt;/math&gt; in the [[w:affine space|affine space]] &lt;math&gt;F^3&lt;/math&gt; so that each consecutive triple of vectors spans a [[w:parallelepiped|parallelepiped]] having [[w:determinant|determinant]] equal to &lt;math&gt;1&lt;/math&gt;. This leads to the relation{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=equation 4.1}}

: &lt;math&gt;V_{k+3} = a_k V_{k+2} + b_k V_{k+1} + V_k.&lt;/math&gt;

This bring out an analogy between twisted polygons and solutions of third order linear [[w:ordinary differential equations|ordinary differential equations]], normalized to have unit [[w:Wronskian|Wronskian]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=remark 6.6}}

They are linked to the corner coordinates by:{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=lemma 4.5}}
: &lt;math&gt;x_k=\frac{a_{k-2}}{b_{k-2}b_{k-1}},&lt;/math&gt;
: &lt;math&gt;y_k=-\frac{b_{k-1}}{a_{k-2}a_{k-1}}.&lt;/math&gt;

==Formulas on the moduli space==

===As a birational map ===
The pentagram map is a [[w:birational map|birational map]] on the moduli space, because it can be decomposed as the [[w:Function composition|composition]] of two [[w:Birational geometry|birational]] [[w:Involution (mathematics)|involutions]].{{Sfn|Schwartz|2008|loc=§1.2 The Pentagram Map}} The corner invariants change in the following way:{{Sfn|Ovsienko|Schwartz|loc=lemma 2.4|Tabachnikov|2010}}

: &lt;math&gt;x_k'=x_k\frac{1-x_{k-1} y_{k-1}}{1-x_{k+1}y_{k+1}},&lt;/math&gt;

: &lt;math&gt;y_k'=y_{k+1}\frac{1-x_{k+2} y_{k+2}}{1-x_k y_k}.&lt;/math&gt;

=== The scaling symmetry ===
The [[w:multiplicative group|multiplicative group]] &lt;math&gt;F\smallsetminus\{0\}&lt;/math&gt; [[w:One-parameter group|acts]] on the moduli space in the following way:

: &lt;math&gt;R_s\cdot(x_1,\dots,x_n,y_1,\dots,y_n)=(sx_1,\dots,sx_n,s^{-1}y_1,\dots,s^{-1}y_n),&lt;/math&gt;

where &lt;math&gt;R&lt;/math&gt; is called the scaling action and &lt;math&gt;s&lt;/math&gt; is the scaling parameter. This action commutes with the pentagram map on the moduli space (as presented in the previous formulas). This property is called the scaling symmetry, and is instrumental in proving the [[w:Pentagram map#Complete integrability|complete integrability]] of the dynamics.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=corollary 2.5}}

==Invariant structures==

===Monodromy invariants===

The monodromy invariants, introduced in {{Harvard citation|Schwartz|2008}}, are a collection of [[w:Function (mathematics)|functions]] on the [[w:moduli space|moduli space]] that are invariant under the pentagram map.{{Sfn|Schwartz|2008|loc=theorem 1.2}} The simplest example of them are 
:&lt;math&gt; O_n= x_1x_2\cdots x_{n}, \quad E_n = y_1y_2\cdots y_n. &lt;/math&gt;
The other monodromy invariants can be retrieved through different points of view: through the [[w:Pentagram map#The scaling symmetry|scaling symmetry]], as [[w:Combinatorics|combinatorial]] objects, or as some [[w:determinant|determinant]]s.{{Sfn|Schwartz|Tabachnikov|2011|loc=§2 The Monodromy Invariants}} The one involving scaling symmetry is presented here.

Let &lt;math&gt;M\in \mathrm{GL}_3&lt;/math&gt; be a [[w:Lift (mathematics)|lift]] of the monodromy of a twisted &lt;math&gt;n&lt;/math&gt;-gon. The quantities

: &lt;math&gt;\Omega_1=\frac{\operatorname{trace}^3(M)}{\det(M)}, \quad \Omega_2=\frac{\operatorname{trace}^3(M^{-1})}{\det(M^{-1})},&lt;/math&gt;

are independent of the choice of lift and are invariant under [[w:Matrix similarity|conjugation]], so they are well defined for the projective class of the polygon. They are invariant under the pentagram map, since the monodromy matrix doesn't change.{{Sfn|Schwartz|Tabachnikov|2011|loc=|p=5}} Now, the quantities

: &lt;math&gt;\tilde{\Omega}_1=O_n^2E_n\Omega_1, \quad \tilde{\Omega}_2=O_nE_n^2\Omega_2,&lt;/math&gt;

have the same properties, but turn out to be polynomials in the corner invariants.{{Efn|Some papers consider the cube roots of this functions, but it doesn't change the following definitions of the monodromy invariants.}} They can be written as{{Sfn|Schwartz|Tabachnikov|2011|loc=|p=5}}

: &lt;math&gt;
\tilde{\Omega}_1=\biggl(\sum_{k=0}^{\lfloor n/2\rfloor}O_k\biggr)^3, \quad
\tilde{\Omega}_2=\biggl(\sum_{k=0}^{\lfloor n/2\rfloor}E_k\biggr)^3,
&lt;/math&gt;

where each &lt;math&gt;O_k&lt;/math&gt; and &lt;math&gt;E_k&lt;/math&gt; are [[w:homogeneous polynomial|homogeneous polynomial]]s respectively of weight &lt;math&gt;k&lt;/math&gt; and &lt;math&gt;-k&lt;/math&gt;,{{Sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=11}} meaning they change under the [[w:Pentagram map#The scaling symmetry|rescaling action]] on variables by{{Sfn|Schwartz|Tabachnikov|2011|p=5}}

: &lt;math&gt; R_s(O_k)= s^k O_k, \quad R_s(E_k)= s^{-k} E_k. &lt;/math&gt;

The quantities &lt;math&gt;O_1,\dots,O_{\lfloor n/2 \rfloor},O_n, E_1,\dots,E_{\lfloor n/2 \rfloor},E_n,&lt;/math&gt; are unchanged by the dynamics, and are called the monodromy invariants. Moreover, they are [[w:algebraically independent|algebraically independent]].{{Sfn|Schwartz|2008|loc=theorem 1.2}}

==== Polygons on conics ====
Whenever &lt;math&gt;P&lt;/math&gt; is [[w:Inscribed figure|inscribed]] on a [[w:conic section|conic section]], one has &lt;math&gt;O_k(P)=E_k(P)&lt;/math&gt; for all &lt;math&gt;k&lt;/math&gt;.{{Sfn|Schwartz|Tabachnikov|2011|loc=theorem 1.1}} Moreover, if &lt;math&gt;P&lt;/math&gt; is circumscribed about another conic,{{Efn|See the paragraph about [[w:Pentagram map#Poncelet polygons|Poncelet polygons]].}} then its monodromy invariants are characterized by the pair of conics.{{Sfn|Schwartz|2015|loc=theorem 1.2}} For such odd-gons, the translation on the [[w:Jacobian variety|Jacobian variety]]{{Efn|See the paragraph about [[w:Pentagram map#Algebro-geometric integrability|algebraic integrability]].}} is restricted to the [[w:Prym variety|Prym variety]] (which is a half-dimensional torus in the Jacobian).{{Sfn|Izosimov|2016|loc=theorem 1.3}}

===Poisson bracket===

An invariant [[w:Poisson bracket|Poisson bracket]] on the space of twisted polygons was found in {{Harvard citation|Ovsienko|Schwartz|Tabachnikov|2010}}. The monodromy invariants [[w:Poisson bracket#Constants of motion|commute]] with respect to it:
&lt;math display=&quot;block&quot;&gt; \{O_i,O_j\}=\{O_i,E_j\}=\{E_i,E_j\}=0 &lt;/math&gt;for all &lt;math&gt;i,j&lt;/math&gt;.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}}

The Poisson bracket is defined in terms of the corner coordinates by:
&lt;math display=&quot;block&quot;&gt; \begin{align}
 \{x_i,x_{i\pm1}\} &amp;= \mp x_i x_{i+1}, \\
 \{y_i,y_{i\pm 1}\} &amp;= \mp y_i y_{i+1}, \\
 \{x_i,x_j\} &amp;= \{y_i,y_j\} = \{x_i,y_j\} = 0
\end{align}&lt;/math&gt;for all other &lt;math&gt; i,j.&lt;/math&gt;{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=equation 2.16}}

=== The spectral curve ===

Let &lt;math&gt;\zeta&lt;/math&gt; be an element of the [[w:multiplicative group|multiplicative group]] and &lt;math&gt;P_\zeta&lt;/math&gt; be the polygon obtained by applying the [[w:Pentagram map#The scaling symmetry|rescaling action]] &lt;math&gt;R_\zeta&lt;/math&gt; on &lt;math&gt;P&lt;/math&gt;.  A [[w:Lax matrix|Lax matrix]] &lt;math&gt;\hat{T}(\zeta) \in \mathrm{GL}_3&lt;/math&gt; is a lift of the monodromy of &lt;math&gt;P_\zeta&lt;/math&gt; satisfying a [[w:Lax pair#Zero-curvature equation|zero-curvature equation]].{{Sfn|Weinreich|2022|loc=§5 The Lax representation}} Then, the spectral function is the [[w:Bivariate polynomial|bivariate]] [[w:characteristic polynomial|characteristic polynomial]]
&lt;math display=&quot;block&quot;&gt; Q(\lambda,\zeta) := \det(\lambda\operatorname{Id}-\hat{T}(\zeta)),&lt;/math&gt;or some renormalization of it. The [[w:spectral curve|spectral curve]] is the [[w:Projective variety#projective completion|projective completion]] of the [[w:Algebraic curve|affine curve]] defined by the equation &lt;math&gt;Q(\lambda,\zeta)=0&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=§6. The geometry of the spectral curve}} It is invariant under the pentagram map, and the monodromy invariants appear as the [[w:coefficient|coefficient]]s of &lt;math&gt;Q&lt;/math&gt;.{{Sfn|Soloviev|2013|loc=theorem 6.4}} Its [[w:geometric genus|geometric genus]] is &lt;math&gt;n-1&lt;/math&gt; if &lt;math&gt;n&lt;/math&gt; is odd, and &lt;math&gt;n-2&lt;/math&gt; if &lt;math&gt;n&lt;/math&gt; is even.{{Sfn|Weinreich|2022|p=|loc=theorem 6.4}}

It was first introduced in {{Harv|Soloviev|2013|ps=|p=}} for his proof of [[w:Pentagram map#Algebro-geometric integrability|algebro-geometric integrability]].{{sfn|Soloviev|2013}}

==Complete integrability==
The pentagram map on the moduli space has been proved to be a [[w:completely integrable|completely integrable]] [[w:discrete dynamical system|discrete dynamical system]], both in the [[w:Integrable system#Hamiltonian systems and Liouville integrability|Arnold-Liouville]]{{Efn|Over the [[w:real number|real number]]s.}} and the [[w:Integrable system#Complete integrability over the complex numbers|algebro-geometric]]{{Efn|Over [[w:algebraically closed field|algebraically closed field]]s of [[w:Characteristic (algebra)|characteristic]] different from 2.}} senses. In any case, this means that the moduli space is [[w:almost everywhere|almost everywhere]] [[w:Foliation|foliated]] by [[w:Torus#Flat torus|flat tori]] (or in the algebraic setting, [[w:Abelian variety|Abelian varieties]]), where the motion is a [[w:Translation (geometry)|translation]]. This [[w:Generic property|generically]] induces a [[w:quasiperiodic motion|quasiperiodic motion]] on the corresponding torus.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2009}}

===Arnold–Liouville integrability===

The proof of the integrability of the pentagram map on a real twisted polygon was achieved in {{Harvard citation|Ovsienko|Schwartz|Tabachnikov|2010}}. This is done by noticing that the monodromy invariants &lt;math&gt;O_n&lt;/math&gt; and &lt;math&gt;E_n&lt;/math&gt; are [[w:Casimir invariant|Casimir invariant]]s for the bracket, meaning (in this context) that&lt;math display=&quot;block&quot;&gt; \{O_n,f\}=\{E_n,f\} = 0 &lt;/math&gt;for all functions &lt;math&gt;f&lt;/math&gt;.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}} When &lt;math&gt;n&lt;/math&gt; is even, this is also true for the monodromy invariants &lt;math&gt;O_{\lfloor n/2 \rfloor }&lt;/math&gt; and &lt;math&gt;E_{\lfloor n/2 \rfloor }&lt;/math&gt;.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}}

This allows to consider the Casimir [[w:level set|level set]], where each Casimir has a specified value. Because of [[w:Sard's theorem|Sard's theorem]], any generic level set is a [[w:smooth manifold|smooth manifold]].{{Sfn|Schwartz|2017|p=44}} They form a [[w:foliation|foliation]] in [[w:Poisson manifold#Symplectic leaves|symplectic leaves]], on which the Poisson bracket gives rise to a [[w:symplectic form|symplectic form]].{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=corollary 2.13}}

Each of these symplectic leaves has an iso-monodromy [[w:foliation|foliation]], namely, a decomposition into the common level sets of the remaining monodromy functions. By using again [[w:Sard's theorem|Sard's theorem]], they are generically [[w:Symplectic manifold#Lagrangian submanifolds|Lagrangian manifolds]].{{Sfn|Schwartz|2017|p=45}} Moreover, they are compact.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=§3.3. Compactness of the level sets}} Since the monodromy invariants Poisson-commute and there are enough of them, the discrete [[w:Liouville–Arnold theorem|Liouville–Arnold theorem]] can be applied to prove that the level sets are [[w:Torus#Flat torus|flat tori]] over which the dynamics is a translation.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=412}}
===Algebro-geometric integrability===

In {{Harvard citation|Soloviev|2013}}, it was shown that the pentagram map admits a [[w:Lax representation|Lax representation]] with a spectral parameter, which allows to prove its algebro-geometric integrability. This means that the space of polygons (either twisted or closed) is parametrized by its spectral data, consisting of [[w:Pentagram map#The spectral curve|its spectral curve]], with marked points and a [[w:Divisor (algebraic geometry)|divisor]] given by a [[w:Floquet theory|Floquet]]–[[w:Bloch's theorem|Bloch]] equation. This gives an embedding to the [[w:Jacobian variety|Jacobian variety]] through the [[w:Abel–Jacobi map|Abel–Jacobi map]], where the motion is expressed in terms of translation.{{sfn|Soloviev|2013|loc=theorems A, B and C}} The previously defined Poisson bracket is also retrieved.{{sfn|Soloviev|2013|loc=theorem D}} 

This integrability was generalized in {{Harvard citation|Weinreich|2022}} from the field of [[w:complex number|complex number]]s to any [[w:algebraically closed field|algebraically closed field]] of [[w:Characteristic (algebra)|characteristic]] different from 2. The translation on a torus is replaced by a translation on an [[w:Abelian variety|Abelian variety]] (in fact, a Jacobian variety again).{{sfn|Weinreich|2022|loc=theorem 1.4}}

=== Dimension of the invariant manifold ===

For twisted &lt;math&gt;n&lt;/math&gt;-gons, the [[w:dimension|dimension]] of the invariant tori (or Jacobian varieties) is{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=421}}

: &lt;math&gt;\begin{cases} 
n-1 &amp; \text{when }n \text{ is odd,}\\
n-2 &amp; \text{when }n \text{ is even.}
\end{cases}&lt;/math&gt;

Moreover, when &lt;math&gt;n&lt;/math&gt; is even, there are two isomorphic Jacobians on which the iterates of the pentagram map alternate. But on each of them, the second iterate is a translation.{{Sfn|Weinreich|2022|loc=theorem 1.4}}

=== For closed polygons ===
There is no Poisson structure on the space of closed polygons.{{Sfn|Soloviev|2013|loc=corollary 4.1}} Nevertheless, the one from twisted polygons can be used to prove integrability.{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=2153}}

Algebro-geometric integrability holds for closed polygons in a same manner as for the twisted ones.{{Sfn|Soloviev|2013|loc=theorem C}} However, Arnold-Liouville integrability is proved for real closed polygons only when they are convex. This is done by restricting the [[w:Hamiltonian vector field|Hamiltonian vector field]]s of monodromy functions to smaller dimensional tori, and showing that enough of them are still independent.{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|loc=corollary 1.1}}

In both situation, the dimension of the invariant manifolds decreases by &lt;math&gt;3&lt;/math&gt; for closed &lt;math&gt;n&lt;/math&gt;-gons (compared to the twisted case), and is equal to{{Sfn|Soloviev|2013|loc=theorem C}}{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|loc=theorem 1}}
:&lt;math&gt;\begin{cases} 
n-4 &amp; \text{when }n \text{ is odd,}\\
n-5 &amp; \text{when }n \text{ is even.}
\end{cases}&lt;/math&gt;

==Connections to other topics==

===The Boussinesq equation===

The continuous limit of a convex polygon is a parametrized convex curve in the plane. When the time parameter is suitably chosen, the [[w:Discretization|continuous limit]] of the pentagram map is the classical [[w:Boussinesq approximation (water waves)|Boussinesq equation]]. This equation is a classical example of an [[w:integrable|integrable]] [[w:partial differential equation|partial differential equation]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 5}}

Here is a description of the geometric action of the Boussinesq equation. Given a [[w:locally convex|locally convex]] curve &lt;math&gt; C:\mathbb R\to \mathbb R^2 &lt;/math&gt; and real numbers &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;t&lt;/math&gt;, consider the [[w:chord (geometry)|chord]] connecting &lt;math&gt; C(x-t) &lt;/math&gt; to &lt;math&gt; C(x+t) &lt;/math&gt;. The [[w:Envelope (mathematics)|envelope]] of all these chords is a new curve &lt;math&gt; C_t(x) &lt;/math&gt;. When &lt;math&gt;t&lt;/math&gt; is extremely small, the curve &lt;math&gt; C_t(x) &lt;/math&gt; is a good model for the time &lt;math&gt;t&lt;/math&gt; evolution of the original curve &lt;math&gt; C_0(x) &lt;/math&gt; under the Boussinesq equation. This construction is also similar to the pentagram map. Moreover, the pentagram invariant bracket is a discretization of a well known invariant Poisson bracket associated to the Boussinesq equation.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=§6.4 Discretization}}

===Cluster algebras===

The pentagram map{{Sfn|Glick|2011}} and some of its generalizations{{Sfn|Gekhtman|Shapiro|Tabachnikov|Vainshtein|2012}}{{Sfn|Glick|Pylyavskyy|2016}} are identified as special cases of discrete dynamical systems powered by [[w:cluster algebra|cluster algebra]]. Using the results from {{Harvard citation|Goncharov|Kenyon|2013}}, this provides a link with the [[w:Poisson–Lie group|Poisson–Lie group]]s, [[w:dimer model|dimer model]]s and other so-called cluster-integrable systems.{{Sfn|Fock|Marshakov|2016}}{{Sfn|Izosimov|2022c}} These methods allow to retrieve the Poisson-bracket and Hamiltonians used to prove complete integrability{{Sfn|Affolter|George|Ramassamy|2025|loc=§5 The pentagram map}} and provide [[w:Lax representation|Lax representation]]s.{{Sfn|Izosimov|2022b}}

=== Octahedron recurrence ===
Using a method to compute [[w:Determinant|determinants]] called [[w:Dodgson_condensation|Dodgson condensation]], {{Harvard citation|Schwartz|2008}} proves that the pentagram map satisfies a property called the &quot;octahedron recurrence&quot;.{{Sfn|Schwartz|2008|loc=§5 The Method of Condensation}} This property turns out to be shared by other dynamical systems defined by geometric constructions similar to one of the pentagram map.{{Sfn|Affolter|de Tilière|Melotti|2025|loc=§9 The Pentagram Map}} It is also shared by higher dimensional pentagram maps defined through cluster algebras mutations, referred as &lt;math&gt;T&lt;/math&gt;-systems.{{Sfn|Kedem|Vichitkunakorn|2015||loc=}}

=== Singularity theory ===
The pentagram map exhibit a property called singularity confinement, which is typical from [[w:integrable system|integrable system]]s.{{Sfn|Grammaticos|Ramani|Papageorgiou|1991}} It states that if a polygon &lt;math&gt;P&lt;/math&gt; is [[w:Singular point of an algebraic variety|singular]] for the pentagram map &lt;math&gt;T&lt;/math&gt;, then there exists an integer &lt;math&gt;m&lt;/math&gt; such that &lt;math&gt;P&lt;/math&gt; not singular for the iterate map &lt;math&gt;T^m&lt;/math&gt;.{{Sfn|Glick|2012}}

Moreover, the pentagram map (along with some of its generalizations and other discrete dynamical systems) exhibit the Devron property.{{Efn|The name comes from an episode of [[w:Star Trek|Star Trek]].{{Sfn|Glick|2015|loc=§1 Introduction}}}} This means that if a polygon &lt;math&gt;P&lt;/math&gt; is singular for some iterate of the pentagram map &lt;math&gt;T^m&lt;/math&gt;, then it will also be singular for some iterate of the inverse map &lt;math&gt;T^{-m'}&lt;/math&gt;.{{Sfn|Glick|2015}}

== Generalizations ==
The definition of twisted polygons still makes sense in any [[w:projective space|projective space]] &lt;math&gt;\mathbb P^d&lt;/math&gt;, under the action of the [[w:Projective linear group|projective group]] &lt;math&gt;\mathbb P \mathrm{GL}_{d+1}&lt;/math&gt;. The pentagram map can be generalized in many ways, and some of them are presented here. Not all of them are integrable.{{Sfn|Khesin|Soloviev|2015|}} Some are [[w:discretization|discretization]]s of [[w:PDEs|PDEs]] from the [[w:KdV hierarchy|KdV hierarchy]], seen as higher dimensional version of [[w:Boussinesq approximation (water waves)|Boussinesq]] or [[w:Kadomtsev–Petviashvili equation|KP]] equations.{{Sfn|Marí-Beffa|2012}}{{Sfn|Wang|2023}} The description of all generalized pentagram maps in terms of [[w:cluster algebra|cluster algebra]]s is still an open question.{{Sfn|Gekhtman|Izosimov|2025|p=14}}

=== Polygons in general positions ===
Let &lt;math&gt;d \geq 2&lt;/math&gt; and &lt;math&gt;P&lt;/math&gt; be a twisted polygon of &lt;math&gt;\mathbb P^d&lt;/math&gt; in [[w:general position|general position]].

==== Short diagonal pentagram maps ====
The &lt;math&gt;k&lt;/math&gt;-th ''short diagonal hyperplane'' &lt;math&gt;H_k^{sh}&lt;/math&gt; is uniquely defined by passing through the vertices &lt;math&gt;v_k,v_{k+2},\dots,v_{k+2d-2}&lt;/math&gt;. [[w:Generic property#In algebraic geometry|Generically]], the intersection of &lt;math&gt;d&lt;/math&gt; consecutive hyperplanes uniquely defines a new point

: &lt;math&gt;T_{sh}v_k:=H_k^{sh}\cap H_{k+1}^{sh}\cap \dots \cap H_{k+d-1}^{sh}.&lt;/math&gt;

Doing this for every vertex defines a new twisted polygon. This map, denoted by &lt;math&gt;T_{sh}&lt;/math&gt;, is again projectively equivariant.{{Sfn|Khesin|Soloviev|2013}}

==== Generalized pentagram maps ====
The previous procedure can be generalized. Let &lt;math&gt;I=(i_1,\dots,i_{d-1}),~J=(j_1,\dots,j_{d-1})&lt;/math&gt; be two sets of integers, respectively called the jump tuple and the intersection tuple. Define the &lt;math&gt;k&lt;/math&gt;-th hyperplane &lt;math&gt;H_k^I&lt;/math&gt; to be passing through the vertices &lt;math&gt;v_k,v_{k+i_1},\dots,v_{k+i_1+\dots+i_{d-1}}&lt;/math&gt;. A new point is given by the intersection

: &lt;math&gt;T_{I,J}v_k:=H_k^I \cap H_{k+j_1}^I \cap \dots \cap H_{k+j_1+\dots +j_{d-1}}^I.&lt;/math&gt;

The map &lt;math&gt;T_{I,J}&lt;/math&gt; is called a generalized pentagram map.{{Sfn|Khesin|Soloviev|2015a}} The original pentagram map is recovered by considering&lt;math&gt;d=2,~I=(2),~J=(1)&lt;/math&gt;.

Integrability can be numerically tested by picking a random polygon &lt;math&gt;P&lt;/math&gt; with [[wikipedia:Rational_point|rational coordinates]] and studying the growth rate of the [[wikipedia:Height_function|height]] of its iterates. This is called the [[wikipedia:Integrable_system#Diophantine_integrability|diophantine integrability]] test, and some generalized pentagram maps don't seem to pass it.{{Sfn|Khesin|Soloviev|2015a|loc=§5 and §6}} However, it is conjectured that the maps &lt;math&gt;T_{I,I}&lt;/math&gt; are integrable for any &lt;math&gt;I&lt;/math&gt;.{{Sfn|Bolsinov|Matveev|Miranda|Tabachnikov|2018|loc=conjecture 4.13 (B. Khesin, F. Soloviev)}}

Some of these maps are [[w:discretization|discretization]]s of higher dimensional counterpart of the [[w:Boussinesq approximation (water waves)|Boussinesq equation]] in the [[w:KdV hierarchy|KdV hierarchy]].{{Sfn|Khesin|Soloviev|2015b|loc=theorem 4.1}}{{Sfn|Izosimov|2022b|loc=theorem 4.1}}

==== Dented pentagram maps ====
Fix an integer &lt;math&gt;m\in \{1,\dots ,d-1\}&lt;/math&gt;. Consider the jump tuple &lt;math&gt;I_m:=(1,\dots,1,2,1,\dots,1)&lt;/math&gt;, where the &lt;math&gt;2&lt;/math&gt; is at the &lt;math&gt;m&lt;/math&gt;-th place, and the intersection tuple &lt;math&gt;J:=(1,\dots,1)&lt;/math&gt;. The dented pentagram map is &lt;math&gt;T_m :=T_{I_m,J}&lt;/math&gt;. They are proved to be integrable.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 2.14}}

For an integer &lt;math&gt;p \geq 2&lt;/math&gt;, the deep dented pentagram map (of depth &lt;math&gt;p&lt;/math&gt;) &lt;math&gt;T_m^p&lt;/math&gt; is the same map as before, but the number &lt;math&gt;2&lt;/math&gt; in the definition of &lt;math&gt;I_m&lt;/math&gt; is replaced by &lt;math&gt;p&lt;/math&gt;. This kind of pentagram maps are again integrable.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 6.2}}

=== Corrugated polygons ===
A twisted polygon &lt;math&gt;P&lt;/math&gt; lying in &lt;math&gt;\mathbb P^d&lt;/math&gt; is said to be corrugated if for any &lt;math&gt;k\in \mathbb Z&lt;/math&gt;, the vertices &lt;math&gt;v_k,v_{k+1},v_{k+d},v_{k+d+1}&lt;/math&gt; span a projective two-dimensional plane. Such polygons are not in [[w:general position|general position]]. A new point is defined by

: &lt;math&gt;T_\text{cor}v_k:=\overline{v_k v_{k+d}}\cap \overline{v_{k+1} v_{k+d+1}}.&lt;/math&gt;

The map &lt;math&gt;T_\text{cor}&lt;/math&gt; yields a new corrugated polygon. They are [[w:Integrable system#Hamiltonian systems and Liouville integrability|completely Liouville-integrable]].{{Sfn|Gekhtman|Shapiro|Tabachnikov|Vainshtein|2012|loc=theorem 4.4}}

In fact, they can be retrieved as some dented pentagram map applied on corrugated polygons.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 5.3}}

=== Grassmannian polygons ===
Let &lt;math&gt;d \geq 3, m \geq 1&lt;/math&gt; be integers. The pentagram map can also be generalized to the [[w:Grassmannian|Grassmannian]] space &lt;math&gt;\mathrm{Gr}(m,md)&lt;/math&gt;, which consists of &lt;math&gt;m&lt;/math&gt;-[[w:Dimension (vector space)|dimensional]] [[w:linear subspace|linear subspace]]s of an &lt;math&gt;md&lt;/math&gt;-dimensional [[w:vector space|vector space]]. When &lt;math&gt;m=1&lt;/math&gt;, the linear subspaces are [[w:Vector space#vector line|lines]], which retrieves the definition of [[w:projective space|projective space]]s &lt;math&gt;\mathbb P^d&lt;/math&gt;.{{Sfn|Felipe|Marí-Beffa|2019|loc=§2 definitions and notations}}

A point &lt;math&gt;v\in\operatorname{Gr}(m,md)&lt;/math&gt; is represented by an &lt;math&gt;md \times m&lt;/math&gt; matrix &lt;math&gt;X_v&lt;/math&gt; such that its columns form a [[w:Basis (linear algebra)|basis]] of &lt;math&gt;v&lt;/math&gt;. Consider the [[w:Group action|action]] of the [[w:general linear group|general linear group]] &lt;math&gt;\mathrm{GL}_{md}&lt;/math&gt; by multiplication on the left of &lt;math&gt;X_v&lt;/math&gt;. This defines an action on the Grassmannian, even though it is not [[w:Faithful action|faithful]].{{Efn|Because there can be many lifts for &lt;math&gt;v&lt;/math&gt;, and because some matrices act trivially.}} Hence, the polygons of &lt;math&gt;\mathrm{Gr}(m,md)&lt;/math&gt; and their moduli spaces are defined as before, after the change of underlying group.{{Sfn|Felipe|Marí-Beffa|2019|loc=§2 definitions and notations}}

Depending on the parity of &lt;math&gt;d&lt;/math&gt;, one can define linear subspaces spanned by some &lt;math&gt;X_{v_k}&lt;/math&gt;'s such that taking their intersection generically defines a new point &lt;math&gt;v\in\mathrm{Gr}(m,md)&lt;/math&gt;.{{Sfn|Felipe|Marí-Beffa|2019|loc=sections 4 and 5}} This generalization of the pentagram map is integrable in a [[w:noncommutative|noncommutative]] sense.{{Sfn|Ovenhouse|2020}}

=== Over rings ===
The pentagram map admits a generalization by considering [[w:Projective space#Generalizations|projective planes]] over [[w:stably finite ring|stably finite ring]]s, instead of [[w:Field (mathematics)|field]]s. In particular, this retrieves the pentagram map over Grassmannians. Again, it admits a [[w:Lax representation|Lax representation]].{{Sfn|Hand|Izosimov|2025}}

== References ==
{{reflist|25em}}

===Notes===
{{notelist}}

==Works cited==
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| last1 = Stiegler
| orcid1 = 0009-0001-5789-6923
| first1 = Jean-Baptiste
| affiliation1 = Université Paris-Saclay
| correspondence1 = jean-baptiste.stiegler@universite-paris-saclay.fr
| journal = WikiJournal of Science
| et_al = true
| w1 = Pentagram map
| from w1 = true
| keywords = Pentagram map, Dynamical system, Projective geometry, Moduli space, Integrable systems
| license = CC-BY-SA 4.0
| submitted = 2025-12-08
| abstract = In [[w:mathematics|mathematics]], the '''pentagram map''' is a [[w:Dynamical system#Discrete dynamical system|discrete dynamical system]] acting on [[w:polygons|polygons]] in the [[w:projective plane|projective plane]]. It defines a new polygon whose vertices are obtained as the intersection points of the shortest [[w:Diagonal|diagonals]] of the initial polygon. This is a [[w:Projective linear group|projectively]] [[w:Equivariant map|equivariant]] procedure, hence it [[w:Quotient space (topology)|descends]] to the [[w:moduli space|moduli space]] of polygons and defines another dynamical system (which is also referred to as the pentagram map). It was first introduced by [[w:Richard Schwartz (mathematician)|Richard Schwartz]] in 1992.{{Sfn|Schwartz|1992}}

The pentagram map on the moduli space is famous for its [[w:Completely integrable|complete integrability]] and its link with [[w:cluster algebra|cluster algebras]].{{sfn|Gekhtman|Izosimov|2025|p=14}}

It admits many generalizations in [[w:Projective space|projective spaces]] and other settings.
}}
== Introduction ==

=== Informal definition ===

==== On polygons ====
[[File:Pentagram pentagon nolabel big.svg|alt=|thumb|300x300px|The pentagram map applied on a [[w:Convex set|convex]] [[w:pentagon|pentagon]].]]
Initially, the pentagram map was defined for [[w:convex polygon|convex polygon]]s (with at least five sides) on the [[w:euclidean plane|Euclidean plane]]. Given such a polygon &lt;math&gt;P&lt;/math&gt; with &lt;math&gt;n&lt;/math&gt; sides, one can draw the &quot;shortest [[w:diagonal|diagonal]]s&quot;, meaning the [[w:Line segment|segments]] whose endpoints are a [[w:Vertex (geometry)|vertex]] and one of its second neighbors (as in Figure 1). The intersections of the shortest diagonals are then taken as the vertices of a new &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;T(P)&lt;/math&gt;; this new polygon is the output of the pentagram map.{{Sfn|Berger|2005}}

The same construction can be done on [[w:Concave polygon|non-convex polygons]], but there are several complications. First, some consecutive short diagonals may not intersect, so one must extend the segments to [[w:Line (geometry)|lines]]. Second, the image &lt;math&gt;T(P)&lt;/math&gt; can fail to be a new &lt;math&gt;n&lt;/math&gt;-gon because some consecutive vertices could coincide. However, this [[w:Generic property|generically]] doesn't happen.{{Sfn|Ovsienko|Schwartz|Tabachnikov|p=411|2009}} Finally, it is possible that two diagonals are [[w:Parallel (geometry)|parallel]] and don't intersect on the [[w:euclidean plane|Euclidean plane]]. This is resolved by extending the Euclidean plane to the [[w:real projective plane|real projective plane]] by the addition of a [[w:line at infinity|line at infinity]], where the [[w:Vanishing point|intersection point]] lies (see Figure 3). Hence, the pentagram map is defined for generic polygons in the real projective plane.{{Sfn|Berger|2005|p=25}}

More generally, the construction of the pentagram map is well defined whenever the concepts of lines and their intersections make sense. This is encompassed by the notion of a general [[w:projective plane|projective plane]], of which the real projective plane is one example; but the pentagram map can also be considered over other [[w:Field (mathematics)|fields]], for instance the [[w:complex number|complex number]]s, which give the [[w:complex projective plane|complex projective plane]].{{Sfn|Weinreich|2022|loc=§3.1.1}}

==== On the moduli space of polygons ====
Since the pentagram map is constructed by drawing lines and marking their intersections, it [[w:Commutative property|commutes]] with any transformation that sends lines to lines. Such maps are called [[w:projective transformations|projective transformations]]. Hence, polygons can be identified [[w:up to|up to]] [[w:Perspectivity#Projectivity|projective transformations]]. This identification gives the [[w:Quotient space (topology)|quotient space]] (technically called a [[w:moduli space|moduli space]]) of [[w:Equivalence class|classes]] of polygons. 

The pentagram map on polygons induces another dynamical system on the moduli space,{{Sfn|Schwartz|1992|loc=§1 Projective geometry}} whose behavior differs quite a lot from the initial one.{{Efn|Compare the paragraph about the [[w:Pentagram map#Collapsing of convex polygons|collapsing of convex polygons]] and the one about [[w:Pentagram map#Complete integrability|complete integrability]].}} The dynamic is trivial for the classes of pentagons and hexagons, but this is no longer the case for polygons with more vertices.{{Efn|See the paragraph about [[w:Pentagram_map#Pentagons_and_hexagons|pentagons and hexagons]].}}

=== Historical elements ===
The pentagram map for general polygons was introduced in {{Harvard citation|Schwartz|1992}}, but the simplest case is the one of [[w:pentagons|pentagons]], hence the name &quot;[[w:pentagram|pentagram]]&quot;.{{Sfn|Marí-Beffa|2014|p=1}} Their study goes back to {{Harvard citation|Clebsch|1871}},{{Sfn|Izosimov|2022a|p=1085}} {{Harvard citation|Kasner|1928}}{{Sfn|Tabachnikov|2019}} and {{Harvard citation|Motzkin|1945}}.{{Sfn|Schwartz|2013|p=1}}

The pentagram map interacts with some classical configuration theorems of [[w:projective geometry|projective geometry]]. It provides results analogous to the ones of [[w:Pascal's theorem|Pascal's theorem]] and [[wikipedia:Brianchon's_theorem|Brianchon's theorem]].{{Sfn|Schwartz|Tabachnikov|2010}} Some specific configurations make [[w:Desargues' theorem|Desargues's theorem]] and [[w:Poncelet's porism|Poncelet's porism]] appear.{{Sfn|Berger|2005|loc=§4 and §5}}{{Efn|See the paragraph about [[w:Pentagram map#Poncelet polygons|Poncelet polygons]].}}

==Definitions and first properties==

=== Definition of the map ===
[[File:Pentagram pentagon label big.svg|alt=|thumb|300x300px|The pentagram map on a convex pentagon, with vertices labeled.]]
[[File:Pentagram on nonconvex pentagon.svg|alt=|thumb|300x300px|The pentagram map applied on a [[w:Self-intersecting polygon|self-intersecting]] (in particular, non-convex) pentagon. The vertex &lt;math&gt;w_2&lt;/math&gt; is on the [[w:line at infinity|line at infinity]], because it is the [[w:Vanishing point|intersection of two parallel lines]].]]
Let &lt;math&gt;n\geq 5&lt;/math&gt; be an integer. A polygon &lt;math&gt;P&lt;/math&gt; with &lt;math&gt;n&lt;/math&gt; sides, or &lt;math&gt;n&lt;/math&gt;-gon, is a tuple of [[w:Vertex (geometry)|vertices]] &lt;math&gt;(v_1,\dots,v_n)&lt;/math&gt; lying in some [[w:projective plane|projective plane]] &lt;math&gt;\mathbb P ^2&lt;/math&gt;,{{Efn|In the following, the figures represent polygons on the real plane, where the intuition is easier to grasp.}} where the indices are understood [[w:Modular arithmetic|modulo]] &lt;math&gt;n&lt;/math&gt;. The [[w:Dimension of an algebraic variety|dimension]] of the space of &lt;math&gt;n&lt;/math&gt;-gons is &lt;math&gt;2n&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=definition 1.1}}

Suppose that the vertices are in sufficiently [[w:general position|general position]], meaning that no consecutive triple of points are [[w:Collinearity|collinear]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=1}} Taking the intersection of two consecutive &quot;shortest&quot; [[w:diagonal|diagonal]]s{{Efn|Meaning the line between a vertex &lt;math&gt;v_k&lt;/math&gt; and a &quot;second neighbour&quot; &lt;math&gt;v_{k\pm 2}&lt;/math&gt;.}} defines a new point&lt;math display=&quot;block&quot;&gt; w_k := \overline{v_{k-1} v_{k+1}} \cap \overline{v_{k} v_{k+2}}. &lt;/math&gt;This procedure defines a new &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;T(P)=(w_1,\dots,w_n)&lt;/math&gt;, as in Figure 2.{{Sfn|Schwartz|1992|p=71}}

The labeling of the indices of &lt;math&gt;T(P)&lt;/math&gt; is not [[w:canonical|canonical]]. In most papers, a choice is made at the beginning of the paper and the formulas are tuned accordingly.{{Sfn|Izosimov|2016|loc=remark 1.5}}

The pentagram map on polygons is a [[w:birational map|birational map]] &lt;math&gt;T:(\mathbb P^2)^n&lt;/math&gt;{{nowrap|{{font|size=145%|⇢}}}}&lt;math&gt;(\mathbb P^2)^n&lt;/math&gt;. Indeed, each [[w:Homogeneous coordinates|coordinate]] of &lt;math&gt;w_k&lt;/math&gt; is given as a [[w:rational function|rational function]] of the coordinates of &lt;math&gt;v_{k-1},\dots,v_{k+2}&lt;/math&gt;, since it is defined as the intersection of lines passing by them. Moreover, the [[w:inverse map|inverse map]] is given by taking the intersections &lt;math&gt;\overline{w_{k-2} w_{k-1}} \cap \overline{w_{k} w_{k+1}} &lt;/math&gt;, which is rational for the same reason.{{Sfnp|Weinreich|2022|loc=definition 1.2}}

=== Moduli space ===
The pentagram map is defined by taking [[w:Line (geometry)|lines]] and intersections of them. The biggest [[w:Group (mathematics)|group]] which maps lines to lines is the one of [[w:projective transformations|projective transformations]], denoted by &lt;math&gt;\mathbb P \mathrm{GL}_{3}&lt;/math&gt;. Such a transformation &lt;math&gt;M&lt;/math&gt; [[w:Group action|acts]] on a polygon &lt;math&gt;P&lt;/math&gt; by sending it to &lt;math&gt;M \cdot P:=(Mv_1,\dots,Mv_n)&lt;/math&gt;. The pentagram map [[w:Commutative property|commutes]] with this action, and thereby induces another [[w:dynamical system|dynamical system]] on the [[w:moduli space|moduli space]] of projective [[w:equivalence classes|equivalence classes]] of polygons, whose [[w:Dimension of an algebraic variety|dimension]] is &lt;math&gt;2n-8&lt;/math&gt;.{{Sfn|Schwartz|1992|loc=§1 Projective geometry}}

===Twisted polygons===
[[File:Twisted heptagon.svg|alt=|thumb|300x300px|An example of twisted [[w:heptagon|heptagon]] on the real plane.]]
The pentagram map naturally generalizes to the larger space of twisted polygons (see example in Figure 4). For any integer &lt;math&gt;n\geq5&lt;/math&gt;, a twisted &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;P&lt;/math&gt; is the data of:

* a [[w:Sequence#Indexing|bi-infinite sequence]] of points &lt;math&gt;(v_k)_{k\in\mathbb Z}&lt;/math&gt; in the projective plane (called the vertices),
* a [[w:projective transformation|projective transformation]] &lt;math&gt;M \in \mathbb P \mathrm{GL}_3&lt;/math&gt; (called the [[w:monodromy|monodromy]]),

such that for any &lt;math&gt;k \in \mathbb Z&lt;/math&gt;, the property &lt;math&gt;v_{k+n}=Mv_k&lt;/math&gt; is satisfied. The dimension of the space of twisted &lt;math&gt;n&lt;/math&gt;-gons is &lt;math&gt;2n+8&lt;/math&gt;.{{Sfn|Schwartz|2008}}

When &lt;math&gt;M&lt;/math&gt; is the [[w:Identity_element|identity]], this gives back the initial definition of polygons (which are said to be closed). The space of closed &lt;math&gt;n&lt;/math&gt;-gons is of [[w:codimension|codimension]] &lt;math&gt;8&lt;/math&gt; in the space of twisted ones.{{Sfn|Soloviev|2013|p=2816}}

The action of projective transformations over the space of closed polygons generalizes to the space of twisted ones (the monodromy is changed by [[w:Matrix similarity|conjugation]]). This provides again a moduli space, of dimension &lt;math&gt;2n&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=definition 1.3}}

== Collapsing of convex polygons ==

=== Exponential shrinking ===
[[File:Pentagram map convex heptagon iterate.svg|alt=|thumb|300x300px|The pentagram map iterated on a convex [[w:heptagon|heptagon]], exhibiting the convergence.]]
Let &lt;math&gt;P&lt;/math&gt; be a closed [[w:Convex polygon#Strictly convex polygon|strictly convex polygon]] lying on the real plane. One of the first results proved by Richard Schwartz it that its iterates under the pentagram map shrink [[w:Exponential growth|exponentially fast]] to a point, as illustrated in Figure 5. This follows from two facts.
# The image of a strictly convex polygon is contained in its [[w:Interior (topology)|interior]], and is also strictly convex.{{Sfn|Glick|2020|p=2818}}
# There exists a constant &lt;math&gt;0&lt; \eta_P&lt;1&lt;/math&gt;, depending on &lt;math&gt;P&lt;/math&gt;, such that for any &lt;math&gt;N \in \mathbb N&lt;/math&gt;, the diameters of the iterates verify the inequality &lt;math display=&quot;inline&quot;&gt;\operatorname{diam}(T^N(P))\leq\eta_P^N \operatorname{diam}(P). &lt;/math&gt;{{Sfn|Schwartz|1992|loc=theorem 3.1}}

Hence, by [[w:Cantor's intersection theorem#Variant in complete metric spaces|Cantor's intersection theorem]], the sequence of polygons collapses toward a point.{{Sfn|Schwartz|1992|loc=§3 Convex polygons}}

The behavior on the moduli space is very different, since the dynamics is [[w:Recurrent point|recurrent]].{{Sfn|Schwartz|2001|loc=theorem 1.1}} It is even a [[w:quasiperiodic motion|quasiperiodic motion]],{{Sfn|Ovsienko|Schwartz|Tabachnikov|2009}} as discussed in [[w:Pentagram map#Complete integrability|the section about integrability]].

=== Coordinates of the limit point ===
The limit point coordinates were given in {{Harvard citation|Glick|2020}}. They satisfy some [[w:Degree of a polynomial|degree]] 3 [[w:polynomial equation|polynomial equations]], whose coefficients are [[w:rational function|rational function]]s in the coordinates of the vertices of the starting polygon. The proof relies on the fact that the limit point must be an [[w:eigenline|eigenline]] of a certain [[w:linear operator|linear operator]] of &lt;math&gt;\mathbb R^3&lt;/math&gt;.{{sfn|Glick|2020}}

This operator was reinterpreted in {{Harvard citation|Aboud|Izosimov|2022}} as the infinitesimal monodromy of the polygon. The [[w:Pentagram map#The scaling symmetry|scaling symmetry]] is used to [[w:Deformation (mathematics)|deform]] a closed polygon &lt;math&gt;P&lt;/math&gt; into a family of twisted ones '''&lt;math&gt;(P_z)_{z\in \mathbb C^*}&lt;/math&gt;''' with monodromy &lt;math&gt;M_z&lt;/math&gt;. The infinitesimal monodromy is defined to be:{{sfn|Aboud|Izosimov|2022}}
&lt;math display=&quot;block&quot;&gt;\left.\frac{dM_z}{dz}\right|_{z=1}.&lt;/math&gt;

=== Generalization ===
The collapsing of polygons may also happen in some [[w:Pentagram map#Generalizations|generalization of the pentagram map]], when considering some specific configurations of polygons in the real plane. The coordinates of the collapse point are given by a formula analogous to the one for the original pentagram map.{{Sfn|Schwartz|2026}}

== Periodic orbits on the moduli space ==
For some configurations of closed polygons, the iterate of the pentagram map will send &lt;math&gt;P&lt;/math&gt; to a projectively equivalent polygon (up to some shift of the indices). This means that, on the moduli space, the orbit of the class of &lt;math&gt;P&lt;/math&gt; is [[w:Periodic orbit|periodic]].

===Pentagons and hexagons===
[[File:penta hexagon.svg|300px|thumb|The outward hexagon is projectively equivalent to the inward one, with respect to their labeling.]]The following two facts are proved by checking [[w:cross-ratio|cross-ratio]] equalities, so they are true for polygons in any [[w:projective plane|projective plane]] (not just the [[w:Real projective plane|real one]]).{{Sfn|Schwartz|1992|loc=§2 Pentagons and hexagons}}

The pentagram map &lt;math&gt;T&lt;/math&gt; is the identity on the moduli space of [[w:pentagon|pentagon]]s.{{Sfn|Schwartz|1992|loc=theorem 2.1}}{{Sfn|Clebsch|1871}}{{Sfn|Motzkin|1945}} The second iterate &lt;math&gt;T^2&lt;/math&gt; is the identity on the space of labeled [[w:hexagon|hexagon]]s, up to a shift of labeling (see Figure 6).{{Sfn|Schwartz|1992|loc=theorem 2.3}} This phenomenon doesn't generalize to generic polygons with at least seven sides, for which the motion is [[w:Quasiperiodic motion|quasi-periodic]].{{Sfn|Tupan|2022}}

==== Generalization ====
The result about pentagons and hexagons generalizes to some [[w:Pentagram map#Generalizations|higher pentagram maps]] in &lt;math&gt;\mathbb P ^k&lt;/math&gt;, for polygons with &lt;math&gt;k+3&lt;/math&gt; or &lt;math&gt;2k+2&lt;/math&gt; sides. The proof uses a generalization of the [[w:Gale transform|Gale transform]].{{Sfn|Dirdak|2024}}

=== Poncelet polygons ===
A polygon is said to be Poncelet{{Efn|The name comes from [[w:Jean-Victor Poncelet|Jean-Victor Poncelet]] and [[w:Poncelet porism|his porism]].{{Sfn|Izosimov|2022a|p=1085}}}} if it is [[w:Inscribed figure|inscribed]] in a [[w:Conic section|conic]] and circumscribed about another one.{{Sfn|Schwartz|2015|loc=|p=433}}{{Efn|In particular, pentagons are Poncelet since [[w:five points determine a conic|five points determine a conic]].{{Sfn|Schwartz|2015|loc=|p=433}}}} For a convex Poncelet &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;P&lt;/math&gt; lying on the [[w:real projective plane|real projective plane]], the polygon &lt;math&gt;T^2(P)&lt;/math&gt; is projectively equivalent to &lt;math&gt;P&lt;/math&gt;.{{Sfn|Schwartz|2015|loc=theorem 1.1}} In fact, when &lt;math&gt;n&lt;/math&gt; is odd, the converse is also true.{{Sfn|Izosimov|2022a|loc=corollary 1.1}}

However, this converse statement is no longer true when the polygons are considered over the [[w:complex projective plane|complex projective plane]].{{Sfn|Izosimov|2022a|loc=remark 1.3}}

==Coordinates for the moduli space==
The moduli space can be described by different [[w:Coordinate_system|coordinate systems]]. The following ones are practical to explicit the dynamic, as presented in the next section.

=== Corner coordinates ===
[[File:Corner coordinates big.svg|thumb|300x300px|The geometric construction of the points defining the corner invariants.]]
Define the [[w:cross-ratio|cross-ratio]] of four [[w:Collinearity|collinear]] points to be

: &lt;math&gt; [a,b,c,d]=\frac{(a-b)(c-d)}{(a-c)(b-d)}. &lt;/math&gt;

The corner invariants are a system of coordinates on the space of twisted polygons, constructed by taking intersections as in Figure 7.{{Sfn|Schwartz|2001|loc=figure 2}} The left and right invariants are respectively defined{{Efn|The ordering of the vertices in the cross-ratios can differ from a paper to another one, which slightly changes the formulas in the following sections.}} as the following cross-ratios:

: &lt;math&gt;x_k:=[v_{k-2},v_{k-1},\overline{v_{k-2}v_{k-1}}\cap\overline{v_{k}v_{k+1}},\overline{v_{k-2}v_{k-1}}\cap\overline{v_{k+1}v_{k+2}}],&lt;/math&gt;

: &lt;math&gt;y_k:=[\overline{v_{k+1}v_{k+2}}\cap\overline{v_{k-2}v_{k-1}}, \overline{v_{k+1}v_{k+2}}\cap\overline{v_{k-1}v_{k}},v_{k+1},v_{k+2}].&lt;/math&gt;

Since the cross-ratio is [[w:Cross-ratio#Projective geometry|projective invariant]], the sequences &lt;math&gt;(x_k)_{k \in \mathbb Z}&lt;/math&gt; and &lt;math&gt;(y_k)_{k \in \mathbb Z}&lt;/math&gt; associated to a twisted &lt;math&gt;n&lt;/math&gt;-gon are &lt;math&gt;n&lt;/math&gt; periodic.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=415}}

The corner invariants are elements of &lt;math&gt;\mathbb{P}^1\smallsetminus\{0,1,\infty\}&lt;/math&gt;, and they realize an [[w:Isomorphism_of_varieties|isomorphism of variety]] between the moduli space of twisted &lt;math&gt;n&lt;/math&gt;-gons and &lt;math&gt;(\mathbb{P}^1\smallsetminus\{0,1,\infty\})^{2n}&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=theorem 3.6}}

===ab-coordinates===

There is a second set of coordinates for the moduli space of twisted &lt;math&gt;n&lt;/math&gt;-gons defined over a [[w:Field (mathematics)|field]] &lt;math&gt;F&lt;/math&gt; satisfying &lt;math&gt;\mathrm{SL}_3(F)\cong \mathbb P\mathrm{GL}_3(F)&lt;/math&gt;,{{Sfn|Weinreich|2022|loc=remark 3.8}} and such that &lt;math&gt;n&lt;/math&gt; is not divisible by &lt;math&gt;3&lt;/math&gt;.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=section 4.1}}

The vertices &lt;math&gt;v_k&lt;/math&gt; in the [[w:projective plane|projective plane]] &lt;math&gt;\mathbb P^2(F)&lt;/math&gt; can be [[w:Lift (mathematics)|lifted]] to [[w:Vector space|vectors]] &lt;math&gt;V_k&lt;/math&gt; in the [[w:affine space|affine space]] &lt;math&gt;F^3&lt;/math&gt; so that each consecutive triple of vectors spans a [[w:parallelepiped|parallelepiped]] having [[w:determinant|determinant]] equal to &lt;math&gt;1&lt;/math&gt;. This leads to the relation{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=equation 4.1}}

: &lt;math&gt;V_{k+3} = a_k V_{k+2} + b_k V_{k+1} + V_k.&lt;/math&gt;

This bring out an analogy between twisted polygons and solutions of third order linear [[w:ordinary differential equations|ordinary differential equations]], normalized to have unit [[w:Wronskian|Wronskian]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=remark 6.6}}

They are linked to the corner coordinates by:{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=lemma 4.5}}
: &lt;math&gt;x_k=\frac{a_{k-2}}{b_{k-2}b_{k-1}},&lt;/math&gt;
: &lt;math&gt;y_k=-\frac{b_{k-1}}{a_{k-2}a_{k-1}}.&lt;/math&gt;

==Formulas on the moduli space==

===As a birational map ===
The pentagram map is a [[w:birational map|birational map]] on the moduli space, because it can be decomposed as the [[w:Function composition|composition]] of two [[w:Birational geometry|birational]] [[w:Involution (mathematics)|involutions]].{{Sfn|Schwartz|2008|loc=§1.2 The Pentagram Map}} The corner invariants change in the following way:{{Sfn|Ovsienko|Schwartz|loc=lemma 2.4|Tabachnikov|2010}}

: &lt;math&gt;x_k'=x_k\frac{1-x_{k-1} y_{k-1}}{1-x_{k+1}y_{k+1}},&lt;/math&gt;

: &lt;math&gt;y_k'=y_{k+1}\frac{1-x_{k+2} y_{k+2}}{1-x_k y_k}.&lt;/math&gt;

=== The scaling symmetry ===
The [[w:multiplicative group|multiplicative group]] &lt;math&gt;F\smallsetminus\{0\}&lt;/math&gt; [[w:One-parameter group|acts]] on the moduli space in the following way:

: &lt;math&gt;R_s\cdot(x_1,\dots,x_n,y_1,\dots,y_n)=(sx_1,\dots,sx_n,s^{-1}y_1,\dots,s^{-1}y_n),&lt;/math&gt;

where &lt;math&gt;R&lt;/math&gt; is called the scaling action and &lt;math&gt;s&lt;/math&gt; is the scaling parameter. This action commutes with the pentagram map on the moduli space (as presented in the previous formulas). This property is called the scaling symmetry, and is instrumental in proving the [[w:Pentagram map#Complete integrability|complete integrability]] of the dynamics.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=corollary 2.5}}

==Invariant structures==

===Monodromy invariants===

The monodromy invariants, introduced in {{Harvard citation|Schwartz|2008}}, are a collection of [[w:Function (mathematics)|functions]] on the [[w:moduli space|moduli space]] that are invariant under the pentagram map.{{Sfn|Schwartz|2008|loc=theorem 1.2}} The simplest example of them are 
:&lt;math&gt; O_n= x_1x_2\cdots x_{n}, \quad E_n = y_1y_2\cdots y_n. &lt;/math&gt;
The other monodromy invariants can be retrieved through different points of view: through the [[w:Pentagram map#The scaling symmetry|scaling symmetry]], as [[w:Combinatorics|combinatorial]] objects, or as some [[w:determinant|determinant]]s.{{Sfn|Schwartz|Tabachnikov|2011|loc=§2 The Monodromy Invariants}} The one involving scaling symmetry is presented here.

Let &lt;math&gt;M\in \mathrm{GL}_3&lt;/math&gt; be a [[w:Lift (mathematics)|lift]] of the monodromy of a twisted &lt;math&gt;n&lt;/math&gt;-gon. The quantities

: &lt;math&gt;\Omega_1=\frac{\operatorname{trace}^3(M)}{\det(M)}, \quad \Omega_2=\frac{\operatorname{trace}^3(M^{-1})}{\det(M^{-1})},&lt;/math&gt;

are independent of the choice of lift and are invariant under [[w:Matrix similarity|conjugation]], so they are well defined for the projective class of the polygon. They are invariant under the pentagram map, since the monodromy matrix doesn't change.{{Sfn|Schwartz|Tabachnikov|2011|loc=|p=5}} Now, the quantities

: &lt;math&gt;\tilde{\Omega}_1=O_n^2E_n\Omega_1, \quad \tilde{\Omega}_2=O_nE_n^2\Omega_2,&lt;/math&gt;

have the same properties, but turn out to be polynomials in the corner invariants.{{Efn|Some papers consider the cube roots of this functions, but it doesn't change the following definitions of the monodromy invariants.}} They can be written as{{Sfn|Schwartz|Tabachnikov|2011|loc=|p=5}}

: &lt;math&gt;
\tilde{\Omega}_1=\biggl(\sum_{k=0}^{\lfloor n/2\rfloor}O_k\biggr)^3, \quad
\tilde{\Omega}_2=\biggl(\sum_{k=0}^{\lfloor n/2\rfloor}E_k\biggr)^3,
&lt;/math&gt;

where each &lt;math&gt;O_k&lt;/math&gt; and &lt;math&gt;E_k&lt;/math&gt; are [[w:homogeneous polynomial|homogeneous polynomial]]s respectively of weight &lt;math&gt;k&lt;/math&gt; and &lt;math&gt;-k&lt;/math&gt;,{{Sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=11}} meaning they change under the [[w:Pentagram map#The scaling symmetry|rescaling action]] on variables by{{Sfn|Schwartz|Tabachnikov|2011|p=5}}

: &lt;math&gt; R_s(O_k)= s^k O_k, \quad R_s(E_k)= s^{-k} E_k. &lt;/math&gt;

The quantities &lt;math&gt;O_1,\dots,O_{\lfloor n/2 \rfloor},O_n, E_1,\dots,E_{\lfloor n/2 \rfloor},E_n,&lt;/math&gt; are unchanged by the dynamics, and are called the monodromy invariants. Moreover, they are [[w:algebraically independent|algebraically independent]].{{Sfn|Schwartz|2008|loc=theorem 1.2}}

==== Polygons on conics ====
Whenever &lt;math&gt;P&lt;/math&gt; is [[w:Inscribed figure|inscribed]] on a [[w:conic section|conic section]], one has &lt;math&gt;O_k(P)=E_k(P)&lt;/math&gt; for all &lt;math&gt;k&lt;/math&gt;.{{Sfn|Schwartz|Tabachnikov|2011|loc=theorem 1.1}} Moreover, if &lt;math&gt;P&lt;/math&gt; is circumscribed about another conic,{{Efn|See the paragraph about [[w:Pentagram map#Poncelet polygons|Poncelet polygons]].}} then its monodromy invariants are characterized by the pair of conics.{{Sfn|Schwartz|2015|loc=theorem 1.2}} For such odd-gons, the translation on the [[w:Jacobian variety|Jacobian variety]]{{Efn|See the paragraph about [[w:Pentagram map#Algebro-geometric integrability|algebraic integrability]].}} is restricted to the [[w:Prym variety|Prym variety]] (which is a half-dimensional torus in the Jacobian).{{Sfn|Izosimov|2016|loc=theorem 1.3}}

===Poisson bracket===

An invariant [[w:Poisson bracket|Poisson bracket]] on the space of twisted polygons was found in {{Harvard citation|Ovsienko|Schwartz|Tabachnikov|2010}}. The monodromy invariants [[w:Poisson bracket#Constants of motion|commute]] with respect to it:
&lt;math display=&quot;block&quot;&gt; \{O_i,O_j\}=\{O_i,E_j\}=\{E_i,E_j\}=0 &lt;/math&gt;for all &lt;math&gt;i,j&lt;/math&gt;.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}}

The Poisson bracket is defined in terms of the corner coordinates by:
&lt;math display=&quot;block&quot;&gt; \begin{align}
 \{x_i,x_{i\pm1}\} &amp;= \mp x_i x_{i+1}, \\
 \{y_i,y_{i\pm 1}\} &amp;= \mp y_i y_{i+1}, \\
 \{x_i,x_j\} &amp;= \{y_i,y_j\} = \{x_i,y_j\} = 0
\end{align}&lt;/math&gt;for all other &lt;math&gt; i,j.&lt;/math&gt;{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=equation 2.16}}

=== The spectral curve ===

Let &lt;math&gt;\zeta&lt;/math&gt; be an element of the [[w:multiplicative group|multiplicative group]] and &lt;math&gt;P_\zeta&lt;/math&gt; be the polygon obtained by applying the [[w:Pentagram map#The scaling symmetry|rescaling action]] &lt;math&gt;R_\zeta&lt;/math&gt; on &lt;math&gt;P&lt;/math&gt;.  A [[w:Lax matrix|Lax matrix]] &lt;math&gt;\hat{T}(\zeta) \in \mathrm{GL}_3&lt;/math&gt; is a lift of the monodromy of &lt;math&gt;P_\zeta&lt;/math&gt; satisfying a [[w:Lax pair#Zero-curvature equation|zero-curvature equation]].{{Sfn|Weinreich|2022|loc=§5 The Lax representation}} Then, the spectral function is the [[w:Bivariate polynomial|bivariate]] [[w:characteristic polynomial|characteristic polynomial]]
&lt;math display=&quot;block&quot;&gt; Q(\lambda,\zeta) := \det(\lambda\operatorname{Id}-\hat{T}(\zeta)),&lt;/math&gt;or some renormalization of it. The [[w:spectral curve|spectral curve]] is the [[w:Projective variety#projective completion|projective completion]] of the [[w:Algebraic curve|affine curve]] defined by the equation &lt;math&gt;Q(\lambda,\zeta)=0&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=§6. The geometry of the spectral curve}} It is invariant under the pentagram map, and the monodromy invariants appear as the [[w:coefficient|coefficient]]s of &lt;math&gt;Q&lt;/math&gt;.{{Sfn|Soloviev|2013|loc=theorem 6.4}} Its [[w:geometric genus|geometric genus]] is &lt;math&gt;n-1&lt;/math&gt; if &lt;math&gt;n&lt;/math&gt; is odd, and &lt;math&gt;n-2&lt;/math&gt; if &lt;math&gt;n&lt;/math&gt; is even.{{Sfn|Weinreich|2022|p=|loc=theorem 6.4}}

It was first introduced in {{Harv|Soloviev|2013|ps=|p=}} for his proof of [[w:Pentagram map#Algebro-geometric integrability|algebro-geometric integrability]].{{sfn|Soloviev|2013}}

==Complete integrability==
The pentagram map on the moduli space has been proved to be a [[w:completely integrable|completely integrable]] [[w:discrete dynamical system|discrete dynamical system]], both in the [[w:Integrable system#Hamiltonian systems and Liouville integrability|Arnold-Liouville]]{{Efn|Over the [[w:real number|real number]]s.}} and the [[w:Integrable system#Complete integrability over the complex numbers|algebro-geometric]]{{Efn|Over [[w:algebraically closed field|algebraically closed field]]s of [[w:Characteristic (algebra)|characteristic]] different from 2.}} senses. In any case, this means that the moduli space is [[w:almost everywhere|almost everywhere]] [[w:Foliation|foliated]] by [[w:Torus#Flat torus|flat tori]] (or in the algebraic setting, [[w:Abelian variety|Abelian varieties]]), where the motion is a [[w:Translation (geometry)|translation]]. This [[w:Generic property|generically]] induces a [[w:quasiperiodic motion|quasiperiodic motion]] on the corresponding torus.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2009}}

===Arnold–Liouville integrability===

The proof of the integrability of the pentagram map on a real twisted polygon was achieved in {{Harvard citation|Ovsienko|Schwartz|Tabachnikov|2010}}. This is done by noticing that the monodromy invariants &lt;math&gt;O_n&lt;/math&gt; and &lt;math&gt;E_n&lt;/math&gt; are [[w:Casimir invariant|Casimir invariant]]s for the bracket, meaning (in this context) that&lt;math display=&quot;block&quot;&gt; \{O_n,f\}=\{E_n,f\} = 0 &lt;/math&gt;for all functions &lt;math&gt;f&lt;/math&gt;.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}} When &lt;math&gt;n&lt;/math&gt; is even, this is also true for the monodromy invariants &lt;math&gt;O_{\lfloor n/2 \rfloor }&lt;/math&gt; and &lt;math&gt;E_{\lfloor n/2 \rfloor }&lt;/math&gt;.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}}

This allows to consider the Casimir [[w:level set|level set]], where each Casimir has a specified value. Because of [[w:Sard's theorem|Sard's theorem]], any generic level set is a [[w:smooth manifold|smooth manifold]].{{Sfn|Schwartz|2017|p=44}} They form a [[w:foliation|foliation]] in [[w:Poisson manifold#Symplectic leaves|symplectic leaves]], on which the Poisson bracket gives rise to a [[w:symplectic form|symplectic form]].{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=corollary 2.13}}

Each of these symplectic leaves has an iso-monodromy [[w:foliation|foliation]], namely, a decomposition into the common level sets of the remaining monodromy functions. By using again [[w:Sard's theorem|Sard's theorem]], they are generically [[w:Symplectic manifold#Lagrangian submanifolds|Lagrangian manifolds]].{{Sfn|Schwartz|2017|p=45}} Moreover, they are compact.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=§3.3. Compactness of the level sets}} Since the monodromy invariants Poisson-commute and there are enough of them, the discrete [[w:Liouville–Arnold theorem|Liouville–Arnold theorem]] can be applied to prove that the level sets are [[w:Torus#Flat torus|flat tori]] over which the dynamics is a translation.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=412}}
===Algebro-geometric integrability===

In {{Harvard citation|Soloviev|2013}}, it was shown that the pentagram map admits a [[w:Lax representation|Lax representation]] with a spectral parameter, which allows to prove its algebro-geometric integrability. This means that the space of polygons (either twisted or closed) is parametrized by its spectral data, consisting of [[w:Pentagram map#The spectral curve|its spectral curve]], with marked points and a [[w:Divisor (algebraic geometry)|divisor]] given by a [[w:Floquet theory|Floquet]]–[[w:Bloch's theorem|Bloch]] equation. This gives an embedding to the [[w:Jacobian variety|Jacobian variety]] through the [[w:Abel–Jacobi map|Abel–Jacobi map]], where the motion is expressed in terms of translation.{{sfn|Soloviev|2013|loc=theorems A, B and C}} The previously defined Poisson bracket is also retrieved.{{sfn|Soloviev|2013|loc=theorem D}} 

This integrability was generalized in {{Harvard citation|Weinreich|2022}} from the field of [[w:complex number|complex number]]s to any [[w:algebraically closed field|algebraically closed field]] of [[w:Characteristic (algebra)|characteristic]] different from 2. The translation on a torus is replaced by a translation on an [[w:Abelian variety|Abelian variety]] (in fact, a Jacobian variety again).{{sfn|Weinreich|2022|loc=theorem 1.4}}

=== Dimension of the invariant manifold ===

For twisted &lt;math&gt;n&lt;/math&gt;-gons, the [[w:dimension|dimension]] of the invariant tori (or Jacobian varieties) is{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=421}}

: &lt;math&gt;\begin{cases} 
n-1 &amp; \text{when }n \text{ is odd,}\\
n-2 &amp; \text{when }n \text{ is even.}
\end{cases}&lt;/math&gt;

Moreover, when &lt;math&gt;n&lt;/math&gt; is even, there are two isomorphic Jacobians on which the iterates of the pentagram map alternate. But on each of them, the second iterate is a translation.{{Sfn|Weinreich|2022|loc=theorem 1.4}}

=== For closed polygons ===
There is no Poisson structure on the space of closed polygons.{{Sfn|Soloviev|2013|loc=corollary 4.1}} Nevertheless, the one from twisted polygons can be used to prove integrability.{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=2153}}

Algebro-geometric integrability holds for closed polygons in a same manner as for the twisted ones.{{Sfn|Soloviev|2013|loc=theorem C}} However, Arnold-Liouville integrability is proved for real closed polygons only when they are convex. This is done by restricting the [[w:Hamiltonian vector field|Hamiltonian vector field]]s of monodromy functions to smaller dimensional tori, and showing that enough of them are still independent.{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|loc=corollary 1.1}}

In both situation, the dimension of the invariant manifolds decreases by &lt;math&gt;3&lt;/math&gt; for closed &lt;math&gt;n&lt;/math&gt;-gons (compared to the twisted case), and is equal to{{Sfn|Soloviev|2013|loc=theorem C}}{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|loc=theorem 1}}
:&lt;math&gt;\begin{cases} 
n-4 &amp; \text{when }n \text{ is odd,}\\
n-5 &amp; \text{when }n \text{ is even.}
\end{cases}&lt;/math&gt;

==Connections to other topics==

===The Boussinesq equation===

The continuous limit of a convex polygon is a parametrized convex curve in the plane. When the time parameter is suitably chosen, the [[w:Discretization|continuous limit]] of the pentagram map is the classical [[w:Boussinesq approximation (water waves)|Boussinesq equation]]. This equation is a classical example of an [[w:integrable|integrable]] [[w:partial differential equation|partial differential equation]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 5}}

Here is a description of the geometric action of the Boussinesq equation. Given a [[w:locally convex|locally convex]] curve &lt;math&gt; C:\mathbb R\to \mathbb R^2 &lt;/math&gt; and real numbers &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;t&lt;/math&gt;, consider the [[w:chord (geometry)|chord]] connecting &lt;math&gt; C(x-t) &lt;/math&gt; to &lt;math&gt; C(x+t) &lt;/math&gt;. The [[w:Envelope (mathematics)|envelope]] of all these chords is a new curve &lt;math&gt; C_t(x) &lt;/math&gt;. When &lt;math&gt;t&lt;/math&gt; is extremely small, the curve &lt;math&gt; C_t(x) &lt;/math&gt; is a good model for the time &lt;math&gt;t&lt;/math&gt; evolution of the original curve &lt;math&gt; C_0(x) &lt;/math&gt; under the Boussinesq equation. This construction is also similar to the pentagram map. Moreover, the pentagram invariant bracket is a discretization of a well known invariant Poisson bracket associated to the Boussinesq equation.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=§6.4 Discretization}}

===Cluster algebras===

The pentagram map{{Sfn|Glick|2011}} and some of its generalizations{{Sfn|Gekhtman|Shapiro|Tabachnikov|Vainshtein|2012}}{{Sfn|Glick|Pylyavskyy|2016}} are identified as special cases of discrete dynamical systems powered by [[w:cluster algebra|cluster algebra]]. Using the results from {{Harvard citation|Goncharov|Kenyon|2013}}, this provides a link with the [[w:Poisson–Lie group|Poisson–Lie group]]s, [[w:dimer model|dimer model]]s and other so-called cluster-integrable systems.{{Sfn|Fock|Marshakov|2016}}{{Sfn|Izosimov|2022c}} These methods allow to retrieve the Poisson-bracket and Hamiltonians used to prove complete integrability{{Sfn|Affolter|George|Ramassamy|2025|loc=§5 The pentagram map}} and provide [[w:Lax representation|Lax representation]]s.{{Sfn|Izosimov|2022b}}

=== Octahedron recurrence ===
Using a method to compute [[w:Determinant|determinants]] called [[w:Dodgson_condensation|Dodgson condensation]], {{Harvard citation|Schwartz|2008}} proves that the pentagram map satisfies a property called the &quot;octahedron recurrence&quot;.{{Sfn|Schwartz|2008|loc=§5 The Method of Condensation}} This property turns out to be shared by other dynamical systems defined by geometric constructions similar to one of the pentagram map.{{Sfn|Affolter|de Tilière|Melotti|2025|loc=§9 The Pentagram Map}} It is also shared by higher dimensional pentagram maps defined through cluster algebras mutations, referred as &lt;math&gt;T&lt;/math&gt;-systems.{{Sfn|Kedem|Vichitkunakorn|2015||loc=}}

=== Singularity theory ===
The pentagram map exhibit a property called singularity confinement, which is typical from [[w:integrable system|integrable system]]s.{{Sfn|Grammaticos|Ramani|Papageorgiou|1991}} It states that if a polygon &lt;math&gt;P&lt;/math&gt; is [[w:Singular point of an algebraic variety|singular]] for the pentagram map &lt;math&gt;T&lt;/math&gt;, then there exists an integer &lt;math&gt;m&lt;/math&gt; such that &lt;math&gt;P&lt;/math&gt; not singular for the iterate map &lt;math&gt;T^m&lt;/math&gt;.{{Sfn|Glick|2012}}

Moreover, the pentagram map (along with some of its generalizations and other discrete dynamical systems) exhibit the Devron property.{{Efn|The name comes from an episode of [[w:Star Trek|Star Trek]].{{Sfn|Glick|2015|loc=§1 Introduction}}}} This means that if a polygon &lt;math&gt;P&lt;/math&gt; is singular for some iterate of the pentagram map &lt;math&gt;T^m&lt;/math&gt;, then it will also be singular for some iterate of the inverse map &lt;math&gt;T^{-m'}&lt;/math&gt;.{{Sfn|Glick|2015}}

== Generalizations ==
The definition of twisted polygons still makes sense in any [[w:projective space|projective space]] &lt;math&gt;\mathbb P^d&lt;/math&gt;, under the action of the [[w:Projective linear group|projective group]] &lt;math&gt;\mathbb P \mathrm{GL}_{d+1}&lt;/math&gt;. The pentagram map can be generalized in many ways, and some of them are presented here. Not all of them are integrable.{{Sfn|Khesin|Soloviev|2015|}} Some are [[w:discretization|discretization]]s of [[w:PDEs|PDEs]] from the [[w:KdV hierarchy|KdV hierarchy]], seen as higher dimensional version of [[w:Boussinesq approximation (water waves)|Boussinesq]] or [[w:Kadomtsev–Petviashvili equation|KP]] equations.{{Sfn|Marí-Beffa|2012}}{{Sfn|Wang|2023}} The description of all generalized pentagram maps in terms of [[w:cluster algebra|cluster algebra]]s is still an open question.{{Sfn|Gekhtman|Izosimov|2025|p=14}}

=== Polygons in general positions ===
Let &lt;math&gt;d \geq 2&lt;/math&gt; and &lt;math&gt;P&lt;/math&gt; be a twisted polygon of &lt;math&gt;\mathbb P^d&lt;/math&gt; in [[w:general position|general position]].

==== Short diagonal pentagram maps ====
The &lt;math&gt;k&lt;/math&gt;-th ''short diagonal hyperplane'' &lt;math&gt;H_k^{sh}&lt;/math&gt; is uniquely defined by passing through the vertices &lt;math&gt;v_k,v_{k+2},\dots,v_{k+2d-2}&lt;/math&gt;. [[w:Generic property#In algebraic geometry|Generically]], the intersection of &lt;math&gt;d&lt;/math&gt; consecutive hyperplanes uniquely defines a new point

: &lt;math&gt;T_{sh}v_k:=H_k^{sh}\cap H_{k+1}^{sh}\cap \dots \cap H_{k+d-1}^{sh}.&lt;/math&gt;

Doing this for every vertex defines a new twisted polygon. This map, denoted by &lt;math&gt;T_{sh}&lt;/math&gt;, is again projectively equivariant.{{Sfn|Khesin|Soloviev|2013}}

==== Generalized pentagram maps ====
The previous procedure can be generalized. Let &lt;math&gt;I=(i_1,\dots,i_{d-1}),~J=(j_1,\dots,j_{d-1})&lt;/math&gt; be two sets of integers, respectively called the jump tuple and the intersection tuple. Define the &lt;math&gt;k&lt;/math&gt;-th hyperplane &lt;math&gt;H_k^I&lt;/math&gt; to be passing through the vertices &lt;math&gt;v_k,v_{k+i_1},\dots,v_{k+i_1+\dots+i_{d-1}}&lt;/math&gt;. A new point is given by the intersection

: &lt;math&gt;T_{I,J}v_k:=H_k^I \cap H_{k+j_1}^I \cap \dots \cap H_{k+j_1+\dots +j_{d-1}}^I.&lt;/math&gt;

The map &lt;math&gt;T_{I,J}&lt;/math&gt; is called a generalized pentagram map.{{Sfn|Khesin|Soloviev|2015a}} The original pentagram map is recovered by considering&lt;math&gt;d=2,~I=(2),~J=(1)&lt;/math&gt;.

Integrability can be numerically tested by picking a random polygon &lt;math&gt;P&lt;/math&gt; with [[wikipedia:Rational_point|rational coordinates]] and studying the growth rate of the [[wikipedia:Height_function|height]] of its iterates. This is called the [[wikipedia:Integrable_system#Diophantine_integrability|diophantine integrability]] test, and some generalized pentagram maps don't seem to pass it.{{Sfn|Khesin|Soloviev|2015a|loc=§5 and §6}} However, it is conjectured that the maps &lt;math&gt;T_{I,I}&lt;/math&gt; are integrable for any &lt;math&gt;I&lt;/math&gt;.{{Sfn|Bolsinov|Matveev|Miranda|Tabachnikov|2018|loc=conjecture 4.13 (B. Khesin, F. Soloviev)}}

Some of these maps are [[w:discretization|discretization]]s of higher dimensional counterpart of the [[w:Boussinesq approximation (water waves)|Boussinesq equation]] in the [[w:KdV hierarchy|KdV hierarchy]].{{Sfn|Khesin|Soloviev|2015b|loc=theorem 4.1}}{{Sfn|Izosimov|2022b|loc=theorem 4.1}}

==== Dented pentagram maps ====
Fix an integer &lt;math&gt;m\in \{1,\dots ,d-1\}&lt;/math&gt;. Consider the jump tuple &lt;math&gt;I_m:=(1,\dots,1,2,1,\dots,1)&lt;/math&gt;, where the &lt;math&gt;2&lt;/math&gt; is at the &lt;math&gt;m&lt;/math&gt;-th place, and the intersection tuple &lt;math&gt;J:=(1,\dots,1)&lt;/math&gt;. The dented pentagram map is &lt;math&gt;T_m :=T_{I_m,J}&lt;/math&gt;. They are proved to be integrable.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 2.14}}

For an integer &lt;math&gt;p \geq 2&lt;/math&gt;, the deep dented pentagram map (of depth &lt;math&gt;p&lt;/math&gt;) &lt;math&gt;T_m^p&lt;/math&gt; is the same map as before, but the number &lt;math&gt;2&lt;/math&gt; in the definition of &lt;math&gt;I_m&lt;/math&gt; is replaced by &lt;math&gt;p&lt;/math&gt;. This kind of pentagram maps are again integrable.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 6.2}}

=== Corrugated polygons ===
A twisted polygon &lt;math&gt;P&lt;/math&gt; lying in &lt;math&gt;\mathbb P^d&lt;/math&gt; is said to be corrugated if for any &lt;math&gt;k\in \mathbb Z&lt;/math&gt;, the vertices &lt;math&gt;v_k,v_{k+1},v_{k+d},v_{k+d+1}&lt;/math&gt; span a projective two-dimensional plane. Such polygons are not in [[w:general position|general position]]. A new point is defined by

: &lt;math&gt;T_\text{cor}v_k:=\overline{v_k v_{k+d}}\cap \overline{v_{k+1} v_{k+d+1}}.&lt;/math&gt;

The map &lt;math&gt;T_\text{cor}&lt;/math&gt; yields a new corrugated polygon. They are [[w:Integrable system#Hamiltonian systems and Liouville integrability|completely Liouville-integrable]].{{Sfn|Gekhtman|Shapiro|Tabachnikov|Vainshtein|2012|loc=theorem 4.4}}

In fact, they can be retrieved as some dented pentagram map applied on corrugated polygons.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 5.3}}

=== Grassmannian polygons ===
Let &lt;math&gt;d \geq 3, m \geq 1&lt;/math&gt; be integers. The pentagram map can also be generalized to the [[w:Grassmannian|Grassmannian]] space &lt;math&gt;\mathrm{Gr}(m,md)&lt;/math&gt;, which consists of &lt;math&gt;m&lt;/math&gt;-[[w:Dimension (vector space)|dimensional]] [[w:linear subspace|linear subspace]]s of an &lt;math&gt;md&lt;/math&gt;-dimensional [[w:vector space|vector space]]. When &lt;math&gt;m=1&lt;/math&gt;, the linear subspaces are [[w:Vector space#vector line|lines]], which retrieves the definition of [[w:projective space|projective space]]s &lt;math&gt;\mathbb P^d&lt;/math&gt;.{{Sfn|Felipe|Marí-Beffa|2019|loc=§2 definitions and notations}}

A point &lt;math&gt;v\in\operatorname{Gr}(m,md)&lt;/math&gt; is represented by an &lt;math&gt;md \times m&lt;/math&gt; matrix &lt;math&gt;X_v&lt;/math&gt; such that its columns form a [[w:Basis (linear algebra)|basis]] of &lt;math&gt;v&lt;/math&gt;. Consider the [[w:Group action|action]] of the [[w:general linear group|general linear group]] &lt;math&gt;\mathrm{GL}_{md}&lt;/math&gt; by multiplication on the left of &lt;math&gt;X_v&lt;/math&gt;. This defines an action on the Grassmannian, even though it is not [[w:Faithful action|faithful]].{{Efn|Because there can be many lifts for &lt;math&gt;v&lt;/math&gt;, and because some matrices act trivially.}} Hence, the polygons of &lt;math&gt;\mathrm{Gr}(m,md)&lt;/math&gt; and their moduli spaces are defined as before, after the change of underlying group.{{Sfn|Felipe|Marí-Beffa|2019|loc=§2 definitions and notations}}

Depending on the parity of &lt;math&gt;d&lt;/math&gt;, one can define linear subspaces spanned by some &lt;math&gt;X_{v_k}&lt;/math&gt;'s such that taking their intersection generically defines a new point &lt;math&gt;v\in\mathrm{Gr}(m,md)&lt;/math&gt;.{{Sfn|Felipe|Marí-Beffa|2019|loc=sections 4 and 5}} This generalization of the pentagram map is integrable in a [[w:noncommutative|noncommutative]] sense.{{Sfn|Ovenhouse|2020}}

=== Over rings ===
The pentagram map admits a generalization by considering [[w:Projective space#Generalizations|projective planes]] over [[w:stably finite ring|stably finite ring]]s, instead of [[w:Field (mathematics)|field]]s. In particular, this retrieves the pentagram map over Grassmannians. Again, it admits a [[w:Lax representation|Lax representation]].{{Sfn|Hand|Izosimov|2025}}

== References ==
{{reflist|25em}}

===Notes===
{{notelist}}

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      <text bytes="60962" sha1="c10ge4dyta5nwfc77vrkirmi766czdi" xml:space="preserve">{{Article info
| last1 = Stiegler
| orcid1 = 0009-0001-5789-6923
| first1 = Jean-Baptiste
| affiliation1 = Université Paris-Saclay
| correspondence1 = jean-baptiste.stiegler@universite-paris-saclay.fr
| journal = WikiJournal of Science
| et_al = true
| w1 = Pentagram map
| from w1 = true
| keywords = Pentagram map, Dynamical system, Projective geometry, Moduli space, Integrable systems
| license = CC-BY-SA 4.0
| submitted = 2025-12-08
| abstract = In [[w:mathematics|mathematics]], the '''pentagram map''' is a [[w:Dynamical system#Discrete dynamical system|discrete dynamical system]] acting on [[w:polygons|polygons]] in the [[w:projective plane|projective plane]]. It defines a new polygon whose vertices are obtained as the intersection points of the shortest [[w:Diagonal|diagonals]] of the initial polygon. This is a [[w:Projective linear group|projectively]] [[w:Equivariant map|equivariant]] procedure, hence it [[w:Quotient space (topology)|descends]] to the [[w:moduli space|moduli space]] of polygons and defines another dynamical system (which is also referred to as the pentagram map). It was first introduced by [[w:Richard Schwartz (mathematician)|Richard Schwartz]] in 1992.{{Sfn|Schwartz|1992}}

The pentagram map on the moduli space is famous for its [[w:Completely integrable|complete integrability]] and its link with [[w:cluster algebra|cluster algebras]].{{sfn|Gekhtman|Izosimov|2025|p=14}}

It admits many generalizations in [[w:Projective space|projective spaces]] and other settings.
}}
== Introduction ==

=== Informal definition ===

==== On polygons ====
[[File:Pentagram pentagon nolabel big.svg|alt=|thumb|300x300px|The pentagram map applied on a [[w:Convex set|convex]] [[w:pentagon|pentagon]].]]
Initially, the pentagram map was defined for [[w:convex polygon|convex polygon]]s (with at least five sides) on the [[w:euclidean plane|Euclidean plane]]. Given such a polygon &lt;math&gt;P&lt;/math&gt; with &lt;math&gt;n&lt;/math&gt; sides, one can draw the &quot;shortest [[w:diagonal|diagonal]]s&quot;, meaning the [[w:Line segment|segments]] whose endpoints are a [[w:Vertex (geometry)|vertex]] and one of its second neighbors (as in Figure 1). The intersections of the shortest diagonals are then taken as the vertices of a new &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;T(P)&lt;/math&gt;; this new polygon is the output of the pentagram map.{{Sfn|Berger|2005}}

The same construction can be done on [[w:Concave polygon|non-convex polygons]], but there are several complications. First, some consecutive short diagonals may not intersect, so one must extend the segments to [[w:Line (geometry)|lines]]. Second, the image &lt;math&gt;T(P)&lt;/math&gt; can fail to be a new &lt;math&gt;n&lt;/math&gt;-gon because some consecutive vertices could coincide. However, this [[w:Generic property|generically]] doesn't happen.{{Sfn|Ovsienko|Schwartz|Tabachnikov|p=411|2009}} Finally, it is possible that two diagonals are [[w:Parallel (geometry)|parallel]] and don't intersect on the [[w:euclidean plane|Euclidean plane]]. This is resolved by extending the Euclidean plane to the [[w:real projective plane|real projective plane]] by the addition of a [[w:line at infinity|line at infinity]], where the [[w:Vanishing point|intersection point]] lies (see Figure 3). Hence, the pentagram map is defined for generic polygons in the real projective plane.{{Sfn|Berger|2005|p=25}}

More generally, the construction of the pentagram map is well defined whenever the concepts of lines and their intersections make sense. This is encompassed by the notion of a general [[w:projective plane|projective plane]], of which the real projective plane is one example; but the pentagram map can also be considered over other [[w:Field (mathematics)|fields]], for instance the [[w:complex number|complex number]]s, which give the [[w:complex projective plane|complex projective plane]].{{Sfn|Weinreich|2022|loc=§3.1.1}}

==== On the moduli space of polygons ====
Since the pentagram map is constructed by drawing lines and marking their intersections, it [[w:Commutative property|commutes]] with any transformation that sends lines to lines. Such maps are called [[w:projective transformations|projective transformations]]. Hence, polygons can be identified [[w:up to|up to]] [[w:Perspectivity#Projectivity|projective transformations]]. This identification gives the [[w:Quotient space (topology)|quotient space]] (technically called a [[w:moduli space|moduli space]]) of [[w:Equivalence class|classes]] of polygons. 

The pentagram map on polygons induces another dynamical system on the moduli space,{{Sfn|Schwartz|1992|loc=§1 Projective geometry}} whose behavior differs quite a lot from the initial one.{{Efn|Compare the paragraph about the [[w:Pentagram map#Collapsing of convex polygons|collapsing of convex polygons]] and the one about [[w:Pentagram map#Complete integrability|complete integrability]].}} The dynamic is trivial for the classes of pentagons and hexagons, but this is no longer the case for polygons with more vertices.{{Efn|See the paragraph about [[w:Pentagram_map#Pentagons_and_hexagons|pentagons and hexagons]].}}

=== Historical elements ===
The pentagram map for general polygons was introduced in {{Harvard citation|Schwartz|1992}}, but the simplest case is the one of [[w:pentagons|pentagons]], hence the name &quot;[[w:pentagram|pentagram]]&quot;.{{Sfn|Marí-Beffa|2014|p=1}} Their study goes back to {{Harvard citation|Clebsch|1871}},{{Sfn|Izosimov|2022a|p=1085}} {{Harvard citation|Kasner|1928}}{{Sfn|Tabachnikov|2019}} and {{Harvard citation|Motzkin|1945}}.{{Sfn|Schwartz|2013|p=1}}

The pentagram map interacts with some classical configuration theorems of [[w:projective geometry|projective geometry]]. It provides results analogous to the ones of [[w:Pascal's theorem|Pascal's theorem]] and [[wikipedia:Brianchon's_theorem|Brianchon's theorem]].{{Sfn|Schwartz|Tabachnikov|2010}} Some specific configurations make [[w:Desargues' theorem|Desargues's theorem]] and [[w:Poncelet's porism|Poncelet's porism]] appear.{{Sfn|Berger|2005|loc=§4 and §5}}{{Efn|See the paragraph about [[w:Pentagram map#Poncelet polygons|Poncelet polygons]].}}

==Definitions and first properties==

=== Definition of the map ===
[[File:Pentagram pentagon label big.svg|alt=|thumb|300x300px|The pentagram map on a convex pentagon, with vertices labeled.]]
[[File:Pentagram on nonconvex pentagon.svg|alt=|thumb|300x300px|The pentagram map applied on a [[w:Self-intersecting polygon|self-intersecting]] (in particular, non-convex) pentagon. The vertex &lt;math&gt;w_2&lt;/math&gt; is on the [[w:line at infinity|line at infinity]], because it is the [[w:Vanishing point|intersection of two parallel lines]].]]
Let &lt;math&gt;n\geq 5&lt;/math&gt; be an integer. A polygon &lt;math&gt;P&lt;/math&gt; with &lt;math&gt;n&lt;/math&gt; sides, or &lt;math&gt;n&lt;/math&gt;-gon, is a tuple of [[w:Vertex (geometry)|vertices]] &lt;math&gt;(v_1,\dots,v_n)&lt;/math&gt; lying in some [[w:projective plane|projective plane]] &lt;math&gt;\mathbb P ^2&lt;/math&gt;,{{Efn|In the following, the figures represent polygons on the real plane, where the intuition is easier to grasp.}} where the indices are understood [[w:Modular arithmetic|modulo]] &lt;math&gt;n&lt;/math&gt;. The [[w:Dimension of an algebraic variety|dimension]] of the space of &lt;math&gt;n&lt;/math&gt;-gons is &lt;math&gt;2n&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=definition 1.1}}

Suppose that the vertices are in sufficiently [[w:general position|general position]], meaning that no consecutive triple of points are [[w:Collinearity|collinear]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=1}} Taking the intersection of two consecutive &quot;shortest&quot; [[w:diagonal|diagonal]]s{{Efn|Meaning the line between a vertex &lt;math&gt;v_k&lt;/math&gt; and a &quot;second neighbour&quot; &lt;math&gt;v_{k\pm 2}&lt;/math&gt;.}} defines a new point&lt;math display=&quot;block&quot;&gt; w_k := \overline{v_{k-1} v_{k+1}} \cap \overline{v_{k} v_{k+2}}. &lt;/math&gt;This procedure defines a new &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;T(P)=(w_1,\dots,w_n)&lt;/math&gt;, as in Figure 2.{{Sfn|Schwartz|1992|p=71}}

The labeling of the indices of &lt;math&gt;T(P)&lt;/math&gt; is not [[w:canonical|canonical]]. In most papers, a choice is made at the beginning of the paper and the formulas are tuned accordingly.{{Sfn|Izosimov|2016|loc=remark 1.5}}

The pentagram map on polygons is a [[w:birational map|birational map]] &lt;math&gt;T:(\mathbb P^2)^n&lt;/math&gt;{{nowrap|{{font|size=145%|⇢}}}}&lt;math&gt;(\mathbb P^2)^n&lt;/math&gt;. Indeed, each [[w:Homogeneous coordinates|coordinate]] of &lt;math&gt;w_k&lt;/math&gt; is given as a [[w:rational function|rational function]] of the coordinates of &lt;math&gt;v_{k-1},\dots,v_{k+2}&lt;/math&gt;, since it is defined as the intersection of lines passing by them. Moreover, the [[w:inverse map|inverse map]] is given by taking the intersections &lt;math&gt;\overline{w_{k-2} w_{k-1}} \cap \overline{w_{k} w_{k+1}} &lt;/math&gt;, which is rational for the same reason.{{Sfnp|Weinreich|2022|loc=definition 1.2}}

=== Moduli space ===
The pentagram map is defined by taking [[w:Line (geometry)|lines]] and intersections of them. The biggest [[w:Group (mathematics)|group]] which maps lines to lines is the one of [[w:projective transformations|projective transformations]], denoted by &lt;math&gt;\mathbb P \mathrm{GL}_{3}&lt;/math&gt;. Such a transformation &lt;math&gt;M&lt;/math&gt; [[w:Group action|acts]] on a polygon &lt;math&gt;P&lt;/math&gt; by sending it to &lt;math&gt;M \cdot P:=(Mv_1,\dots,Mv_n)&lt;/math&gt;. The pentagram map [[w:Commutative property|commutes]] with this action, and thereby induces another [[w:dynamical system|dynamical system]] on the [[w:moduli space|moduli space]] of projective [[w:equivalence classes|equivalence classes]] of polygons, whose [[w:Dimension of an algebraic variety|dimension]] is &lt;math&gt;2n-8&lt;/math&gt;.{{Sfn|Schwartz|1992|loc=§1 Projective geometry}}

===Twisted polygons===
[[File:Twisted heptagon.svg|alt=|thumb|300x300px|An example of twisted [[w:heptagon|heptagon]] on the real plane.]]
The pentagram map naturally generalizes to the larger space of twisted polygons (see example in Figure 4). For any integer &lt;math&gt;n\geq5&lt;/math&gt;, a twisted &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;P&lt;/math&gt; is the data of:

* a [[w:Sequence#Indexing|bi-infinite sequence]] of points &lt;math&gt;(v_k)_{k\in\mathbb Z}&lt;/math&gt; in the projective plane (called the vertices),
* a [[w:projective transformation|projective transformation]] &lt;math&gt;M \in \mathbb P \mathrm{GL}_3&lt;/math&gt; (called the [[w:monodromy|monodromy]]),

such that for any &lt;math&gt;k \in \mathbb Z&lt;/math&gt;, the property &lt;math&gt;v_{k+n}=Mv_k&lt;/math&gt; is satisfied. The dimension of the space of twisted &lt;math&gt;n&lt;/math&gt;-gons is &lt;math&gt;2n+8&lt;/math&gt;.{{Sfn|Schwartz|2008}}

When &lt;math&gt;M&lt;/math&gt; is the [[w:Identity_element|identity]], this gives back the initial definition of polygons (which are said to be closed). The space of closed &lt;math&gt;n&lt;/math&gt;-gons is of [[w:codimension|codimension]] &lt;math&gt;8&lt;/math&gt; in the space of twisted ones.{{Sfn|Soloviev|2013|p=2816}}

The action of projective transformations over the space of closed polygons generalizes to the space of twisted ones (the monodromy is changed by [[w:Matrix similarity|conjugation]]). This provides again a moduli space, of dimension &lt;math&gt;2n&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=definition 1.3}}

== Collapsing of convex polygons ==

=== Exponential shrinking ===
[[File:Pentagram map convex heptagon iterate.svg|alt=|thumb|300x300px|The pentagram map iterated on a convex [[w:heptagon|heptagon]], exhibiting the convergence.]]
Let &lt;math&gt;P&lt;/math&gt; be a closed [[w:Convex polygon#Strictly convex polygon|strictly convex polygon]] lying on the real plane. One of the first results proved by Richard Schwartz it that its iterates under the pentagram map shrink [[w:Exponential growth|exponentially fast]] to a point, as illustrated in Figure 5. This follows from two facts.
# The image of a strictly convex polygon is contained in its [[w:Interior (topology)|interior]], and is also strictly convex.{{Sfn|Glick|2020|p=2818}}
# There exists a constant &lt;math&gt;0&lt; \eta_P&lt;1&lt;/math&gt;, depending on &lt;math&gt;P&lt;/math&gt;, such that for any &lt;math&gt;N \in \mathbb N&lt;/math&gt;, the diameters of the iterates verify the inequality &lt;math display=&quot;inline&quot;&gt;\operatorname{diam}(T^N(P))\leq\eta_P^N \operatorname{diam}(P). &lt;/math&gt;{{Sfn|Schwartz|1992|loc=theorem 3.1}}

Hence, by [[w:Cantor's intersection theorem#Variant in complete metric spaces|Cantor's intersection theorem]], the sequence of polygons collapses toward a point.{{Sfn|Schwartz|1992|loc=§3 Convex polygons}}

The behavior on the moduli space is very different, since the dynamics is [[w:Recurrent point|recurrent]].{{Sfn|Schwartz|2001|loc=theorem 1.1}} It is even a [[w:quasiperiodic motion|quasiperiodic motion]],{{Sfn|Ovsienko|Schwartz|Tabachnikov|2009}} as discussed in [[w:Pentagram map#Complete integrability|the section about integrability]].

=== Coordinates of the limit point ===
The limit point coordinates were given in {{Harvard citation|Glick|2020}}. They satisfy some [[w:Degree of a polynomial|degree]] 3 [[w:polynomial equation|polynomial equations]], whose coefficients are [[w:rational function|rational function]]s in the coordinates of the vertices of the starting polygon. The proof relies on the fact that the limit point must be an [[w:eigenline|eigenline]] of a certain [[w:linear operator|linear operator]] of &lt;math&gt;\mathbb R^3&lt;/math&gt;.{{sfn|Glick|2020}}

This operator was reinterpreted in {{Harvard citation|Aboud|Izosimov|2022}} as the infinitesimal monodromy of the polygon. The [[w:Pentagram map#The scaling symmetry|scaling symmetry]] is used to [[w:Deformation (mathematics)|deform]] a closed polygon &lt;math&gt;P&lt;/math&gt; into a family of twisted ones '''&lt;math&gt;(P_z)_{z\in \mathbb C^*}&lt;/math&gt;''' with monodromy &lt;math&gt;M_z&lt;/math&gt;. The infinitesimal monodromy is defined to be:{{sfn|Aboud|Izosimov|2022}}
&lt;math display=&quot;block&quot;&gt;\left.\frac{dM_z}{dz}\right|_{z=1}.&lt;/math&gt;

=== Generalization ===
The collapsing of polygons may also happen in some [[w:Pentagram map#Generalizations|generalization of the pentagram map]], when considering some specific configurations of polygons in the real plane. The coordinates of the collapse point are given by a formula analogous to the one for the original pentagram map.{{Sfn|Schwartz|2026}}

== Periodic orbits on the moduli space ==
For some configurations of closed polygons, the iterate of the pentagram map will send &lt;math&gt;P&lt;/math&gt; to a projectively equivalent polygon (up to some shift of the indices). This means that, on the moduli space, the orbit of the class of &lt;math&gt;P&lt;/math&gt; is [[w:Periodic orbit|periodic]].

===Pentagons and hexagons===
[[File:penta hexagon.svg|300px|thumb|The outward hexagon is projectively equivalent to the inward one, with respect to their labeling.]]The following two facts are proved by checking [[w:cross-ratio|cross-ratio]] equalities, so they are true for polygons in any [[w:projective plane|projective plane]] (not just the [[w:Real projective plane|real one]]).{{Sfn|Schwartz|1992|loc=§2 Pentagons and hexagons}}

The pentagram map &lt;math&gt;T&lt;/math&gt; is the identity on the moduli space of [[w:pentagon|pentagon]]s.{{Sfn|Schwartz|1992|loc=theorem 2.1}}{{Sfn|Clebsch|1871}}{{Sfn|Motzkin|1945}} The second iterate &lt;math&gt;T^2&lt;/math&gt; is the identity on the space of labeled [[w:hexagon|hexagon]]s, up to a shift of labeling (see Figure 6).{{Sfn|Schwartz|1992|loc=theorem 2.3}} This phenomenon doesn't generalize to generic polygons with at least seven sides, for which the motion is [[w:Quasiperiodic motion|quasiperiodic]].{{Sfn|Tupan|2022}}

==== Generalization ====
The result about pentagons and hexagons generalizes to some [[w:Pentagram map#Generalizations|higher pentagram maps]] in &lt;math&gt;\mathbb P ^k&lt;/math&gt;, for polygons with &lt;math&gt;k+3&lt;/math&gt; or &lt;math&gt;2k+2&lt;/math&gt; sides. The proof uses a generalization of the [[w:Gale transform|Gale transform]].{{Sfn|Dirdak|2024}}

=== Poncelet polygons ===
A polygon is said to be Poncelet{{Efn|The name comes from [[w:Jean-Victor Poncelet|Jean-Victor Poncelet]] and [[w:Poncelet porism|his porism]].{{Sfn|Izosimov|2022a|p=1085}}}} if it is [[w:Inscribed figure|inscribed]] in a [[w:Conic section|conic]] and circumscribed about another one.{{Sfn|Schwartz|2015|loc=|p=433}}{{Efn|In particular, pentagons are Poncelet since [[w:five points determine a conic|five points determine a conic]].{{Sfn|Schwartz|2015|loc=|p=433}}}} For a convex Poncelet &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;P&lt;/math&gt; lying on the [[w:real projective plane|real projective plane]], the polygon &lt;math&gt;T^2(P)&lt;/math&gt; is projectively equivalent to &lt;math&gt;P&lt;/math&gt;.{{Sfn|Schwartz|2015|loc=theorem 1.1}} In fact, when &lt;math&gt;n&lt;/math&gt; is odd, the converse is also true.{{Sfn|Izosimov|2022a|loc=corollary 1.1}}

However, this converse statement is no longer true when the polygons are considered over the [[w:complex projective plane|complex projective plane]].{{Sfn|Izosimov|2022a|loc=remark 1.3}}

==Coordinates for the moduli space==
The moduli space can be described by different [[w:Coordinate_system|coordinate systems]]. The following ones are practical to explicit the dynamic, as presented in the next section.

=== Corner coordinates ===
[[File:Corner coordinates big.svg|thumb|300x300px|The geometric construction of the points defining the corner invariants.]]
Define the [[w:cross-ratio|cross-ratio]] of four [[w:Collinearity|collinear]] points to be

: &lt;math&gt; [a,b,c,d]=\frac{(a-b)(c-d)}{(a-c)(b-d)}. &lt;/math&gt;

The corner invariants are a system of coordinates on the space of twisted polygons, constructed by taking intersections as in Figure 7.{{Sfn|Schwartz|2001|loc=figure 2}} The left and right invariants are respectively defined{{Efn|The ordering of the vertices in the cross-ratios can differ from a paper to another one, which slightly changes the formulas in the following sections.}} as the following cross-ratios:

: &lt;math&gt;x_k:=[v_{k-2},v_{k-1},\overline{v_{k-2}v_{k-1}}\cap\overline{v_{k}v_{k+1}},\overline{v_{k-2}v_{k-1}}\cap\overline{v_{k+1}v_{k+2}}],&lt;/math&gt;

: &lt;math&gt;y_k:=[\overline{v_{k+1}v_{k+2}}\cap\overline{v_{k-2}v_{k-1}}, \overline{v_{k+1}v_{k+2}}\cap\overline{v_{k-1}v_{k}},v_{k+1},v_{k+2}].&lt;/math&gt;

Since the cross-ratio is [[w:Cross-ratio#Projective geometry|projective invariant]], the sequences &lt;math&gt;(x_k)_{k \in \mathbb Z}&lt;/math&gt; and &lt;math&gt;(y_k)_{k \in \mathbb Z}&lt;/math&gt; associated to a twisted &lt;math&gt;n&lt;/math&gt;-gon are &lt;math&gt;n&lt;/math&gt;-periodic.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=415}}

The corner invariants are elements of &lt;math&gt;\mathbb{P}^1\smallsetminus\{0,1,\infty\}&lt;/math&gt;, and they realize an [[w:Isomorphism_of_varieties|isomorphism of varieties]] between the moduli space of twisted &lt;math&gt;n&lt;/math&gt;-gons and &lt;math&gt;(\mathbb{P}^1\smallsetminus\{0,1,\infty\})^{2n}&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=theorem 3.6}}

===ab-coordinates===

There is a second set of coordinates for the moduli space of twisted &lt;math&gt;n&lt;/math&gt;-gons defined over a [[w:Field (mathematics)|field]] &lt;math&gt;F&lt;/math&gt; satisfying &lt;math&gt;\mathrm{SL}_3(F)\cong \mathbb P\mathrm{GL}_3(F)&lt;/math&gt;,{{Sfn|Weinreich|2022|loc=remark 3.8}} and such that &lt;math&gt;n&lt;/math&gt; is not divisible by &lt;math&gt;3&lt;/math&gt;.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=section 4.1}}

The vertices &lt;math&gt;v_k&lt;/math&gt; in the [[w:projective plane|projective plane]] &lt;math&gt;\mathbb P^2(F)&lt;/math&gt; can be [[w:Lift (mathematics)|lifted]] to [[w:Vector space|vectors]] &lt;math&gt;V_k&lt;/math&gt; in the [[w:affine space|affine space]] &lt;math&gt;F^3&lt;/math&gt; so that each consecutive triple of vectors spans a [[w:parallelepiped|parallelepiped]] having [[w:determinant|determinant]] equal to &lt;math&gt;1&lt;/math&gt;. This leads to the relation{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=equation 4.1}}

: &lt;math&gt;V_{k+3} = a_k V_{k+2} + b_k V_{k+1} + V_k.&lt;/math&gt;

This bring out an analogy between twisted polygons and solutions of third order linear [[w:ordinary differential equations|ordinary differential equations]], normalized to have unit [[w:Wronskian|Wronskian]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=remark 6.6}}

They are linked to the corner coordinates by:{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=lemma 4.5}}
: &lt;math&gt;x_k=\frac{a_{k-2}}{b_{k-2}b_{k-1}},&lt;/math&gt;
: &lt;math&gt;y_k=-\frac{b_{k-1}}{a_{k-2}a_{k-1}}.&lt;/math&gt;

==Formulas on the moduli space==

===As a birational map ===
The pentagram map is a [[w:birational map|birational map]] on the moduli space, because it can be decomposed as the [[w:Function composition|composition]] of two [[w:Birational geometry|birational]] [[w:Involution (mathematics)|involutions]].{{Sfn|Schwartz|2008|loc=§1.2 The Pentagram Map}} The corner invariants change in the following way:{{Sfn|Ovsienko|Schwartz|loc=lemma 2.4|Tabachnikov|2010}}

: &lt;math&gt;x_k'=x_k\frac{1-x_{k-1} y_{k-1}}{1-x_{k+1}y_{k+1}},&lt;/math&gt;

: &lt;math&gt;y_k'=y_{k+1}\frac{1-x_{k+2} y_{k+2}}{1-x_k y_k}.&lt;/math&gt;

=== The scaling symmetry ===
The [[w:multiplicative group|multiplicative group]] &lt;math&gt;F\smallsetminus\{0\}&lt;/math&gt; [[w:One-parameter group|acts]] on the moduli space in the following way:

: &lt;math&gt;R_s\cdot(x_1,\dots,x_n,y_1,\dots,y_n)=(sx_1,\dots,sx_n,s^{-1}y_1,\dots,s^{-1}y_n),&lt;/math&gt;

where &lt;math&gt;R&lt;/math&gt; is called the scaling action and &lt;math&gt;s&lt;/math&gt; is the scaling parameter. This action commutes with the pentagram map on the moduli space (as presented in the previous formulas). This property is called the scaling symmetry, and is instrumental in proving the [[w:Pentagram map#Complete integrability|complete integrability]] of the dynamics.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=corollary 2.5}}

==Invariant structures==

===Monodromy invariants===

The monodromy invariants, introduced in {{Harvard citation|Schwartz|2008}}, are a collection of [[w:Function (mathematics)|functions]] on the [[w:moduli space|moduli space]] that are invariant under the pentagram map.{{Sfn|Schwartz|2008|loc=theorem 1.2}} The simplest example of them are 
:&lt;math&gt; O_n= x_1x_2\cdots x_{n}, \quad E_n = y_1y_2\cdots y_n. &lt;/math&gt;
The other monodromy invariants can be retrieved through different points of view: through the [[w:Pentagram map#The scaling symmetry|scaling symmetry]], as [[w:Combinatorics|combinatorial]] objects, or as some [[w:determinant|determinant]]s.{{Sfn|Schwartz|Tabachnikov|2011|loc=§2 The Monodromy Invariants}} The one involving scaling symmetry is presented here.

Let &lt;math&gt;M\in \mathrm{GL}_3&lt;/math&gt; be a [[w:Lift (mathematics)|lift]] of the monodromy of a twisted &lt;math&gt;n&lt;/math&gt;-gon. The quantities

: &lt;math&gt;\Omega_1=\frac{\operatorname{trace}^3(M)}{\det(M)}, \quad \Omega_2=\frac{\operatorname{trace}^3(M^{-1})}{\det(M^{-1})},&lt;/math&gt;

are independent of the choice of lift and are invariant under [[w:Matrix similarity|conjugation]], so they are well defined for the projective class of the polygon. They are invariant under the pentagram map, since the monodromy matrix doesn't change.{{Sfn|Schwartz|Tabachnikov|2011|loc=|p=5}} Now, the quantities

: &lt;math&gt;\tilde{\Omega}_1=O_n^2E_n\Omega_1, \quad \tilde{\Omega}_2=O_nE_n^2\Omega_2,&lt;/math&gt;

have the same properties, but turn out to be polynomials in the corner invariants.{{Efn|Some papers consider the cube roots of this functions, but it doesn't change the following definitions of the monodromy invariants.}} They can be written as{{Sfn|Schwartz|Tabachnikov|2011|loc=|p=5}}

: &lt;math&gt;
\tilde{\Omega}_1=\biggl(\sum_{k=0}^{\lfloor n/2\rfloor}O_k\biggr)^3, \quad
\tilde{\Omega}_2=\biggl(\sum_{k=0}^{\lfloor n/2\rfloor}E_k\biggr)^3,
&lt;/math&gt;

where each &lt;math&gt;O_k&lt;/math&gt; and &lt;math&gt;E_k&lt;/math&gt; are [[w:homogeneous polynomial|homogeneous polynomial]]s respectively of weight &lt;math&gt;k&lt;/math&gt; and &lt;math&gt;-k&lt;/math&gt;,{{Sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=11}} meaning they change under the [[w:Pentagram map#The scaling symmetry|rescaling action]] on variables by{{Sfn|Schwartz|Tabachnikov|2011|p=5}}

: &lt;math&gt; R_s(O_k)= s^k O_k, \quad R_s(E_k)= s^{-k} E_k. &lt;/math&gt;

The quantities &lt;math&gt;O_1,\dots,O_{\lfloor n/2 \rfloor},O_n, E_1,\dots,E_{\lfloor n/2 \rfloor},E_n,&lt;/math&gt; are unchanged by the dynamics, and are called the monodromy invariants. Moreover, they are [[w:algebraically independent|algebraically independent]].{{Sfn|Schwartz|2008|loc=theorem 1.2}}

==== Polygons on conics ====
Whenever &lt;math&gt;P&lt;/math&gt; is [[w:Inscribed figure|inscribed]] on a [[w:conic section|conic section]], one has &lt;math&gt;O_k(P)=E_k(P)&lt;/math&gt; for all &lt;math&gt;k&lt;/math&gt;.{{Sfn|Schwartz|Tabachnikov|2011|loc=theorem 1.1}} Moreover, if &lt;math&gt;P&lt;/math&gt; is circumscribed about another conic,{{Efn|See the paragraph about [[w:Pentagram map#Poncelet polygons|Poncelet polygons]].}} then its monodromy invariants are characterized by the pair of conics.{{Sfn|Schwartz|2015|loc=theorem 1.2}} For such odd-gons, the translation on the [[w:Jacobian variety|Jacobian variety]]{{Efn|See the paragraph about [[w:Pentagram map#Algebro-geometric integrability|algebraic integrability]].}} is restricted to the [[w:Prym variety|Prym variety]] (which is a half-dimensional torus in the Jacobian).{{Sfn|Izosimov|2016|loc=theorem 1.3}}

===Poisson bracket===

An invariant [[w:Poisson bracket|Poisson bracket]] on the space of twisted polygons was found in {{Harvard citation|Ovsienko|Schwartz|Tabachnikov|2010}}. The monodromy invariants [[w:Poisson bracket#Constants of motion|commute]] with respect to it:
&lt;math display=&quot;block&quot;&gt; \{O_i,O_j\}=\{O_i,E_j\}=\{E_i,E_j\}=0 &lt;/math&gt;for all &lt;math&gt;i,j&lt;/math&gt;.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}}

The Poisson bracket is defined in terms of the corner coordinates by:
&lt;math display=&quot;block&quot;&gt; \begin{align}
 \{x_i,x_{i\pm1}\} &amp;= \mp x_i x_{i+1}, \\
 \{y_i,y_{i\pm 1}\} &amp;= \mp y_i y_{i+1}, \\
 \{x_i,x_j\} &amp;= \{y_i,y_j\} = \{x_i,y_j\} = 0
\end{align}&lt;/math&gt;for all other &lt;math&gt; i,j.&lt;/math&gt;{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=equation 2.16}}

=== The spectral curve ===

Let &lt;math&gt;\zeta&lt;/math&gt; be an element of the [[w:multiplicative group|multiplicative group]] and &lt;math&gt;P_\zeta&lt;/math&gt; be the polygon obtained by applying the [[w:Pentagram map#The scaling symmetry|rescaling action]] &lt;math&gt;R_\zeta&lt;/math&gt; on &lt;math&gt;P&lt;/math&gt;.  A [[w:Lax matrix|Lax matrix]] &lt;math&gt;\hat{T}(\zeta) \in \mathrm{GL}_3&lt;/math&gt; is a lift of the monodromy of &lt;math&gt;P_\zeta&lt;/math&gt; satisfying a [[w:Lax pair#Zero-curvature equation|zero-curvature equation]].{{Sfn|Weinreich|2022|loc=§5 The Lax representation}} Then, the spectral function is the [[w:Bivariate polynomial|bivariate]] [[w:characteristic polynomial|characteristic polynomial]]
&lt;math display=&quot;block&quot;&gt; Q(\lambda,\zeta) := \det(\lambda\operatorname{Id}-\hat{T}(\zeta)),&lt;/math&gt;or some renormalization of it. The [[w:spectral curve|spectral curve]] is the [[w:Projective variety#projective completion|projective completion]] of the [[w:Algebraic curve|affine curve]] defined by the equation &lt;math&gt;Q(\lambda,\zeta)=0&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=§6. The geometry of the spectral curve}} It is invariant under the pentagram map, and the monodromy invariants appear as the [[w:coefficient|coefficient]]s of &lt;math&gt;Q&lt;/math&gt;.{{Sfn|Soloviev|2013|loc=theorem 6.4}} Its [[w:geometric genus|geometric genus]] is &lt;math&gt;n-1&lt;/math&gt; if &lt;math&gt;n&lt;/math&gt; is odd, and &lt;math&gt;n-2&lt;/math&gt; if &lt;math&gt;n&lt;/math&gt; is even.{{Sfn|Weinreich|2022|p=|loc=theorem 6.4}}

It was first introduced in {{Harv|Soloviev|2013|ps=|p=}} for his proof of [[w:Pentagram map#Algebro-geometric integrability|algebro-geometric integrability]].{{sfn|Soloviev|2013}}

==Complete integrability==
The pentagram map on the moduli space has been proved to be a [[w:completely integrable|completely integrable]] [[w:discrete dynamical system|discrete dynamical system]], both in the [[w:Integrable system#Hamiltonian systems and Liouville integrability|Arnold-Liouville]]{{Efn|Over the [[w:real number|real number]]s.}} and the [[w:Integrable system#Complete integrability over the complex numbers|algebro-geometric]]{{Efn|Over [[w:algebraically closed field|algebraically closed field]]s of [[w:Characteristic (algebra)|characteristic]] different from 2.}} senses. In any case, this means that the moduli space is [[w:almost everywhere|almost everywhere]] [[w:Foliation|foliated]] by [[w:Torus#Flat torus|flat tori]] (or in the algebraic setting, [[w:Abelian variety|Abelian varieties]]), where the motion is a [[w:Translation (geometry)|translation]]. This [[w:Generic property|generically]] induces a [[w:quasiperiodic motion|quasiperiodic motion]] on the corresponding torus.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2009}}

===Arnold–Liouville integrability===

The proof of the integrability of the pentagram map on a real twisted polygon was achieved in {{Harvard citation|Ovsienko|Schwartz|Tabachnikov|2010}}. This is done by noticing that the monodromy invariants &lt;math&gt;O_n&lt;/math&gt; and &lt;math&gt;E_n&lt;/math&gt; are [[w:Casimir invariant|Casimir invariant]]s for the bracket, meaning (in this context) that&lt;math display=&quot;block&quot;&gt; \{O_n,f\}=\{E_n,f\} = 0 &lt;/math&gt;for all functions &lt;math&gt;f&lt;/math&gt;.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}} When &lt;math&gt;n&lt;/math&gt; is even, this is also true for the monodromy invariants &lt;math&gt;O_{\lfloor n/2 \rfloor }&lt;/math&gt; and &lt;math&gt;E_{\lfloor n/2 \rfloor }&lt;/math&gt;.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}}

This allows to consider the Casimir [[w:level set|level set]], where each Casimir has a specified value. Because of [[w:Sard's theorem|Sard's theorem]], any generic level set is a [[w:smooth manifold|smooth manifold]].{{Sfn|Schwartz|2017|p=44}} They form a [[w:foliation|foliation]] in [[w:Poisson manifold#Symplectic leaves|symplectic leaves]], on which the Poisson bracket gives rise to a [[w:symplectic form|symplectic form]].{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=corollary 2.13}}

Each of these symplectic leaves has an iso-monodromy [[w:foliation|foliation]], namely, a decomposition into the common level sets of the remaining monodromy functions. By using again [[w:Sard's theorem|Sard's theorem]], they are generically [[w:Symplectic manifold#Lagrangian submanifolds|Lagrangian manifolds]].{{Sfn|Schwartz|2017|p=45}} Moreover, they are compact.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=§3.3. Compactness of the level sets}} Since the monodromy invariants Poisson-commute and there are enough of them, the discrete [[w:Liouville–Arnold theorem|Liouville–Arnold theorem]] can be applied to prove that the level sets are [[w:Torus#Flat torus|flat tori]] over which the dynamics is a translation.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=412}}
===Algebro-geometric integrability===

In {{Harvard citation|Soloviev|2013}}, it was shown that the pentagram map admits a [[w:Lax representation|Lax representation]] with a spectral parameter, which allows to prove its algebro-geometric integrability. This means that the space of polygons (either twisted or closed) is parametrized by its spectral data, consisting of [[w:Pentagram map#The spectral curve|its spectral curve]], with marked points and a [[w:Divisor (algebraic geometry)|divisor]] given by a [[w:Floquet theory|Floquet]]–[[w:Bloch's theorem|Bloch]] equation. This gives an embedding to the [[w:Jacobian variety|Jacobian variety]] through the [[w:Abel–Jacobi map|Abel–Jacobi map]], where the motion is expressed in terms of translation.{{sfn|Soloviev|2013|loc=theorems A, B and C}} The previously defined Poisson bracket is also retrieved.{{sfn|Soloviev|2013|loc=theorem D}} 

This integrability was generalized in {{Harvard citation|Weinreich|2022}} from the field of [[w:complex number|complex number]]s to any [[w:algebraically closed field|algebraically closed field]] of [[w:Characteristic (algebra)|characteristic]] different from 2. The translation on a torus is replaced by a translation on an [[w:Abelian variety|Abelian variety]] (in fact, a Jacobian variety again).{{sfn|Weinreich|2022|loc=theorem 1.4}}

=== Dimension of the invariant manifold ===

For twisted &lt;math&gt;n&lt;/math&gt;-gons, the [[w:dimension|dimension]] of the invariant tori (or Jacobian varieties) is{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=421}}

: &lt;math&gt;\begin{cases} 
n-1 &amp; \text{when }n \text{ is odd,}\\
n-2 &amp; \text{when }n \text{ is even.}
\end{cases}&lt;/math&gt;

Moreover, when &lt;math&gt;n&lt;/math&gt; is even, there are two isomorphic Jacobians on which the iterates of the pentagram map alternate. But on each of them, the second iterate is a translation.{{Sfn|Weinreich|2022|loc=theorem 1.4}}

=== For closed polygons ===
There is no Poisson structure on the space of closed polygons.{{Sfn|Soloviev|2013|loc=corollary 4.1}} Nevertheless, the one from twisted polygons can be used to prove integrability.{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=2153}}

Algebro-geometric integrability holds for closed polygons in a same manner as for the twisted ones.{{Sfn|Soloviev|2013|loc=theorem C}} However, Arnold-Liouville integrability is proved for real closed polygons only when they are convex. This is done by restricting the [[w:Hamiltonian vector field|Hamiltonian vector field]]s of monodromy functions to smaller dimensional tori, and showing that enough of them are still independent.{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|loc=corollary 1.1}}

In both situation, the dimension of the invariant manifolds decreases by &lt;math&gt;3&lt;/math&gt; for closed &lt;math&gt;n&lt;/math&gt;-gons (compared to the twisted case), and is equal to{{Sfn|Soloviev|2013|loc=theorem C}}{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|loc=theorem 1}}
:&lt;math&gt;\begin{cases} 
n-4 &amp; \text{when }n \text{ is odd,}\\
n-5 &amp; \text{when }n \text{ is even.}
\end{cases}&lt;/math&gt;

==Connections to other topics==

===The Boussinesq equation===

The continuous limit of a convex polygon is a parametrized convex curve in the plane. When the time parameter is suitably chosen, the [[w:Discretization|continuous limit]] of the pentagram map is the classical [[w:Boussinesq approximation (water waves)|Boussinesq equation]]. This equation is a classical example of an [[w:integrable|integrable]] [[w:partial differential equation|partial differential equation]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 5}}

Here is a description of the geometric action of the Boussinesq equation. Given a [[w:locally convex|locally convex]] curve &lt;math&gt; C:\mathbb R\to \mathbb R^2 &lt;/math&gt; and real numbers &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;t&lt;/math&gt;, consider the [[w:chord (geometry)|chord]] connecting &lt;math&gt; C(x-t) &lt;/math&gt; to &lt;math&gt; C(x+t) &lt;/math&gt;. The [[w:Envelope (mathematics)|envelope]] of all these chords is a new curve &lt;math&gt; C_t(x) &lt;/math&gt;. When &lt;math&gt;t&lt;/math&gt; is extremely small, the curve &lt;math&gt; C_t(x) &lt;/math&gt; is a good model for the time &lt;math&gt;t&lt;/math&gt; evolution of the original curve &lt;math&gt; C_0(x) &lt;/math&gt; under the Boussinesq equation. This construction is also similar to the pentagram map. Moreover, the pentagram invariant bracket is a discretization of a well known invariant Poisson bracket associated to the Boussinesq equation.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=§6.4 Discretization}}

===Cluster algebras===

The pentagram map{{Sfn|Glick|2011}} and some of its generalizations{{Sfn|Gekhtman|Shapiro|Tabachnikov|Vainshtein|2012}}{{Sfn|Glick|Pylyavskyy|2016}} are identified as special cases of discrete dynamical systems powered by [[w:cluster algebra|cluster algebra]]. Using the results from {{Harvard citation|Goncharov|Kenyon|2013}}, this provides a link with the [[w:Poisson–Lie group|Poisson–Lie group]]s, [[w:dimer model|dimer model]]s and other so-called cluster-integrable systems.{{Sfn|Fock|Marshakov|2016}}{{Sfn|Izosimov|2022c}} These methods allow to retrieve the Poisson-bracket and Hamiltonians used to prove complete integrability{{Sfn|Affolter|George|Ramassamy|2025|loc=§5 The pentagram map}} and provide [[w:Lax representation|Lax representation]]s.{{Sfn|Izosimov|2022b}}

=== Octahedron recurrence ===
Using a method to compute [[w:Determinant|determinants]] called [[w:Dodgson_condensation|Dodgson condensation]], {{Harvard citation|Schwartz|2008}} proves that the pentagram map satisfies a property called the &quot;octahedron recurrence&quot;.{{Sfn|Schwartz|2008|loc=§5 The Method of Condensation}} This property turns out to be shared by other dynamical systems defined by geometric constructions similar to one of the pentagram map.{{Sfn|Affolter|de Tilière|Melotti|2025|loc=§9 The Pentagram Map}} It is also shared by higher dimensional pentagram maps defined through cluster algebras mutations, referred as &lt;math&gt;T&lt;/math&gt;-systems.{{Sfn|Kedem|Vichitkunakorn|2015||loc=}}

=== Singularity theory ===
The pentagram map exhibit a property called singularity confinement, which is typical from [[w:integrable system|integrable system]]s.{{Sfn|Grammaticos|Ramani|Papageorgiou|1991}} It states that if a polygon &lt;math&gt;P&lt;/math&gt; is [[w:Singular point of an algebraic variety|singular]] for the pentagram map &lt;math&gt;T&lt;/math&gt;, then there exists an integer &lt;math&gt;m&lt;/math&gt; such that &lt;math&gt;P&lt;/math&gt; not singular for the iterate map &lt;math&gt;T^m&lt;/math&gt;.{{Sfn|Glick|2012}}

Moreover, the pentagram map (along with some of its generalizations and other discrete dynamical systems) exhibit the Devron property.{{Efn|The name comes from an episode of [[w:Star Trek|Star Trek]].{{Sfn|Glick|2015|loc=§1 Introduction}}}} This means that if a polygon &lt;math&gt;P&lt;/math&gt; is singular for some iterate of the pentagram map &lt;math&gt;T^m&lt;/math&gt;, then it will also be singular for some iterate of the inverse map &lt;math&gt;T^{-m'}&lt;/math&gt;.{{Sfn|Glick|2015}}

== Generalizations ==
The definition of twisted polygons still makes sense in any [[w:projective space|projective space]] &lt;math&gt;\mathbb P^d&lt;/math&gt;, under the action of the [[w:Projective linear group|projective group]] &lt;math&gt;\mathbb P \mathrm{GL}_{d+1}&lt;/math&gt;. The pentagram map can be generalized in many ways, and some of them are presented here. Not all of them are integrable.{{Sfn|Khesin|Soloviev|2015|}} Some are [[w:discretization|discretization]]s of [[w:PDEs|PDEs]] from the [[w:KdV hierarchy|KdV hierarchy]], seen as higher dimensional version of [[w:Boussinesq approximation (water waves)|Boussinesq]] or [[w:Kadomtsev–Petviashvili equation|KP]] equations.{{Sfn|Marí-Beffa|2012}}{{Sfn|Wang|2023}} The description of all generalized pentagram maps in terms of [[w:cluster algebra|cluster algebra]]s is still an open question.{{Sfn|Gekhtman|Izosimov|2025|p=14}}

=== Polygons in general positions ===
Let &lt;math&gt;d \geq 2&lt;/math&gt; and &lt;math&gt;P&lt;/math&gt; be a twisted polygon of &lt;math&gt;\mathbb P^d&lt;/math&gt; in [[w:general position|general position]].

==== Short diagonal pentagram maps ====
The &lt;math&gt;k&lt;/math&gt;-th ''short diagonal hyperplane'' &lt;math&gt;H_k^{sh}&lt;/math&gt; is uniquely defined by passing through the vertices &lt;math&gt;v_k,v_{k+2},\dots,v_{k+2d-2}&lt;/math&gt;. [[w:Generic property#In algebraic geometry|Generically]], the intersection of &lt;math&gt;d&lt;/math&gt; consecutive hyperplanes uniquely defines a new point

: &lt;math&gt;T_{sh}v_k:=H_k^{sh}\cap H_{k+1}^{sh}\cap \dots \cap H_{k+d-1}^{sh}.&lt;/math&gt;

Doing this for every vertex defines a new twisted polygon. This map, denoted by &lt;math&gt;T_{sh}&lt;/math&gt;, is again projectively equivariant.{{Sfn|Khesin|Soloviev|2013}}

==== Generalized pentagram maps ====
The previous procedure can be generalized. Let &lt;math&gt;I=(i_1,\dots,i_{d-1}),~J=(j_1,\dots,j_{d-1})&lt;/math&gt; be two sets of integers, respectively called the jump tuple and the intersection tuple. Define the &lt;math&gt;k&lt;/math&gt;-th hyperplane &lt;math&gt;H_k^I&lt;/math&gt; to be passing through the vertices &lt;math&gt;v_k,v_{k+i_1},\dots,v_{k+i_1+\dots+i_{d-1}}&lt;/math&gt;. A new point is given by the intersection

: &lt;math&gt;T_{I,J}v_k:=H_k^I \cap H_{k+j_1}^I \cap \dots \cap H_{k+j_1+\dots +j_{d-1}}^I.&lt;/math&gt;

The map &lt;math&gt;T_{I,J}&lt;/math&gt; is called a generalized pentagram map.{{Sfn|Khesin|Soloviev|2015a}} The original pentagram map is recovered by considering&lt;math&gt;d=2,~I=(2),~J=(1)&lt;/math&gt;.

Integrability can be numerically tested by picking a random polygon &lt;math&gt;P&lt;/math&gt; with [[wikipedia:Rational_point|rational coordinates]] and studying the growth rate of the [[wikipedia:Height_function|height]] of its iterates. This is called the [[wikipedia:Integrable_system#Diophantine_integrability|diophantine integrability]] test, and some generalized pentagram maps don't seem to pass it.{{Sfn|Khesin|Soloviev|2015a|loc=§5 and §6}} However, it is conjectured that the maps &lt;math&gt;T_{I,I}&lt;/math&gt; are integrable for any &lt;math&gt;I&lt;/math&gt;.{{Sfn|Bolsinov|Matveev|Miranda|Tabachnikov|2018|loc=conjecture 4.13 (B. Khesin, F. Soloviev)}}

Some of these maps are [[w:discretization|discretization]]s of higher dimensional counterpart of the [[w:Boussinesq approximation (water waves)|Boussinesq equation]] in the [[w:KdV hierarchy|KdV hierarchy]].{{Sfn|Khesin|Soloviev|2015b|loc=theorem 4.1}}{{Sfn|Izosimov|2022b|loc=theorem 4.1}}

==== Dented pentagram maps ====
Fix an integer &lt;math&gt;m\in \{1,\dots ,d-1\}&lt;/math&gt;. Consider the jump tuple &lt;math&gt;I_m:=(1,\dots,1,2,1,\dots,1)&lt;/math&gt;, where the &lt;math&gt;2&lt;/math&gt; is at the &lt;math&gt;m&lt;/math&gt;-th place, and the intersection tuple &lt;math&gt;J:=(1,\dots,1)&lt;/math&gt;. The dented pentagram map is &lt;math&gt;T_m :=T_{I_m,J}&lt;/math&gt;. They are proved to be integrable.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 2.14}}

For an integer &lt;math&gt;p \geq 2&lt;/math&gt;, the deep dented pentagram map (of depth &lt;math&gt;p&lt;/math&gt;) &lt;math&gt;T_m^p&lt;/math&gt; is the same map as before, but the number &lt;math&gt;2&lt;/math&gt; in the definition of &lt;math&gt;I_m&lt;/math&gt; is replaced by &lt;math&gt;p&lt;/math&gt;. This kind of pentagram maps are again integrable.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 6.2}}

=== Corrugated polygons ===
A twisted polygon &lt;math&gt;P&lt;/math&gt; lying in &lt;math&gt;\mathbb P^d&lt;/math&gt; is said to be corrugated if for any &lt;math&gt;k\in \mathbb Z&lt;/math&gt;, the vertices &lt;math&gt;v_k,v_{k+1},v_{k+d},v_{k+d+1}&lt;/math&gt; span a projective two-dimensional plane. Such polygons are not in [[w:general position|general position]]. A new point is defined by

: &lt;math&gt;T_\text{cor}v_k:=\overline{v_k v_{k+d}}\cap \overline{v_{k+1} v_{k+d+1}}.&lt;/math&gt;

The map &lt;math&gt;T_\text{cor}&lt;/math&gt; yields a new corrugated polygon. They are [[w:Integrable system#Hamiltonian systems and Liouville integrability|completely Liouville-integrable]].{{Sfn|Gekhtman|Shapiro|Tabachnikov|Vainshtein|2012|loc=theorem 4.4}}

In fact, they can be retrieved as some dented pentagram map applied on corrugated polygons.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 5.3}}

=== Grassmannian polygons ===
Let &lt;math&gt;d \geq 3, m \geq 1&lt;/math&gt; be integers. The pentagram map can also be generalized to the [[w:Grassmannian|Grassmannian]] space &lt;math&gt;\mathrm{Gr}(m,md)&lt;/math&gt;, which consists of &lt;math&gt;m&lt;/math&gt;-[[w:Dimension (vector space)|dimensional]] [[w:linear subspace|linear subspace]]s of an &lt;math&gt;md&lt;/math&gt;-dimensional [[w:vector space|vector space]]. When &lt;math&gt;m=1&lt;/math&gt;, the linear subspaces are [[w:Vector space#vector line|lines]], which retrieves the definition of [[w:projective space|projective space]]s &lt;math&gt;\mathbb P^d&lt;/math&gt;.{{Sfn|Felipe|Marí-Beffa|2019|loc=§2 definitions and notations}}

A point &lt;math&gt;v\in\operatorname{Gr}(m,md)&lt;/math&gt; is represented by an &lt;math&gt;md \times m&lt;/math&gt; matrix &lt;math&gt;X_v&lt;/math&gt; such that its columns form a [[w:Basis (linear algebra)|basis]] of &lt;math&gt;v&lt;/math&gt;. Consider the [[w:Group action|action]] of the [[w:general linear group|general linear group]] &lt;math&gt;\mathrm{GL}_{md}&lt;/math&gt; by multiplication on the left of &lt;math&gt;X_v&lt;/math&gt;. This defines an action on the Grassmannian, even though it is not [[w:Faithful action|faithful]].{{Efn|Because there can be many lifts for &lt;math&gt;v&lt;/math&gt;, and because some matrices act trivially.}} Hence, the polygons of &lt;math&gt;\mathrm{Gr}(m,md)&lt;/math&gt; and their moduli spaces are defined as before, after the change of underlying group.{{Sfn|Felipe|Marí-Beffa|2019|loc=§2 definitions and notations}}

Depending on the parity of &lt;math&gt;d&lt;/math&gt;, one can define linear subspaces spanned by some &lt;math&gt;X_{v_k}&lt;/math&gt;'s such that taking their intersection generically defines a new point &lt;math&gt;v\in\mathrm{Gr}(m,md)&lt;/math&gt;.{{Sfn|Felipe|Marí-Beffa|2019|loc=sections 4 and 5}} This generalization of the pentagram map is integrable in a [[w:noncommutative|noncommutative]] sense.{{Sfn|Ovenhouse|2020}}

=== Over rings ===
The pentagram map admits a generalization by considering [[w:Projective space#Generalizations|projective planes]] over [[w:stably finite ring|stably finite ring]]s, instead of [[w:Field (mathematics)|field]]s. In particular, this retrieves the pentagram map over Grassmannians. Again, it admits a [[w:Lax representation|Lax representation]].{{Sfn|Hand|Izosimov|2025}}

== References ==
{{reflist|25em}}

===Notes===
{{notelist}}

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      <text bytes="60965" sha1="0kltcp76shkt96hal8z00ial9mc4wn5" xml:space="preserve">{{Article info
| last1 = Stiegler
| orcid1 = 0009-0001-5789-6923
| first1 = Jean-Baptiste
| affiliation1 = Université Paris-Saclay
| correspondence1 = jean-baptiste.stiegler@universite-paris-saclay.fr
| journal = WikiJournal of Science
| et_al = true
| w1 = Pentagram map
| from w1 = true
| keywords = Pentagram map, Dynamical system, Projective geometry, Moduli space, Integrable systems
| license = CC-BY-SA 4.0
| submitted = 2025-12-08
| abstract = In [[w:mathematics|mathematics]], the '''pentagram map''' is a [[w:Dynamical system#Discrete dynamical system|discrete dynamical system]] acting on [[w:polygons|polygons]] in the [[w:projective plane|projective plane]]. It defines a new polygon whose vertices are obtained as the intersection points of the shortest [[w:Diagonal|diagonals]] of the initial polygon. This is a [[w:Projective linear group|projectively]] [[w:Equivariant map|equivariant]] procedure, hence it [[w:Quotient space (topology)|descends]] to the [[w:moduli space|moduli space]] of polygons and defines another dynamical system (which is also referred to as the pentagram map). It was first introduced by [[w:Richard Schwartz (mathematician)|Richard Schwartz]] in 1992.{{Sfn|Schwartz|1992}}

The pentagram map on the moduli space is famous for its [[w:Completely integrable|complete integrability]] and its link with [[w:cluster algebra|cluster algebras]].{{sfn|Gekhtman|Izosimov|2025|p=14}}

It admits many generalizations in [[w:Projective space|projective spaces]] and other settings.
}}
== Introduction ==

=== Informal definition ===

==== On polygons ====
[[File:Pentagram pentagon nolabel big.svg|alt=|thumb|300x300px|The pentagram map applied on a [[w:Convex set|convex]] [[w:pentagon|pentagon]].]]
Initially, the pentagram map was defined for [[w:convex polygon|convex polygon]]s (with at least five sides) on the [[w:euclidean plane|Euclidean plane]]. Given such a polygon &lt;math&gt;P&lt;/math&gt; with &lt;math&gt;n&lt;/math&gt; sides, one can draw the &quot;shortest [[w:diagonal|diagonal]]s&quot;, meaning the [[w:Line segment|segments]] whose endpoints are a [[w:Vertex (geometry)|vertex]] and one of its second neighbors (as in Figure 1). The intersections of the shortest diagonals are then taken as the vertices of a new &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;T(P)&lt;/math&gt;; this new polygon is the output of the pentagram map.{{Sfn|Berger|2005}}

The same construction can be done on [[w:Concave polygon|non-convex polygons]], but there are several complications. First, some consecutive short diagonals may not intersect, so one must extend the segments to [[w:Line (geometry)|lines]]. Second, the image &lt;math&gt;T(P)&lt;/math&gt; can fail to be a new &lt;math&gt;n&lt;/math&gt;-gon because some consecutive vertices could coincide. However, this [[w:Generic property|generically]] doesn't happen.{{Sfn|Ovsienko|Schwartz|Tabachnikov|p=411|2009}} Finally, it is possible that two diagonals are [[w:Parallel (geometry)|parallel]] and don't intersect on the [[w:euclidean plane|Euclidean plane]]. This is resolved by extending the Euclidean plane to the [[w:real projective plane|real projective plane]] by the addition of a [[w:line at infinity|line at infinity]], where the [[w:Vanishing point|intersection point]] lies (see Figure 3). Hence, the pentagram map is defined for generic polygons in the real projective plane.{{Sfn|Berger|2005|p=25}}

More generally, the construction of the pentagram map is well defined whenever the concepts of lines and their intersections make sense. This is encompassed by the notion of a general [[w:projective plane|projective plane]], of which the real projective plane is one example; but the pentagram map can also be considered over other [[w:Field (mathematics)|fields]], for instance the [[w:complex number|complex number]]s, which give the [[w:complex projective plane|complex projective plane]].{{Sfn|Weinreich|2022|loc=§3.1.1}}

==== On the moduli space of polygons ====
Since the pentagram map is constructed by drawing lines and marking their intersections, it [[w:Commutative property|commutes]] with any transformation that sends lines to lines. Such maps are called [[w:projective transformations|projective transformations]]. Hence, polygons can be identified [[w:up to|up to]] [[w:Perspectivity#Projectivity|projective transformations]]. This identification gives the [[w:Quotient space (topology)|quotient space]] (technically called a [[w:moduli space|moduli space]]) of [[w:Equivalence class|classes]] of polygons. 

The pentagram map on polygons induces another dynamical system on the moduli space,{{Sfn|Schwartz|1992|loc=§1 Projective geometry}} whose behavior differs quite a lot from the initial one.{{Efn|Compare the paragraph about the [[w:Pentagram map#Collapsing of convex polygons|collapsing of convex polygons]] and the one about [[w:Pentagram map#Complete integrability|complete integrability]].}} The dynamic is trivial for the classes of pentagons and hexagons, but this is no longer the case for polygons with more vertices.{{Efn|See the paragraph about [[w:Pentagram_map#Pentagons_and_hexagons|pentagons and hexagons]].}}

=== Historical elements ===
The pentagram map for general polygons was introduced in {{Harvard citation|Schwartz|1992}}, but the simplest case is the one of [[w:pentagons|pentagons]], hence the name &quot;[[w:pentagram|pentagram]]&quot;.{{Sfn|Marí-Beffa|2014|p=1}} Their study goes back to {{Harvard citation|Clebsch|1871}},{{Sfn|Izosimov|2022a|p=1085}} {{Harvard citation|Kasner|1928}}{{Sfn|Tabachnikov|2019}} and {{Harvard citation|Motzkin|1945}}.{{Sfn|Schwartz|2013|p=1}}

The pentagram map interacts with some classical configuration theorems of [[w:projective geometry|projective geometry]]. It provides results analogous to the ones of [[w:Pascal's theorem|Pascal's theorem]] and [[wikipedia:Brianchon's_theorem|Brianchon's theorem]].{{Sfn|Schwartz|Tabachnikov|2010}} Some specific configurations make [[w:Desargues' theorem|Desargues's theorem]] and [[w:Poncelet's porism|Poncelet's porism]] appear.{{Sfn|Berger|2005|loc=§4 and §5}}{{Efn|See the paragraph about [[w:Pentagram map#Poncelet polygons|Poncelet polygons]].}}

==Definitions and first properties==

=== Definition of the map ===
[[File:Pentagram pentagon label big.svg|alt=|thumb|300x300px|The pentagram map on a convex pentagon, with vertices labeled.]]
[[File:Pentagram on nonconvex pentagon.svg|alt=|thumb|300x300px|The pentagram map applied on a [[w:Self-intersecting polygon|self-intersecting]] (in particular, non-convex) pentagon. The vertex &lt;math&gt;w_2&lt;/math&gt; is on the [[w:line at infinity|line at infinity]], because it is the [[w:Vanishing point|intersection of two parallel lines]].]]
Let &lt;math&gt;n\geq 5&lt;/math&gt; be an integer. A polygon &lt;math&gt;P&lt;/math&gt; with &lt;math&gt;n&lt;/math&gt; sides, or &lt;math&gt;n&lt;/math&gt;-gon, is a tuple of [[w:Vertex (geometry)|vertices]] &lt;math&gt;(v_1,\dots,v_n)&lt;/math&gt; lying in some [[w:projective plane|projective plane]] &lt;math&gt;\mathbb P ^2&lt;/math&gt;,{{Efn|In the following, the figures represent polygons on the real plane, where the intuition is easier to grasp.}} where the indices are understood [[w:Modular arithmetic|modulo]] &lt;math&gt;n&lt;/math&gt;. The [[w:Dimension of an algebraic variety|dimension]] of the space of &lt;math&gt;n&lt;/math&gt;-gons is &lt;math&gt;2n&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=definition 1.1}}

Suppose that the vertices are in sufficiently [[w:general position|general position]], meaning that no consecutive triple of points are [[w:Collinearity|collinear]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=1}} Taking the intersection of two consecutive &quot;shortest&quot; [[w:diagonal|diagonal]]s{{Efn|Meaning the line between a vertex &lt;math&gt;v_k&lt;/math&gt; and a &quot;second neighbour&quot; &lt;math&gt;v_{k\pm 2}&lt;/math&gt;.}} defines a new point&lt;math display=&quot;block&quot;&gt; w_k := \overline{v_{k-1} v_{k+1}} \cap \overline{v_{k} v_{k+2}}. &lt;/math&gt;This procedure defines a new &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;T(P)=(w_1,\dots,w_n)&lt;/math&gt;, as in Figure 2.{{Sfn|Schwartz|1992|p=71}}

The labeling of the indices of &lt;math&gt;T(P)&lt;/math&gt; is not [[w:canonical|canonical]]. In most papers, a choice is made at the beginning of the paper and the formulas are tuned accordingly.{{Sfn|Izosimov|2016|loc=remark 1.5}}

The pentagram map on polygons is a [[w:birational map|birational map]] &lt;math&gt;T:(\mathbb P^2)^n&lt;/math&gt;{{nowrap|{{font|size=145%|⇢}}}}&lt;math&gt;(\mathbb P^2)^n&lt;/math&gt;. Indeed, each [[w:Homogeneous coordinates|coordinate]] of &lt;math&gt;w_k&lt;/math&gt; is given as a [[w:rational function|rational function]] of the coordinates of &lt;math&gt;v_{k-1},\dots,v_{k+2}&lt;/math&gt;, since it is defined as the intersection of lines passing by them. Moreover, the [[w:inverse map|inverse map]] is given by taking the intersections &lt;math&gt;\overline{w_{k-2} w_{k-1}} \cap \overline{w_{k} w_{k+1}} &lt;/math&gt;, which is rational for the same reason.{{Sfnp|Weinreich|2022|loc=definition 1.2}}

=== Moduli space ===
The pentagram map is defined by taking [[w:Line (geometry)|lines]] and intersections of them. The biggest [[w:Group (mathematics)|group]] which maps lines to lines is the one of [[w:projective transformations|projective transformations]], denoted by &lt;math&gt;\mathbb P \mathrm{GL}_{3}&lt;/math&gt;. Such a transformation &lt;math&gt;M&lt;/math&gt; [[w:Group action|acts]] on a polygon &lt;math&gt;P&lt;/math&gt; by sending it to &lt;math&gt;M \cdot P:=(Mv_1,\dots,Mv_n)&lt;/math&gt;. The pentagram map [[w:Commutative property|commutes]] with this action, and thereby induces another [[w:dynamical system|dynamical system]] on the [[w:moduli space|moduli space]] of projective [[w:equivalence classes|equivalence classes]] of polygons, whose [[w:Dimension of an algebraic variety|dimension]] is &lt;math&gt;2n-8&lt;/math&gt;.{{Sfn|Schwartz|1992|loc=§1 Projective geometry}}

===Twisted polygons===
[[File:Twisted heptagon.svg|alt=|thumb|300x300px|An example of twisted [[w:heptagon|heptagon]] on the real plane.]]
The pentagram map naturally generalizes to the larger space of twisted polygons (see example in Figure 4). For any integer &lt;math&gt;n\geq5&lt;/math&gt;, a twisted &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;P&lt;/math&gt; is the data of:

* a [[w:Sequence#Indexing|bi-infinite sequence]] of points &lt;math&gt;(v_k)_{k\in\mathbb Z}&lt;/math&gt; in the projective plane (called the vertices),
* a [[w:projective transformation|projective transformation]] &lt;math&gt;M \in \mathbb P \mathrm{GL}_3&lt;/math&gt; (called the [[w:monodromy|monodromy]]),

such that for any &lt;math&gt;k \in \mathbb Z&lt;/math&gt;, the property &lt;math&gt;v_{k+n}=Mv_k&lt;/math&gt; is satisfied. The dimension of the space of twisted &lt;math&gt;n&lt;/math&gt;-gons is &lt;math&gt;2n+8&lt;/math&gt;.{{Sfn|Schwartz|2008}}

When &lt;math&gt;M&lt;/math&gt; is the [[w:Identity_element|identity]], this gives back the initial definition of polygons (which are said to be closed). The space of closed &lt;math&gt;n&lt;/math&gt;-gons is of [[w:codimension|codimension]] &lt;math&gt;8&lt;/math&gt; in the space of twisted ones.{{Sfn|Soloviev|2013|p=2816}}

The action of projective transformations over the space of closed polygons generalizes to the space of twisted ones (the monodromy is changed by [[w:Matrix similarity|conjugation]]). This provides again a moduli space, of dimension &lt;math&gt;2n&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=definition 1.3}}

== Collapsing of convex polygons ==

=== Exponential shrinking ===
[[File:Pentagram map convex heptagon iterate.svg|alt=|thumb|300x300px|The pentagram map iterated on a convex [[w:heptagon|heptagon]], exhibiting the convergence.]]
Let &lt;math&gt;P&lt;/math&gt; be a closed [[w:Convex polygon#Strictly convex polygon|strictly convex polygon]] lying on the real plane. One of the first results proved by Richard Schwartz it that its iterates under the pentagram map shrink [[w:Exponential growth|exponentially fast]] to a point, as illustrated in Figure 5. This follows from two facts.
# The image of a strictly convex polygon is contained in its [[w:Interior (topology)|interior]], and is also strictly convex.{{Sfn|Glick|2020|p=2818}}
# There exists a constant &lt;math&gt;0&lt; \eta_P&lt;1&lt;/math&gt;, depending on &lt;math&gt;P&lt;/math&gt;, such that for any &lt;math&gt;N \in \mathbb N&lt;/math&gt;, the diameters of the iterates verify the inequality &lt;math display=&quot;inline&quot;&gt;\operatorname{diam}(T^N(P))\leq\eta_P^N \operatorname{diam}(P). &lt;/math&gt;{{Sfn|Schwartz|1992|loc=theorem 3.1}}

Hence, by [[w:Cantor's intersection theorem#Variant in complete metric spaces|Cantor's intersection theorem]], the sequence of polygons collapses toward a point.{{Sfn|Schwartz|1992|loc=§3 Convex polygons}}

The behavior on the moduli space is very different, since the dynamics is [[w:Recurrent point|recurrent]].{{Sfn|Schwartz|2001|loc=theorem 1.1}} It is even a [[w:quasiperiodic motion|quasiperiodic motion]],{{Sfn|Ovsienko|Schwartz|Tabachnikov|2009}} as discussed in [[w:Pentagram map#Complete integrability|the section about integrability]].

=== Coordinates of the limit point ===
The limit point coordinates were given in {{Harvard citation|Glick|2020}}. They satisfy some [[w:Degree of a polynomial|degree]] 3 [[w:polynomial equation|polynomial equations]], whose coefficients are [[w:rational function|rational function]]s in the coordinates of the vertices of the starting polygon. The proof relies on the fact that the limit point must be an [[w:eigenline|eigenline]] of a certain [[w:linear operator|linear operator]] of &lt;math&gt;\mathbb R^3&lt;/math&gt;.{{sfn|Glick|2020}}

This operator was reinterpreted in {{Harvard citation|Aboud|Izosimov|2022}} as the infinitesimal monodromy of the polygon. The [[w:Pentagram map#The scaling symmetry|scaling symmetry]] is used to [[w:Deformation (mathematics)|deform]] a closed polygon &lt;math&gt;P&lt;/math&gt; into a family of twisted ones '''&lt;math&gt;(P_z)_{z\in \mathbb C^*}&lt;/math&gt;''' with monodromy &lt;math&gt;M_z&lt;/math&gt;. The infinitesimal monodromy is defined to be:{{sfn|Aboud|Izosimov|2022}}
&lt;math display=&quot;block&quot;&gt;\left.\frac{dM_z}{dz}\right|_{z=1}.&lt;/math&gt;

=== Generalization ===
The collapsing of polygons may also happen in some [[w:Pentagram map#Generalizations|generalization of the pentagram map]], when considering some specific configurations of polygons in the real plane. The coordinates of the collapse point are given by a formula analogous to the one for the original pentagram map.{{Sfn|Schwartz|2026}}

== Periodic orbits on the moduli space ==
For some configurations of closed polygons, the iterate of the pentagram map will send &lt;math&gt;P&lt;/math&gt; to a projectively equivalent polygon (up to some shift of the indices). This means that, on the moduli space, the orbit of the class of &lt;math&gt;P&lt;/math&gt; is [[w:Periodic orbit|periodic]].

===Pentagons and hexagons===
[[File:penta hexagon.svg|300px|thumb|The outward hexagon is projectively equivalent to the inward one, with respect to their labeling.]]The following two facts are proved by checking [[w:cross-ratio|cross-ratio]] equalities, so they are true for polygons in any [[w:projective plane|projective plane]] (not just the [[w:Real projective plane|real one]]).{{Sfn|Schwartz|1992|loc=§2 Pentagons and hexagons}}

The pentagram map &lt;math&gt;T&lt;/math&gt; is the identity on the moduli space of [[w:pentagon|pentagon]]s.{{Sfn|Schwartz|1992|loc=theorem 2.1}}{{Sfn|Clebsch|1871}}{{Sfn|Motzkin|1945}} The second iterate &lt;math&gt;T^2&lt;/math&gt; is the identity on the space of labeled [[w:hexagon|hexagon]]s, up to a shift of labeling (see Figure 6).{{Sfn|Schwartz|1992|loc=theorem 2.3}} This phenomenon doesn't generalize to generic polygons with at least seven sides, for which the motion is [[w:Quasiperiodic motion|quasiperiodic]].{{Sfn|Tupan|2022}}

==== Generalization ====
The result about pentagons and hexagons generalizes to some [[w:Pentagram map#Generalizations|higher pentagram maps]] in &lt;math&gt;\mathbb P ^k&lt;/math&gt;, for polygons with &lt;math&gt;k+3&lt;/math&gt; or &lt;math&gt;2k+2&lt;/math&gt; sides. The proof uses a generalization of the [[w:Gale transform|Gale transform]].{{Sfn|Dirdak|2024}}

=== Poncelet polygons ===
A polygon is said to be Poncelet{{Efn|The name comes from [[w:Jean-Victor Poncelet|Jean-Victor Poncelet]] and [[w:Poncelet porism|his porism]].{{Sfn|Izosimov|2022a|p=1085}}}} if it is [[w:Inscribed figure|inscribed]] in a [[w:Conic section|conic]] and circumscribed about another one.{{Sfn|Schwartz|2015|loc=|p=433}}{{Efn|In particular, pentagons are Poncelet since [[w:five points determine a conic|five points determine a conic]].{{Sfn|Schwartz|2015|loc=|p=433}}}} For a convex Poncelet &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;P&lt;/math&gt; lying on the [[w:real projective plane|real projective plane]], the polygon &lt;math&gt;T^2(P)&lt;/math&gt; is projectively equivalent to &lt;math&gt;P&lt;/math&gt;.{{Sfn|Schwartz|2015|loc=theorem 1.1}} In fact, when &lt;math&gt;n&lt;/math&gt; is odd, the converse is also true.{{Sfn|Izosimov|2022a|loc=corollary 1.1}}

However, this converse statement is no longer true when the polygons are considered over the [[w:complex projective plane|complex projective plane]].{{Sfn|Izosimov|2022a|loc=remark 1.3}}

==Coordinates for the moduli space==
The moduli space can be described by different [[w:Coordinate_system|coordinate systems]]. The following ones give simple expressions for the dynamics, as presented in the next section.

=== Corner coordinates ===
[[File:Corner coordinates big.svg|thumb|300x300px|The geometric construction of the points defining the corner invariants.]]
Define the [[w:cross-ratio|cross-ratio]] of four [[w:Collinearity|collinear]] points to be

: &lt;math&gt; [a,b,c,d]=\frac{(a-b)(c-d)}{(a-c)(b-d)}. &lt;/math&gt;

The corner invariants are a system of coordinates on the space of twisted polygons, constructed by taking intersections as in Figure 7.{{Sfn|Schwartz|2001|loc=figure 2}} The left and right invariants are respectively defined{{Efn|The ordering of the vertices in the cross-ratios can differ from a paper to another one, which slightly changes the formulas in the following sections.}} as the following cross-ratios:

: &lt;math&gt;x_k:=[v_{k-2},v_{k-1},\overline{v_{k-2}v_{k-1}}\cap\overline{v_{k}v_{k+1}},\overline{v_{k-2}v_{k-1}}\cap\overline{v_{k+1}v_{k+2}}],&lt;/math&gt;

: &lt;math&gt;y_k:=[\overline{v_{k+1}v_{k+2}}\cap\overline{v_{k-2}v_{k-1}}, \overline{v_{k+1}v_{k+2}}\cap\overline{v_{k-1}v_{k}},v_{k+1},v_{k+2}].&lt;/math&gt;

Since the cross-ratio is [[w:Cross-ratio#Projective geometry|projective invariant]], the sequences &lt;math&gt;(x_k)_{k \in \mathbb Z}&lt;/math&gt; and &lt;math&gt;(y_k)_{k \in \mathbb Z}&lt;/math&gt; associated to a twisted &lt;math&gt;n&lt;/math&gt;-gon are &lt;math&gt;n&lt;/math&gt;-periodic.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=415}}

The corner invariants are elements of &lt;math&gt;\mathbb{P}^1\smallsetminus\{0,1,\infty\}&lt;/math&gt;, and they realize an [[w:Isomorphism_of_varieties|isomorphism of varieties]] between the moduli space of twisted &lt;math&gt;n&lt;/math&gt;-gons and &lt;math&gt;(\mathbb{P}^1\smallsetminus\{0,1,\infty\})^{2n}&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=theorem 3.6}}

===ab-coordinates===

There is a second set of coordinates for the moduli space of twisted &lt;math&gt;n&lt;/math&gt;-gons defined over a [[w:Field (mathematics)|field]] &lt;math&gt;F&lt;/math&gt; satisfying &lt;math&gt;\mathrm{SL}_3(F)\cong \mathbb P\mathrm{GL}_3(F)&lt;/math&gt;,{{Sfn|Weinreich|2022|loc=remark 3.8}} and such that &lt;math&gt;n&lt;/math&gt; is not divisible by &lt;math&gt;3&lt;/math&gt;.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=section 4.1}}

The vertices &lt;math&gt;v_k&lt;/math&gt; in the [[w:projective plane|projective plane]] &lt;math&gt;\mathbb P^2(F)&lt;/math&gt; can be [[w:Lift (mathematics)|lifted]] to [[w:Vector space|vectors]] &lt;math&gt;V_k&lt;/math&gt; in the [[w:affine space|affine space]] &lt;math&gt;F^3&lt;/math&gt; so that each consecutive triple of vectors spans a [[w:parallelepiped|parallelepiped]] having [[w:determinant|determinant]] equal to &lt;math&gt;1&lt;/math&gt;. This leads to the relation{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=equation 4.1}}

: &lt;math&gt;V_{k+3} = a_k V_{k+2} + b_k V_{k+1} + V_k.&lt;/math&gt;

This bring out an analogy between twisted polygons and solutions of third order linear [[w:ordinary differential equations|ordinary differential equations]], normalized to have unit [[w:Wronskian|Wronskian]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=remark 6.6}}

They are linked to the corner coordinates by:{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=lemma 4.5}}
: &lt;math&gt;x_k=\frac{a_{k-2}}{b_{k-2}b_{k-1}},&lt;/math&gt;
: &lt;math&gt;y_k=-\frac{b_{k-1}}{a_{k-2}a_{k-1}}.&lt;/math&gt;

==Formulas on the moduli space==

===As a birational map ===
The pentagram map is a [[w:birational map|birational map]] on the moduli space, because it can be decomposed as the [[w:Function composition|composition]] of two [[w:Birational geometry|birational]] [[w:Involution (mathematics)|involutions]].{{Sfn|Schwartz|2008|loc=§1.2 The Pentagram Map}} The corner invariants change in the following way:{{Sfn|Ovsienko|Schwartz|loc=lemma 2.4|Tabachnikov|2010}}

: &lt;math&gt;x_k'=x_k\frac{1-x_{k-1} y_{k-1}}{1-x_{k+1}y_{k+1}},&lt;/math&gt;

: &lt;math&gt;y_k'=y_{k+1}\frac{1-x_{k+2} y_{k+2}}{1-x_k y_k}.&lt;/math&gt;

=== The scaling symmetry ===
The [[w:multiplicative group|multiplicative group]] &lt;math&gt;F\smallsetminus\{0\}&lt;/math&gt; [[w:One-parameter group|acts]] on the moduli space in the following way:

: &lt;math&gt;R_s\cdot(x_1,\dots,x_n,y_1,\dots,y_n)=(sx_1,\dots,sx_n,s^{-1}y_1,\dots,s^{-1}y_n),&lt;/math&gt;

where &lt;math&gt;R&lt;/math&gt; is called the scaling action and &lt;math&gt;s&lt;/math&gt; is the scaling parameter. This action commutes with the pentagram map on the moduli space (as presented in the previous formulas). This property is called the scaling symmetry, and is instrumental in proving the [[w:Pentagram map#Complete integrability|complete integrability]] of the dynamics.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=corollary 2.5}}

==Invariant structures==

===Monodromy invariants===

The monodromy invariants, introduced in {{Harvard citation|Schwartz|2008}}, are a collection of [[w:Function (mathematics)|functions]] on the [[w:moduli space|moduli space]] that are invariant under the pentagram map.{{Sfn|Schwartz|2008|loc=theorem 1.2}} The simplest example of them are 
:&lt;math&gt; O_n= x_1x_2\cdots x_{n}, \quad E_n = y_1y_2\cdots y_n. &lt;/math&gt;
The other monodromy invariants can be retrieved through different points of view: through the [[w:Pentagram map#The scaling symmetry|scaling symmetry]], as [[w:Combinatorics|combinatorial]] objects, or as some [[w:determinant|determinant]]s.{{Sfn|Schwartz|Tabachnikov|2011|loc=§2 The Monodromy Invariants}} The one involving scaling symmetry is presented here.

Let &lt;math&gt;M\in \mathrm{GL}_3&lt;/math&gt; be a [[w:Lift (mathematics)|lift]] of the monodromy of a twisted &lt;math&gt;n&lt;/math&gt;-gon. The quantities

: &lt;math&gt;\Omega_1=\frac{\operatorname{trace}^3(M)}{\det(M)}, \quad \Omega_2=\frac{\operatorname{trace}^3(M^{-1})}{\det(M^{-1})},&lt;/math&gt;

are independent of the choice of lift and are invariant under [[w:Matrix similarity|conjugation]], so they are well defined for the projective class of the polygon. They are invariant under the pentagram map, since the monodromy matrix doesn't change.{{Sfn|Schwartz|Tabachnikov|2011|loc=|p=5}} Now, the quantities

: &lt;math&gt;\tilde{\Omega}_1=O_n^2E_n\Omega_1, \quad \tilde{\Omega}_2=O_nE_n^2\Omega_2,&lt;/math&gt;

have the same properties, but turn out to be polynomials in the corner invariants.{{Efn|Some papers consider the cube roots of this functions, but it doesn't change the following definitions of the monodromy invariants.}} They can be written as{{Sfn|Schwartz|Tabachnikov|2011|loc=|p=5}}

: &lt;math&gt;
\tilde{\Omega}_1=\biggl(\sum_{k=0}^{\lfloor n/2\rfloor}O_k\biggr)^3, \quad
\tilde{\Omega}_2=\biggl(\sum_{k=0}^{\lfloor n/2\rfloor}E_k\biggr)^3,
&lt;/math&gt;

where each &lt;math&gt;O_k&lt;/math&gt; and &lt;math&gt;E_k&lt;/math&gt; are [[w:homogeneous polynomial|homogeneous polynomial]]s respectively of weight &lt;math&gt;k&lt;/math&gt; and &lt;math&gt;-k&lt;/math&gt;,{{Sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=11}} meaning they change under the [[w:Pentagram map#The scaling symmetry|rescaling action]] on variables by{{Sfn|Schwartz|Tabachnikov|2011|p=5}}

: &lt;math&gt; R_s(O_k)= s^k O_k, \quad R_s(E_k)= s^{-k} E_k. &lt;/math&gt;

The quantities &lt;math&gt;O_1,\dots,O_{\lfloor n/2 \rfloor},O_n, E_1,\dots,E_{\lfloor n/2 \rfloor},E_n,&lt;/math&gt; are unchanged by the dynamics, and are called the monodromy invariants. Moreover, they are [[w:algebraically independent|algebraically independent]].{{Sfn|Schwartz|2008|loc=theorem 1.2}}

==== Polygons on conics ====
Whenever &lt;math&gt;P&lt;/math&gt; is [[w:Inscribed figure|inscribed]] on a [[w:conic section|conic section]], one has &lt;math&gt;O_k(P)=E_k(P)&lt;/math&gt; for all &lt;math&gt;k&lt;/math&gt;.{{Sfn|Schwartz|Tabachnikov|2011|loc=theorem 1.1}} Moreover, if &lt;math&gt;P&lt;/math&gt; is circumscribed about another conic,{{Efn|See the paragraph about [[w:Pentagram map#Poncelet polygons|Poncelet polygons]].}} then its monodromy invariants are characterized by the pair of conics.{{Sfn|Schwartz|2015|loc=theorem 1.2}} For such odd-gons, the translation on the [[w:Jacobian variety|Jacobian variety]]{{Efn|See the paragraph about [[w:Pentagram map#Algebro-geometric integrability|algebraic integrability]].}} is restricted to the [[w:Prym variety|Prym variety]] (which is a half-dimensional torus in the Jacobian).{{Sfn|Izosimov|2016|loc=theorem 1.3}}

===Poisson bracket===

An invariant [[w:Poisson bracket|Poisson bracket]] on the space of twisted polygons was found in {{Harvard citation|Ovsienko|Schwartz|Tabachnikov|2010}}. The monodromy invariants [[w:Poisson bracket#Constants of motion|commute]] with respect to it:
&lt;math display=&quot;block&quot;&gt; \{O_i,O_j\}=\{O_i,E_j\}=\{E_i,E_j\}=0 &lt;/math&gt;for all &lt;math&gt;i,j&lt;/math&gt;.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}}

The Poisson bracket is defined in terms of the corner coordinates by:
&lt;math display=&quot;block&quot;&gt; \begin{align}
 \{x_i,x_{i\pm1}\} &amp;= \mp x_i x_{i+1}, \\
 \{y_i,y_{i\pm 1}\} &amp;= \mp y_i y_{i+1}, \\
 \{x_i,x_j\} &amp;= \{y_i,y_j\} = \{x_i,y_j\} = 0
\end{align}&lt;/math&gt;for all other &lt;math&gt; i,j.&lt;/math&gt;{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=equation 2.16}}

=== The spectral curve ===

Let &lt;math&gt;\zeta&lt;/math&gt; be an element of the [[w:multiplicative group|multiplicative group]] and &lt;math&gt;P_\zeta&lt;/math&gt; be the polygon obtained by applying the [[w:Pentagram map#The scaling symmetry|rescaling action]] &lt;math&gt;R_\zeta&lt;/math&gt; on &lt;math&gt;P&lt;/math&gt;.  A [[w:Lax matrix|Lax matrix]] &lt;math&gt;\hat{T}(\zeta) \in \mathrm{GL}_3&lt;/math&gt; is a lift of the monodromy of &lt;math&gt;P_\zeta&lt;/math&gt; satisfying a [[w:Lax pair#Zero-curvature equation|zero-curvature equation]].{{Sfn|Weinreich|2022|loc=§5 The Lax representation}} Then, the spectral function is the [[w:Bivariate polynomial|bivariate]] [[w:characteristic polynomial|characteristic polynomial]]
&lt;math display=&quot;block&quot;&gt; Q(\lambda,\zeta) := \det(\lambda\operatorname{Id}-\hat{T}(\zeta)),&lt;/math&gt;or some renormalization of it. The [[w:spectral curve|spectral curve]] is the [[w:Projective variety#projective completion|projective completion]] of the [[w:Algebraic curve|affine curve]] defined by the equation &lt;math&gt;Q(\lambda,\zeta)=0&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=§6. The geometry of the spectral curve}} It is invariant under the pentagram map, and the monodromy invariants appear as the [[w:coefficient|coefficient]]s of &lt;math&gt;Q&lt;/math&gt;.{{Sfn|Soloviev|2013|loc=theorem 6.4}} Its [[w:geometric genus|geometric genus]] is &lt;math&gt;n-1&lt;/math&gt; if &lt;math&gt;n&lt;/math&gt; is odd, and &lt;math&gt;n-2&lt;/math&gt; if &lt;math&gt;n&lt;/math&gt; is even.{{Sfn|Weinreich|2022|p=|loc=theorem 6.4}}

It was first introduced in {{Harv|Soloviev|2013|ps=|p=}} for his proof of [[w:Pentagram map#Algebro-geometric integrability|algebro-geometric integrability]].{{sfn|Soloviev|2013}}

==Complete integrability==
The pentagram map on the moduli space has been proved to be a [[w:completely integrable|completely integrable]] [[w:discrete dynamical system|discrete dynamical system]], both in the [[w:Integrable system#Hamiltonian systems and Liouville integrability|Arnold-Liouville]]{{Efn|Over the [[w:real number|real number]]s.}} and the [[w:Integrable system#Complete integrability over the complex numbers|algebro-geometric]]{{Efn|Over [[w:algebraically closed field|algebraically closed field]]s of [[w:Characteristic (algebra)|characteristic]] different from 2.}} senses. In any case, this means that the moduli space is [[w:almost everywhere|almost everywhere]] [[w:Foliation|foliated]] by [[w:Torus#Flat torus|flat tori]] (or in the algebraic setting, [[w:Abelian variety|Abelian varieties]]), where the motion is a [[w:Translation (geometry)|translation]]. This [[w:Generic property|generically]] induces a [[w:quasiperiodic motion|quasiperiodic motion]] on the corresponding torus.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2009}}

===Arnold–Liouville integrability===

The proof of the integrability of the pentagram map on a real twisted polygon was achieved in {{Harvard citation|Ovsienko|Schwartz|Tabachnikov|2010}}. This is done by noticing that the monodromy invariants &lt;math&gt;O_n&lt;/math&gt; and &lt;math&gt;E_n&lt;/math&gt; are [[w:Casimir invariant|Casimir invariant]]s for the bracket, meaning (in this context) that&lt;math display=&quot;block&quot;&gt; \{O_n,f\}=\{E_n,f\} = 0 &lt;/math&gt;for all functions &lt;math&gt;f&lt;/math&gt;.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}} When &lt;math&gt;n&lt;/math&gt; is even, this is also true for the monodromy invariants &lt;math&gt;O_{\lfloor n/2 \rfloor }&lt;/math&gt; and &lt;math&gt;E_{\lfloor n/2 \rfloor }&lt;/math&gt;.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}}

This allows to consider the Casimir [[w:level set|level set]], where each Casimir has a specified value. Because of [[w:Sard's theorem|Sard's theorem]], any generic level set is a [[w:smooth manifold|smooth manifold]].{{Sfn|Schwartz|2017|p=44}} They form a [[w:foliation|foliation]] in [[w:Poisson manifold#Symplectic leaves|symplectic leaves]], on which the Poisson bracket gives rise to a [[w:symplectic form|symplectic form]].{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=corollary 2.13}}

Each of these symplectic leaves has an iso-monodromy [[w:foliation|foliation]], namely, a decomposition into the common level sets of the remaining monodromy functions. By using again [[w:Sard's theorem|Sard's theorem]], they are generically [[w:Symplectic manifold#Lagrangian submanifolds|Lagrangian manifolds]].{{Sfn|Schwartz|2017|p=45}} Moreover, they are compact.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=§3.3. Compactness of the level sets}} Since the monodromy invariants Poisson-commute and there are enough of them, the discrete [[w:Liouville–Arnold theorem|Liouville–Arnold theorem]] can be applied to prove that the level sets are [[w:Torus#Flat torus|flat tori]] over which the dynamics is a translation.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=412}}
===Algebro-geometric integrability===

In {{Harvard citation|Soloviev|2013}}, it was shown that the pentagram map admits a [[w:Lax representation|Lax representation]] with a spectral parameter, which allows to prove its algebro-geometric integrability. This means that the space of polygons (either twisted or closed) is parametrized by its spectral data, consisting of [[w:Pentagram map#The spectral curve|its spectral curve]], with marked points and a [[w:Divisor (algebraic geometry)|divisor]] given by a [[w:Floquet theory|Floquet]]–[[w:Bloch's theorem|Bloch]] equation. This gives an embedding to the [[w:Jacobian variety|Jacobian variety]] through the [[w:Abel–Jacobi map|Abel–Jacobi map]], where the motion is expressed in terms of translation.{{sfn|Soloviev|2013|loc=theorems A, B and C}} The previously defined Poisson bracket is also retrieved.{{sfn|Soloviev|2013|loc=theorem D}} 

This integrability was generalized in {{Harvard citation|Weinreich|2022}} from the field of [[w:complex number|complex number]]s to any [[w:algebraically closed field|algebraically closed field]] of [[w:Characteristic (algebra)|characteristic]] different from 2. The translation on a torus is replaced by a translation on an [[w:Abelian variety|Abelian variety]] (in fact, a Jacobian variety again).{{sfn|Weinreich|2022|loc=theorem 1.4}}

=== Dimension of the invariant manifold ===

For twisted &lt;math&gt;n&lt;/math&gt;-gons, the [[w:dimension|dimension]] of the invariant tori (or Jacobian varieties) is{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=421}}

: &lt;math&gt;\begin{cases} 
n-1 &amp; \text{when }n \text{ is odd,}\\
n-2 &amp; \text{when }n \text{ is even.}
\end{cases}&lt;/math&gt;

Moreover, when &lt;math&gt;n&lt;/math&gt; is even, there are two isomorphic Jacobians on which the iterates of the pentagram map alternate. But on each of them, the second iterate is a translation.{{Sfn|Weinreich|2022|loc=theorem 1.4}}

=== For closed polygons ===
There is no Poisson structure on the space of closed polygons.{{Sfn|Soloviev|2013|loc=corollary 4.1}} Nevertheless, the one from twisted polygons can be used to prove integrability.{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=2153}}

Algebro-geometric integrability holds for closed polygons in a same manner as for the twisted ones.{{Sfn|Soloviev|2013|loc=theorem C}} However, Arnold-Liouville integrability is proved for real closed polygons only when they are convex. This is done by restricting the [[w:Hamiltonian vector field|Hamiltonian vector field]]s of monodromy functions to smaller dimensional tori, and showing that enough of them are still independent.{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|loc=corollary 1.1}}

In both situation, the dimension of the invariant manifolds decreases by &lt;math&gt;3&lt;/math&gt; for closed &lt;math&gt;n&lt;/math&gt;-gons (compared to the twisted case), and is equal to{{Sfn|Soloviev|2013|loc=theorem C}}{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|loc=theorem 1}}
:&lt;math&gt;\begin{cases} 
n-4 &amp; \text{when }n \text{ is odd,}\\
n-5 &amp; \text{when }n \text{ is even.}
\end{cases}&lt;/math&gt;

==Connections to other topics==

===The Boussinesq equation===

The continuous limit of a convex polygon is a parametrized convex curve in the plane. When the time parameter is suitably chosen, the [[w:Discretization|continuous limit]] of the pentagram map is the classical [[w:Boussinesq approximation (water waves)|Boussinesq equation]]. This equation is a classical example of an [[w:integrable|integrable]] [[w:partial differential equation|partial differential equation]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 5}}

Here is a description of the geometric action of the Boussinesq equation. Given a [[w:locally convex|locally convex]] curve &lt;math&gt; C:\mathbb R\to \mathbb R^2 &lt;/math&gt; and real numbers &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;t&lt;/math&gt;, consider the [[w:chord (geometry)|chord]] connecting &lt;math&gt; C(x-t) &lt;/math&gt; to &lt;math&gt; C(x+t) &lt;/math&gt;. The [[w:Envelope (mathematics)|envelope]] of all these chords is a new curve &lt;math&gt; C_t(x) &lt;/math&gt;. When &lt;math&gt;t&lt;/math&gt; is extremely small, the curve &lt;math&gt; C_t(x) &lt;/math&gt; is a good model for the time &lt;math&gt;t&lt;/math&gt; evolution of the original curve &lt;math&gt; C_0(x) &lt;/math&gt; under the Boussinesq equation. This construction is also similar to the pentagram map. Moreover, the pentagram invariant bracket is a discretization of a well known invariant Poisson bracket associated to the Boussinesq equation.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=§6.4 Discretization}}

===Cluster algebras===

The pentagram map{{Sfn|Glick|2011}} and some of its generalizations{{Sfn|Gekhtman|Shapiro|Tabachnikov|Vainshtein|2012}}{{Sfn|Glick|Pylyavskyy|2016}} are identified as special cases of discrete dynamical systems powered by [[w:cluster algebra|cluster algebra]]. Using the results from {{Harvard citation|Goncharov|Kenyon|2013}}, this provides a link with the [[w:Poisson–Lie group|Poisson–Lie group]]s, [[w:dimer model|dimer model]]s and other so-called cluster-integrable systems.{{Sfn|Fock|Marshakov|2016}}{{Sfn|Izosimov|2022c}} These methods allow to retrieve the Poisson-bracket and Hamiltonians used to prove complete integrability{{Sfn|Affolter|George|Ramassamy|2025|loc=§5 The pentagram map}} and provide [[w:Lax representation|Lax representation]]s.{{Sfn|Izosimov|2022b}}

=== Octahedron recurrence ===
Using a method to compute [[w:Determinant|determinants]] called [[w:Dodgson_condensation|Dodgson condensation]], {{Harvard citation|Schwartz|2008}} proves that the pentagram map satisfies a property called the &quot;octahedron recurrence&quot;.{{Sfn|Schwartz|2008|loc=§5 The Method of Condensation}} This property turns out to be shared by other dynamical systems defined by geometric constructions similar to one of the pentagram map.{{Sfn|Affolter|de Tilière|Melotti|2025|loc=§9 The Pentagram Map}} It is also shared by higher dimensional pentagram maps defined through cluster algebras mutations, referred as &lt;math&gt;T&lt;/math&gt;-systems.{{Sfn|Kedem|Vichitkunakorn|2015||loc=}}

=== Singularity theory ===
The pentagram map exhibit a property called singularity confinement, which is typical from [[w:integrable system|integrable system]]s.{{Sfn|Grammaticos|Ramani|Papageorgiou|1991}} It states that if a polygon &lt;math&gt;P&lt;/math&gt; is [[w:Singular point of an algebraic variety|singular]] for the pentagram map &lt;math&gt;T&lt;/math&gt;, then there exists an integer &lt;math&gt;m&lt;/math&gt; such that &lt;math&gt;P&lt;/math&gt; not singular for the iterate map &lt;math&gt;T^m&lt;/math&gt;.{{Sfn|Glick|2012}}

Moreover, the pentagram map (along with some of its generalizations and other discrete dynamical systems) exhibit the Devron property.{{Efn|The name comes from an episode of [[w:Star Trek|Star Trek]].{{Sfn|Glick|2015|loc=§1 Introduction}}}} This means that if a polygon &lt;math&gt;P&lt;/math&gt; is singular for some iterate of the pentagram map &lt;math&gt;T^m&lt;/math&gt;, then it will also be singular for some iterate of the inverse map &lt;math&gt;T^{-m'}&lt;/math&gt;.{{Sfn|Glick|2015}}

== Generalizations ==
The definition of twisted polygons still makes sense in any [[w:projective space|projective space]] &lt;math&gt;\mathbb P^d&lt;/math&gt;, under the action of the [[w:Projective linear group|projective group]] &lt;math&gt;\mathbb P \mathrm{GL}_{d+1}&lt;/math&gt;. The pentagram map can be generalized in many ways, and some of them are presented here. Not all of them are integrable.{{Sfn|Khesin|Soloviev|2015|}} Some are [[w:discretization|discretization]]s of [[w:PDEs|PDEs]] from the [[w:KdV hierarchy|KdV hierarchy]], seen as higher dimensional version of [[w:Boussinesq approximation (water waves)|Boussinesq]] or [[w:Kadomtsev–Petviashvili equation|KP]] equations.{{Sfn|Marí-Beffa|2012}}{{Sfn|Wang|2023}} The description of all generalized pentagram maps in terms of [[w:cluster algebra|cluster algebra]]s is still an open question.{{Sfn|Gekhtman|Izosimov|2025|p=14}}

=== Polygons in general positions ===
Let &lt;math&gt;d \geq 2&lt;/math&gt; and &lt;math&gt;P&lt;/math&gt; be a twisted polygon of &lt;math&gt;\mathbb P^d&lt;/math&gt; in [[w:general position|general position]].

==== Short diagonal pentagram maps ====
The &lt;math&gt;k&lt;/math&gt;-th ''short diagonal hyperplane'' &lt;math&gt;H_k^{sh}&lt;/math&gt; is uniquely defined by passing through the vertices &lt;math&gt;v_k,v_{k+2},\dots,v_{k+2d-2}&lt;/math&gt;. [[w:Generic property#In algebraic geometry|Generically]], the intersection of &lt;math&gt;d&lt;/math&gt; consecutive hyperplanes uniquely defines a new point

: &lt;math&gt;T_{sh}v_k:=H_k^{sh}\cap H_{k+1}^{sh}\cap \dots \cap H_{k+d-1}^{sh}.&lt;/math&gt;

Doing this for every vertex defines a new twisted polygon. This map, denoted by &lt;math&gt;T_{sh}&lt;/math&gt;, is again projectively equivariant.{{Sfn|Khesin|Soloviev|2013}}

==== Generalized pentagram maps ====
The previous procedure can be generalized. Let &lt;math&gt;I=(i_1,\dots,i_{d-1}),~J=(j_1,\dots,j_{d-1})&lt;/math&gt; be two sets of integers, respectively called the jump tuple and the intersection tuple. Define the &lt;math&gt;k&lt;/math&gt;-th hyperplane &lt;math&gt;H_k^I&lt;/math&gt; to be passing through the vertices &lt;math&gt;v_k,v_{k+i_1},\dots,v_{k+i_1+\dots+i_{d-1}}&lt;/math&gt;. A new point is given by the intersection

: &lt;math&gt;T_{I,J}v_k:=H_k^I \cap H_{k+j_1}^I \cap \dots \cap H_{k+j_1+\dots +j_{d-1}}^I.&lt;/math&gt;

The map &lt;math&gt;T_{I,J}&lt;/math&gt; is called a generalized pentagram map.{{Sfn|Khesin|Soloviev|2015a}} The original pentagram map is recovered by considering&lt;math&gt;d=2,~I=(2),~J=(1)&lt;/math&gt;.

Integrability can be numerically tested by picking a random polygon &lt;math&gt;P&lt;/math&gt; with [[wikipedia:Rational_point|rational coordinates]] and studying the growth rate of the [[wikipedia:Height_function|height]] of its iterates. This is called the [[wikipedia:Integrable_system#Diophantine_integrability|diophantine integrability]] test, and some generalized pentagram maps don't seem to pass it.{{Sfn|Khesin|Soloviev|2015a|loc=§5 and §6}} However, it is conjectured that the maps &lt;math&gt;T_{I,I}&lt;/math&gt; are integrable for any &lt;math&gt;I&lt;/math&gt;.{{Sfn|Bolsinov|Matveev|Miranda|Tabachnikov|2018|loc=conjecture 4.13 (B. Khesin, F. Soloviev)}}

Some of these maps are [[w:discretization|discretization]]s of higher dimensional counterpart of the [[w:Boussinesq approximation (water waves)|Boussinesq equation]] in the [[w:KdV hierarchy|KdV hierarchy]].{{Sfn|Khesin|Soloviev|2015b|loc=theorem 4.1}}{{Sfn|Izosimov|2022b|loc=theorem 4.1}}

==== Dented pentagram maps ====
Fix an integer &lt;math&gt;m\in \{1,\dots ,d-1\}&lt;/math&gt;. Consider the jump tuple &lt;math&gt;I_m:=(1,\dots,1,2,1,\dots,1)&lt;/math&gt;, where the &lt;math&gt;2&lt;/math&gt; is at the &lt;math&gt;m&lt;/math&gt;-th place, and the intersection tuple &lt;math&gt;J:=(1,\dots,1)&lt;/math&gt;. The dented pentagram map is &lt;math&gt;T_m :=T_{I_m,J}&lt;/math&gt;. They are proved to be integrable.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 2.14}}

For an integer &lt;math&gt;p \geq 2&lt;/math&gt;, the deep dented pentagram map (of depth &lt;math&gt;p&lt;/math&gt;) &lt;math&gt;T_m^p&lt;/math&gt; is the same map as before, but the number &lt;math&gt;2&lt;/math&gt; in the definition of &lt;math&gt;I_m&lt;/math&gt; is replaced by &lt;math&gt;p&lt;/math&gt;. This kind of pentagram maps are again integrable.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 6.2}}

=== Corrugated polygons ===
A twisted polygon &lt;math&gt;P&lt;/math&gt; lying in &lt;math&gt;\mathbb P^d&lt;/math&gt; is said to be corrugated if for any &lt;math&gt;k\in \mathbb Z&lt;/math&gt;, the vertices &lt;math&gt;v_k,v_{k+1},v_{k+d},v_{k+d+1}&lt;/math&gt; span a projective two-dimensional plane. Such polygons are not in [[w:general position|general position]]. A new point is defined by

: &lt;math&gt;T_\text{cor}v_k:=\overline{v_k v_{k+d}}\cap \overline{v_{k+1} v_{k+d+1}}.&lt;/math&gt;

The map &lt;math&gt;T_\text{cor}&lt;/math&gt; yields a new corrugated polygon. They are [[w:Integrable system#Hamiltonian systems and Liouville integrability|completely Liouville-integrable]].{{Sfn|Gekhtman|Shapiro|Tabachnikov|Vainshtein|2012|loc=theorem 4.4}}

In fact, they can be retrieved as some dented pentagram map applied on corrugated polygons.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 5.3}}

=== Grassmannian polygons ===
Let &lt;math&gt;d \geq 3, m \geq 1&lt;/math&gt; be integers. The pentagram map can also be generalized to the [[w:Grassmannian|Grassmannian]] space &lt;math&gt;\mathrm{Gr}(m,md)&lt;/math&gt;, which consists of &lt;math&gt;m&lt;/math&gt;-[[w:Dimension (vector space)|dimensional]] [[w:linear subspace|linear subspace]]s of an &lt;math&gt;md&lt;/math&gt;-dimensional [[w:vector space|vector space]]. When &lt;math&gt;m=1&lt;/math&gt;, the linear subspaces are [[w:Vector space#vector line|lines]], which retrieves the definition of [[w:projective space|projective space]]s &lt;math&gt;\mathbb P^d&lt;/math&gt;.{{Sfn|Felipe|Marí-Beffa|2019|loc=§2 definitions and notations}}

A point &lt;math&gt;v\in\operatorname{Gr}(m,md)&lt;/math&gt; is represented by an &lt;math&gt;md \times m&lt;/math&gt; matrix &lt;math&gt;X_v&lt;/math&gt; such that its columns form a [[w:Basis (linear algebra)|basis]] of &lt;math&gt;v&lt;/math&gt;. Consider the [[w:Group action|action]] of the [[w:general linear group|general linear group]] &lt;math&gt;\mathrm{GL}_{md}&lt;/math&gt; by multiplication on the left of &lt;math&gt;X_v&lt;/math&gt;. This defines an action on the Grassmannian, even though it is not [[w:Faithful action|faithful]].{{Efn|Because there can be many lifts for &lt;math&gt;v&lt;/math&gt;, and because some matrices act trivially.}} Hence, the polygons of &lt;math&gt;\mathrm{Gr}(m,md)&lt;/math&gt; and their moduli spaces are defined as before, after the change of underlying group.{{Sfn|Felipe|Marí-Beffa|2019|loc=§2 definitions and notations}}

Depending on the parity of &lt;math&gt;d&lt;/math&gt;, one can define linear subspaces spanned by some &lt;math&gt;X_{v_k}&lt;/math&gt;'s such that taking their intersection generically defines a new point &lt;math&gt;v\in\mathrm{Gr}(m,md)&lt;/math&gt;.{{Sfn|Felipe|Marí-Beffa|2019|loc=sections 4 and 5}} This generalization of the pentagram map is integrable in a [[w:noncommutative|noncommutative]] sense.{{Sfn|Ovenhouse|2020}}

=== Over rings ===
The pentagram map admits a generalization by considering [[w:Projective space#Generalizations|projective planes]] over [[w:stably finite ring|stably finite ring]]s, instead of [[w:Field (mathematics)|field]]s. In particular, this retrieves the pentagram map over Grassmannians. Again, it admits a [[w:Lax representation|Lax representation]].{{Sfn|Hand|Izosimov|2025}}

== References ==
{{reflist|25em}}

===Notes===
{{notelist}}

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*{{Cite journal |ref=harv |last=Tupan |first=Alexandru |date=2022-07-03 |title=Pentagram Configurations for Pentagons and Hexagons |url=https://www.tandfonline.com/doi/full/10.1080/00029890.2022.2060695 |journal=The American Mathematical Monthly |language=en |volume=129 |issue=6 |pages=554–565 |doi=10.1080/00029890.2022.2060695 |issn=0002-9890}}
*{{Cite journal|ref=harv |title=Pentagram-Type Maps and the Discrete KP Equation|url=https://link.springer.com/10.1007/s00332-023-09961-7|journal=Journal of Nonlinear Science|date=2023|issn=0938-8974|volume=33|issue=6|doi=10.1007/s00332-023-09961-7|language=en|first=Bao|last=Wang |article-number=101 |bibcode=2023JNS....33..101W }}
*{{Cite journal |ref=harv |title=The algebraic dynamics of the pentagram map |journal=Ergodic Theory and Dynamical Systems |date=2022-11-25 |issn=0143-3857 |pages=3460–3505 |volume=43 |issue=10 |doi=10.1017/etds.2022.82 |first=Max H. |last=Weinreich}}</text>
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| last1 = Stiegler
| orcid1 = 0009-0001-5789-6923
| first1 = Jean-Baptiste
| affiliation1 = Université Paris-Saclay
| correspondence1 = jean-baptiste.stiegler@universite-paris-saclay.fr
| journal = WikiJournal of Science
| et_al = true
| w1 = Pentagram map
| from w1 = true
| keywords = Pentagram map, Dynamical system, Projective geometry, Moduli space, Integrable systems
| license = CC-BY-SA 4.0
| submitted = 2025-12-08
| abstract = In [[w:mathematics|mathematics]], the '''pentagram map''' is a [[w:Dynamical system#Discrete dynamical system|discrete dynamical system]] acting on [[w:polygons|polygons]] in the [[w:projective plane|projective plane]]. It defines a new polygon whose vertices are obtained as the intersection points of the shortest [[w:Diagonal|diagonals]] of the initial polygon. This is a [[w:Projective linear group|projectively]] [[w:Equivariant map|equivariant]] procedure, hence it [[w:Quotient space (topology)|descends]] to the [[w:moduli space|moduli space]] of polygons and defines another dynamical system (which is also referred to as the pentagram map). It was first introduced by [[w:Richard Schwartz (mathematician)|Richard Schwartz]] in 1992.{{Sfn|Schwartz|1992}}

The pentagram map on the moduli space is famous for its [[w:Completely integrable|complete integrability]] and its link with [[w:cluster algebra|cluster algebras]].{{sfn|Gekhtman|Izosimov|2025|p=14}}

It admits many generalizations in [[w:Projective space|projective spaces]] and other settings.
}}
== Introduction ==

=== Informal definition ===

==== On polygons ====
[[File:Pentagram pentagon nolabel big.svg|alt=|thumb|300x300px|The pentagram map applied on a [[w:Convex set|convex]] [[w:pentagon|pentagon]].]]
Initially, the pentagram map was defined for [[w:convex polygon|convex polygon]]s (with at least five sides) on the [[w:euclidean plane|Euclidean plane]]. Given such a polygon &lt;math&gt;P&lt;/math&gt; with &lt;math&gt;n&lt;/math&gt; sides, one can draw the &quot;shortest [[w:diagonal|diagonal]]s&quot;, meaning the [[w:Line segment|segments]] whose endpoints are a [[w:Vertex (geometry)|vertex]] and one of its second neighbors (as in Figure 1). The intersections of the shortest diagonals are then taken as the vertices of a new &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;T(P)&lt;/math&gt;; this new polygon is the output of the pentagram map.{{Sfn|Berger|2005}}

The same construction can be done on [[w:Concave polygon|non-convex polygons]], but there are several complications. First, some consecutive short diagonals may not intersect, so one must extend the segments to [[w:Line (geometry)|lines]]. Second, the image &lt;math&gt;T(P)&lt;/math&gt; can fail to be a new &lt;math&gt;n&lt;/math&gt;-gon because some consecutive vertices could coincide. However, this [[w:Generic property|generically]] doesn't happen.{{Sfn|Ovsienko|Schwartz|Tabachnikov|p=411|2009}} Finally, it is possible that two diagonals are [[w:Parallel (geometry)|parallel]] and don't intersect on the [[w:euclidean plane|Euclidean plane]]. This is resolved by extending the Euclidean plane to the [[w:real projective plane|real projective plane]] by the addition of a [[w:line at infinity|line at infinity]], where the [[w:Vanishing point|intersection point]] lies (see Figure 3). Hence, the pentagram map is defined for generic polygons in the real projective plane.{{Sfn|Berger|2005|p=25}}

More generally, the construction of the pentagram map is well defined whenever the concepts of lines and their intersections make sense. This is encompassed by the notion of a general [[w:projective plane|projective plane]], of which the real projective plane is one example; but the pentagram map can also be considered over other [[w:Field (mathematics)|fields]], for instance the [[w:complex number|complex number]]s, which give the [[w:complex projective plane|complex projective plane]].{{Sfn|Weinreich|2022|loc=§3.1.1}}

==== On the moduli space of polygons ====
Since the pentagram map is constructed by drawing lines and marking their intersections, it [[w:Commutative property|commutes]] with any transformation that sends lines to lines. Such maps are called [[w:projective transformations|projective transformations]]. Hence, polygons can be identified [[w:up to|up to]] [[w:Perspectivity#Projectivity|projective transformations]]. This identification gives the [[w:Quotient space (topology)|quotient space]] (technically called a [[w:moduli space|moduli space]]) of [[w:Equivalence class|classes]] of polygons. 

The pentagram map on polygons induces another dynamical system on the moduli space,{{Sfn|Schwartz|1992|loc=§1 Projective geometry}} whose behavior differs quite a lot from the initial one.{{Efn|Compare the paragraph about the [[w:Pentagram map#Collapsing of convex polygons|collapsing of convex polygons]] and the one about [[w:Pentagram map#Complete integrability|complete integrability]].}} The dynamic is trivial for the classes of pentagons and hexagons, but this is no longer the case for polygons with more vertices.{{Efn|See the paragraph about [[w:Pentagram_map#Pentagons_and_hexagons|pentagons and hexagons]].}}

=== Historical elements ===
The pentagram map for general polygons was introduced in {{Harvard citation|Schwartz|1992}}, but the simplest case is the one of [[w:pentagons|pentagons]], hence the name &quot;[[w:pentagram|pentagram]]&quot;.{{Sfn|Marí-Beffa|2014|p=1}} Their study goes back to {{Harvard citation|Clebsch|1871}},{{Sfn|Izosimov|2022a|p=1085}} {{Harvard citation|Kasner|1928}}{{Sfn|Tabachnikov|2019}} and {{Harvard citation|Motzkin|1945}}.{{Sfn|Schwartz|2013|p=1}}

The pentagram map interacts with some classical configuration theorems of [[w:projective geometry|projective geometry]]. It provides results analogous to the ones of [[w:Pascal's theorem|Pascal's theorem]] and [[wikipedia:Brianchon's_theorem|Brianchon's theorem]].{{Sfn|Schwartz|Tabachnikov|2010}} Some specific configurations make [[w:Desargues' theorem|Desargues's theorem]] and [[w:Poncelet's porism|Poncelet's porism]] appear.{{Sfn|Berger|2005|loc=§4 and §5}}{{Efn|See the paragraph about [[w:Pentagram map#Poncelet polygons|Poncelet polygons]].}}

==Definitions and first properties==

=== Definition of the map ===
[[File:Pentagram pentagon label big.svg|alt=|thumb|300x300px|The pentagram map on a convex pentagon, with vertices labeled.]]
[[File:Pentagram on nonconvex pentagon.svg|alt=|thumb|300x300px|The pentagram map applied on a [[w:Self-intersecting polygon|self-intersecting]] (in particular, non-convex) pentagon. The vertex &lt;math&gt;w_2&lt;/math&gt; is on the [[w:line at infinity|line at infinity]], because it is the [[w:Vanishing point|intersection of two parallel lines]].]]
Let &lt;math&gt;n\geq 5&lt;/math&gt; be an integer. A polygon &lt;math&gt;P&lt;/math&gt; with &lt;math&gt;n&lt;/math&gt; sides, or &lt;math&gt;n&lt;/math&gt;-gon, is a tuple of [[w:Vertex (geometry)|vertices]] &lt;math&gt;(v_1,\dots,v_n)&lt;/math&gt; lying in some [[w:projective plane|projective plane]] &lt;math&gt;\mathbb P ^2&lt;/math&gt;,{{Efn|In the following, the figures represent polygons on the real plane, where the intuition is easier to grasp.}} where the indices are understood [[w:Modular arithmetic|modulo]] &lt;math&gt;n&lt;/math&gt;. The [[w:Dimension of an algebraic variety|dimension]] of the space of &lt;math&gt;n&lt;/math&gt;-gons is &lt;math&gt;2n&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=definition 1.1}}

Suppose that the vertices are in sufficiently [[w:general position|general position]], meaning that no consecutive triple of points are [[w:Collinearity|collinear]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=1}} Taking the intersection of two consecutive &quot;shortest&quot; [[w:diagonal|diagonal]]s{{Efn|Meaning the line between a vertex &lt;math&gt;v_k&lt;/math&gt; and a &quot;second neighbour&quot; &lt;math&gt;v_{k\pm 2}&lt;/math&gt;.}} defines a new point&lt;math display=&quot;block&quot;&gt; w_k := \overline{v_{k-1} v_{k+1}} \cap \overline{v_{k} v_{k+2}}. &lt;/math&gt;This procedure defines a new &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;T(P)=(w_1,\dots,w_n)&lt;/math&gt;, as in Figure 2.{{Sfn|Schwartz|1992|p=71}}

The labeling of the indices of &lt;math&gt;T(P)&lt;/math&gt; is not [[w:canonical|canonical]]. In most papers, a choice is made at the beginning of the paper and the formulas are tuned accordingly.{{Sfn|Izosimov|2016|loc=remark 1.5}}

The pentagram map on polygons is a [[w:birational map|birational map]] &lt;math&gt;T:(\mathbb P^2)^n&lt;/math&gt;{{nowrap|{{font|size=145%|⇢}}}}&lt;math&gt;(\mathbb P^2)^n&lt;/math&gt;. Indeed, each [[w:Homogeneous coordinates|coordinate]] of &lt;math&gt;w_k&lt;/math&gt; is given as a [[w:rational function|rational function]] of the coordinates of &lt;math&gt;v_{k-1},\dots,v_{k+2}&lt;/math&gt;, since it is defined as the intersection of lines passing by them. Moreover, the [[w:inverse map|inverse map]] is given by taking the intersections &lt;math&gt;\overline{w_{k-2} w_{k-1}} \cap \overline{w_{k} w_{k+1}} &lt;/math&gt;, which is rational for the same reason.{{Sfnp|Weinreich|2022|loc=definition 1.2}}

=== Moduli space ===
The pentagram map is defined by taking [[w:Line (geometry)|lines]] and intersections of them. The biggest [[w:Group (mathematics)|group]] which maps lines to lines is the one of [[w:projective transformations|projective transformations]], denoted by &lt;math&gt;\mathbb P \mathrm{GL}_{3}&lt;/math&gt;. Such a transformation &lt;math&gt;M&lt;/math&gt; [[w:Group action|acts]] on a polygon &lt;math&gt;P&lt;/math&gt; by sending it to &lt;math&gt;M \cdot P:=(Mv_1,\dots,Mv_n)&lt;/math&gt;. The pentagram map [[w:Commutative property|commutes]] with this action, and thereby induces another [[w:dynamical system|dynamical system]] on the [[w:moduli space|moduli space]] of projective [[w:equivalence classes|equivalence classes]] of polygons, whose [[w:Dimension of an algebraic variety|dimension]] is &lt;math&gt;2n-8&lt;/math&gt;.{{Sfn|Schwartz|1992|loc=§1 Projective geometry}}

===Twisted polygons===
[[File:Twisted heptagon.svg|alt=|thumb|300x300px|An example of twisted [[w:heptagon|heptagon]] on the real plane.]]
The pentagram map naturally generalizes to the larger space of twisted polygons (see example in Figure 4). For any integer &lt;math&gt;n\geq5&lt;/math&gt;, a twisted &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;P&lt;/math&gt; is the data of:

* a [[w:Sequence#Indexing|bi-infinite sequence]] of points &lt;math&gt;(v_k)_{k\in\mathbb Z}&lt;/math&gt; in the projective plane (called the vertices),
* a [[w:projective transformation|projective transformation]] &lt;math&gt;M \in \mathbb P \mathrm{GL}_3&lt;/math&gt; (called the [[w:monodromy|monodromy]]),

such that for any &lt;math&gt;k \in \mathbb Z&lt;/math&gt;, the property &lt;math&gt;v_{k+n}=Mv_k&lt;/math&gt; is satisfied. The dimension of the space of twisted &lt;math&gt;n&lt;/math&gt;-gons is &lt;math&gt;2n+8&lt;/math&gt;.{{Sfn|Schwartz|2008}}

When &lt;math&gt;M&lt;/math&gt; is the [[w:Identity_element|identity]], this gives back the initial definition of polygons (which are said to be closed). The space of closed &lt;math&gt;n&lt;/math&gt;-gons is of [[w:codimension|codimension]] &lt;math&gt;8&lt;/math&gt; in the space of twisted ones.{{Sfn|Soloviev|2013|p=2816}}

The action of projective transformations over the space of closed polygons generalizes to the space of twisted ones (the monodromy is changed by [[w:Matrix similarity|conjugation]]). This provides again a moduli space, of dimension &lt;math&gt;2n&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=definition 1.3}}

== Collapsing of convex polygons ==

=== Exponential shrinking ===
[[File:Pentagram map convex heptagon iterate.svg|alt=|thumb|300x300px|The pentagram map iterated on a convex [[w:heptagon|heptagon]], exhibiting the convergence.]]
Let &lt;math&gt;P&lt;/math&gt; be a closed [[w:Convex polygon#Strictly convex polygon|strictly convex polygon]] lying on the real plane. One of the first results proved by Richard Schwartz it that its iterates under the pentagram map shrink [[w:Exponential growth|exponentially fast]] to a point, as illustrated in Figure 5. This follows from two facts.
# The image of a strictly convex polygon is contained in its [[w:Interior (topology)|interior]], and is also strictly convex.{{Sfn|Glick|2020|p=2818}}
# There exists a constant &lt;math&gt;0&lt; \eta_P&lt;1&lt;/math&gt;, depending on &lt;math&gt;P&lt;/math&gt;, such that for any &lt;math&gt;N \in \mathbb N&lt;/math&gt;, the diameters of the iterates verify the inequality &lt;math display=&quot;inline&quot;&gt;\operatorname{diam}(T^N(P))\leq\eta_P^N \operatorname{diam}(P). &lt;/math&gt;{{Sfn|Schwartz|1992|loc=theorem 3.1}}

Hence, by [[w:Cantor's intersection theorem#Variant in complete metric spaces|Cantor's intersection theorem]], the sequence of polygons collapses toward a point.{{Sfn|Schwartz|1992|loc=§3 Convex polygons}}

The behavior on the moduli space is very different, since the dynamics is [[w:Recurrent point|recurrent]].{{Sfn|Schwartz|2001|loc=theorem 1.1}} It is even a [[w:quasiperiodic motion|quasiperiodic motion]],{{Sfn|Ovsienko|Schwartz|Tabachnikov|2009}} as discussed in [[w:Pentagram map#Complete integrability|the section about integrability]].

=== Coordinates of the limit point ===
The limit point coordinates were given in {{Harvard citation|Glick|2020}}. They satisfy some [[w:Degree of a polynomial|degree]] 3 [[w:polynomial equation|polynomial equations]], whose coefficients are [[w:rational function|rational function]]s in the coordinates of the vertices of the starting polygon. The proof relies on the fact that the limit point must be an [[w:eigenline|eigenline]] of a certain [[w:linear operator|linear operator]] of &lt;math&gt;\mathbb R^3&lt;/math&gt;.{{sfn|Glick|2020}}

This operator was reinterpreted in {{Harvard citation|Aboud|Izosimov|2022}} as the infinitesimal monodromy of the polygon. The [[w:Pentagram map#The scaling symmetry|scaling symmetry]] is used to [[w:Deformation (mathematics)|deform]] a closed polygon &lt;math&gt;P&lt;/math&gt; into a family of twisted ones '''&lt;math&gt;(P_z)_{z\in \mathbb C^*}&lt;/math&gt;''' with monodromy &lt;math&gt;M_z&lt;/math&gt;. The infinitesimal monodromy is defined to be:{{sfn|Aboud|Izosimov|2022}}
&lt;math display=&quot;block&quot;&gt;\left.\frac{dM_z}{dz}\right|_{z=1}.&lt;/math&gt;

=== Generalization ===
The collapsing of polygons may also happen in some [[w:Pentagram map#Generalizations|generalization of the pentagram map]], when considering some specific configurations of polygons in the real plane. The coordinates of the collapse point are given by a formula analogous to the one for the original pentagram map.{{Sfn|Schwartz|2026}}

== Periodic orbits on the moduli space ==
For some configurations of closed polygons, the iterate of the pentagram map will send &lt;math&gt;P&lt;/math&gt; to a projectively equivalent polygon (up to some shift of the indices). This means that, on the moduli space, the orbit of the class of &lt;math&gt;P&lt;/math&gt; is [[w:Periodic orbit|periodic]].

===Pentagons and hexagons===
[[File:penta hexagon.svg|300px|thumb|The outward hexagon is projectively equivalent to the inward one, with respect to their labeling.]]The following two facts are proved by checking [[w:cross-ratio|cross-ratio]] equalities, so they are true for polygons in any [[w:projective plane|projective plane]] (not just the [[w:Real projective plane|real one]]).{{Sfn|Schwartz|1992|loc=§2 Pentagons and hexagons}}

The pentagram map &lt;math&gt;T&lt;/math&gt; is the identity on the moduli space of [[w:pentagon|pentagon]]s.{{Sfn|Schwartz|1992|loc=theorem 2.1}}{{Sfn|Clebsch|1871}}{{Sfn|Motzkin|1945}} The second iterate &lt;math&gt;T^2&lt;/math&gt; is the identity on the space of labeled [[w:hexagon|hexagon]]s, up to a shift of labeling (see Figure 6).{{Sfn|Schwartz|1992|loc=theorem 2.3}} This phenomenon doesn't generalize to generic polygons with at least seven sides, for which the motion is [[w:Quasiperiodic motion|quasiperiodic]].{{Sfn|Tupan|2022}}

==== Generalization ====
The result about pentagons and hexagons generalizes to some [[w:Pentagram map#Generalizations|higher pentagram maps]] in &lt;math&gt;\mathbb P ^k&lt;/math&gt;, for polygons with &lt;math&gt;k+3&lt;/math&gt; or &lt;math&gt;2k+2&lt;/math&gt; sides. The proof uses a generalization of the [[w:Gale transform|Gale transform]].{{Sfn|Dirdak|2024}}

=== Poncelet polygons ===
A polygon is said to be Poncelet{{Efn|The name comes from [[w:Jean-Victor Poncelet|Jean-Victor Poncelet]] and [[w:Poncelet porism|his porism]].{{Sfn|Izosimov|2022a|p=1085}}}} if it is [[w:Inscribed figure|inscribed]] in a [[w:Conic section|conic]] and circumscribed about another one.{{Sfn|Schwartz|2015|loc=|p=433}}{{Efn|In particular, pentagons are Poncelet since [[w:five points determine a conic|five points determine a conic]].{{Sfn|Schwartz|2015|loc=|p=433}}}} For a convex Poncelet &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;P&lt;/math&gt; lying on the [[w:real projective plane|real projective plane]], the polygon &lt;math&gt;T^2(P)&lt;/math&gt; is projectively equivalent to &lt;math&gt;P&lt;/math&gt;.{{Sfn|Schwartz|2015|loc=theorem 1.1}} In fact, when &lt;math&gt;n&lt;/math&gt; is odd, the converse is also true.{{Sfn|Izosimov|2022a|loc=corollary 1.1}}

However, this converse statement is no longer true when the polygons are considered over the [[w:complex projective plane|complex projective plane]].{{Sfn|Izosimov|2022a|loc=remark 1.3}}

==Coordinates for the moduli space==
The moduli space can be described by different [[w:Coordinate_system|coordinate systems]]. The following ones give simple expressions for the dynamics, as presented in the next section.

=== Corner coordinates ===
[[File:Corner coordinates big.svg|thumb|300x300px|The geometric construction of the points defining the corner invariants.]]
Define the [[w:cross-ratio|cross-ratio]] of four [[w:Collinearity|collinear]] points to be

: &lt;math&gt; [a,b,c,d]=\frac{(a-b)(c-d)}{(a-c)(b-d)}. &lt;/math&gt;

The corner invariants are a system of coordinates on the space of twisted polygons, constructed by taking intersections as in Figure 7.{{Sfn|Schwartz|2001|loc=figure 2}} The left and right invariants are respectively defined{{Efn|The ordering of the vertices in the cross-ratios can differ from a paper to another one, which slightly changes the formulas in the following sections.}} as the following cross-ratios:

: &lt;math&gt;x_k:=[v_{k-2},v_{k-1},\overline{v_{k-2}v_{k-1}}\cap\overline{v_{k}v_{k+1}},\overline{v_{k-2}v_{k-1}}\cap\overline{v_{k+1}v_{k+2}}],&lt;/math&gt;

: &lt;math&gt;y_k:=[\overline{v_{k+1}v_{k+2}}\cap\overline{v_{k-2}v_{k-1}}, \overline{v_{k+1}v_{k+2}}\cap\overline{v_{k-1}v_{k}},v_{k+1},v_{k+2}].&lt;/math&gt;

Since the cross-ratio is [[w:Cross-ratio#Projective geometry|projective invariant]], the sequences &lt;math&gt;(x_k)_{k \in \mathbb Z}&lt;/math&gt; and &lt;math&gt;(y_k)_{k \in \mathbb Z}&lt;/math&gt; associated to a twisted &lt;math&gt;n&lt;/math&gt;-gon are &lt;math&gt;n&lt;/math&gt;-periodic.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=415}}

The corner invariants are elements of &lt;math&gt;\mathbb{P}^1\smallsetminus\{0,1,\infty\}&lt;/math&gt;, and they realize an [[w:Isomorphism_of_varieties|isomorphism of varieties]] between the moduli space of twisted &lt;math&gt;n&lt;/math&gt;-gons and &lt;math&gt;(\mathbb{P}^1\smallsetminus\{0,1,\infty\})^{2n}&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=theorem 3.6}}

===ab-coordinates===

There is a second set of coordinates for the moduli space of twisted &lt;math&gt;n&lt;/math&gt;-gons defined over a [[w:Field (mathematics)|field]] &lt;math&gt;F&lt;/math&gt; satisfying &lt;math&gt;\mathrm{SL}_3(F)\cong \mathbb P\mathrm{GL}_3(F)&lt;/math&gt;,{{Sfn|Weinreich|2022|loc=remark 3.8}} and such that &lt;math&gt;n&lt;/math&gt; is not divisible by &lt;math&gt;3&lt;/math&gt;.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=section 4.1}}

The vertices &lt;math&gt;v_k&lt;/math&gt; in the [[w:projective plane|projective plane]] &lt;math&gt;\mathbb P^2(F)&lt;/math&gt; can be [[w:Lift (mathematics)|lifted]] to [[w:Vector space|vectors]] &lt;math&gt;V_k&lt;/math&gt; in the [[w:affine space|affine space]] &lt;math&gt;F^3&lt;/math&gt; so that each consecutive triple of vectors spans a [[w:parallelepiped|parallelepiped]] having [[w:determinant|determinant]] equal to &lt;math&gt;1&lt;/math&gt;. This leads to the relation defining the &lt;math&gt;ab&lt;/math&gt;-coordinates:{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=equation 4.1}}

: &lt;math&gt;V_{k+3} = a_k V_{k+2} + b_k V_{k+1} + V_k.&lt;/math&gt;

This bring out an analogy between twisted polygons and solutions of third order linear [[w:ordinary differential equations|ordinary differential equations]], normalized to have unit [[w:Wronskian|Wronskian]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=remark 6.6}}

They are linked to the corner coordinates by:{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=lemma 4.5}}
: &lt;math&gt;x_k=\frac{a_{k-2}}{b_{k-2}b_{k-1}},&lt;/math&gt;
: &lt;math&gt;y_k=-\frac{b_{k-1}}{a_{k-2}a_{k-1}}.&lt;/math&gt;

==Formulas on the moduli space==

===As a birational map ===
The pentagram map is a [[w:birational map|birational map]] on the moduli space, because it can be decomposed as the [[w:Function composition|composition]] of two [[w:Birational geometry|birational]] [[w:Involution (mathematics)|involutions]].{{Sfn|Schwartz|2008|loc=§1.2 The Pentagram Map}} The corner invariants change in the following way:{{Sfn|Ovsienko|Schwartz|loc=lemma 2.4|Tabachnikov|2010}}

: &lt;math&gt;x_k'=x_k\frac{1-x_{k-1} y_{k-1}}{1-x_{k+1}y_{k+1}},&lt;/math&gt;

: &lt;math&gt;y_k'=y_{k+1}\frac{1-x_{k+2} y_{k+2}}{1-x_k y_k}.&lt;/math&gt;

=== The scaling symmetry ===
The [[w:multiplicative group|multiplicative group]] &lt;math&gt;F\smallsetminus\{0\}&lt;/math&gt; [[w:One-parameter group|acts]] on the moduli space in the following way:

: &lt;math&gt;R_s\cdot(x_1,\dots,x_n,y_1,\dots,y_n)=(sx_1,\dots,sx_n,s^{-1}y_1,\dots,s^{-1}y_n),&lt;/math&gt;

where &lt;math&gt;R&lt;/math&gt; is called the scaling action and &lt;math&gt;s&lt;/math&gt; is the scaling parameter. This action commutes with the pentagram map on the moduli space (as presented in the previous formulas). This property is called the scaling symmetry, and is instrumental in proving the [[w:Pentagram map#Complete integrability|complete integrability]] of the dynamics.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=corollary 2.5}}

==Invariant structures==

===Monodromy invariants===

The monodromy invariants, introduced in {{Harvard citation|Schwartz|2008}}, are a collection of [[w:Function (mathematics)|functions]] on the [[w:moduli space|moduli space]] that are invariant under the pentagram map.{{Sfn|Schwartz|2008|loc=theorem 1.2}} The simplest example of them are 
:&lt;math&gt; O_n= x_1x_2\cdots x_{n}, \quad E_n = y_1y_2\cdots y_n. &lt;/math&gt;
The other monodromy invariants can be retrieved through different points of view: through the [[w:Pentagram map#The scaling symmetry|scaling symmetry]], as [[w:Combinatorics|combinatorial]] objects, or as some [[w:determinant|determinant]]s.{{Sfn|Schwartz|Tabachnikov|2011|loc=§2 The Monodromy Invariants}} The one involving scaling symmetry is presented here.

Let &lt;math&gt;M\in \mathrm{GL}_3&lt;/math&gt; be a [[w:Lift (mathematics)|lift]] of the monodromy of a twisted &lt;math&gt;n&lt;/math&gt;-gon. The quantities

: &lt;math&gt;\Omega_1=\frac{\operatorname{trace}^3(M)}{\det(M)}, \quad \Omega_2=\frac{\operatorname{trace}^3(M^{-1})}{\det(M^{-1})},&lt;/math&gt;

are independent of the choice of lift and are invariant under [[w:Matrix similarity|conjugation]], so they are well defined for the projective class of the polygon. They are invariant under the pentagram map, since the monodromy matrix doesn't change.{{Sfn|Schwartz|Tabachnikov|2011|loc=|p=5}} Now, the quantities

: &lt;math&gt;\tilde{\Omega}_1=O_n^2E_n\Omega_1, \quad \tilde{\Omega}_2=O_nE_n^2\Omega_2,&lt;/math&gt;

have the same properties, but turn out to be polynomials in the corner invariants.{{Efn|Some papers consider the cube roots of this functions, but it doesn't change the following definitions of the monodromy invariants.}} They can be written as{{Sfn|Schwartz|Tabachnikov|2011|loc=|p=5}}

: &lt;math&gt;
\tilde{\Omega}_1=\biggl(\sum_{k=0}^{\lfloor n/2\rfloor}O_k\biggr)^3, \quad
\tilde{\Omega}_2=\biggl(\sum_{k=0}^{\lfloor n/2\rfloor}E_k\biggr)^3,
&lt;/math&gt;

where each &lt;math&gt;O_k&lt;/math&gt; and &lt;math&gt;E_k&lt;/math&gt; are [[w:homogeneous polynomial|homogeneous polynomial]]s respectively of weight &lt;math&gt;k&lt;/math&gt; and &lt;math&gt;-k&lt;/math&gt;,{{Sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=11}} meaning they change under the [[w:Pentagram map#The scaling symmetry|rescaling action]] on variables by{{Sfn|Schwartz|Tabachnikov|2011|p=5}}

: &lt;math&gt; R_s(O_k)= s^k O_k, \quad R_s(E_k)= s^{-k} E_k. &lt;/math&gt;

The quantities &lt;math&gt;O_1,\dots,O_{\lfloor n/2 \rfloor},O_n, E_1,\dots,E_{\lfloor n/2 \rfloor},E_n,&lt;/math&gt; are unchanged by the dynamics, and are called the monodromy invariants. Moreover, they are [[w:algebraically independent|algebraically independent]].{{Sfn|Schwartz|2008|loc=theorem 1.2}}

==== Polygons on conics ====
Whenever &lt;math&gt;P&lt;/math&gt; is [[w:Inscribed figure|inscribed]] on a [[w:conic section|conic section]], one has &lt;math&gt;O_k(P)=E_k(P)&lt;/math&gt; for all &lt;math&gt;k&lt;/math&gt;.{{Sfn|Schwartz|Tabachnikov|2011|loc=theorem 1.1}} Moreover, if &lt;math&gt;P&lt;/math&gt; is circumscribed about another conic,{{Efn|See the paragraph about [[w:Pentagram map#Poncelet polygons|Poncelet polygons]].}} then its monodromy invariants are characterized by the pair of conics.{{Sfn|Schwartz|2015|loc=theorem 1.2}} For such odd-gons, the translation on the [[w:Jacobian variety|Jacobian variety]]{{Efn|See the paragraph about [[w:Pentagram map#Algebro-geometric integrability|algebraic integrability]].}} is restricted to the [[w:Prym variety|Prym variety]] (which is a half-dimensional torus in the Jacobian).{{Sfn|Izosimov|2016|loc=theorem 1.3}}

===Poisson bracket===

An invariant [[w:Poisson bracket|Poisson bracket]] on the space of twisted polygons was found in {{Harvard citation|Ovsienko|Schwartz|Tabachnikov|2010}}. The monodromy invariants [[w:Poisson bracket#Constants of motion|commute]] with respect to it:
&lt;math display=&quot;block&quot;&gt; \{O_i,O_j\}=\{O_i,E_j\}=\{E_i,E_j\}=0 &lt;/math&gt;for all &lt;math&gt;i,j&lt;/math&gt;.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}}

The Poisson bracket is defined in terms of the corner coordinates by:
&lt;math display=&quot;block&quot;&gt; \begin{align}
 \{x_i,x_{i\pm1}\} &amp;= \mp x_i x_{i+1}, \\
 \{y_i,y_{i\pm 1}\} &amp;= \mp y_i y_{i+1}, \\
 \{x_i,x_j\} &amp;= \{y_i,y_j\} = \{x_i,y_j\} = 0
\end{align}&lt;/math&gt;for all other &lt;math&gt; i,j.&lt;/math&gt;{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=equation 2.16}}

=== The spectral curve ===

Let &lt;math&gt;\zeta&lt;/math&gt; be an element of the [[w:multiplicative group|multiplicative group]] and &lt;math&gt;P_\zeta&lt;/math&gt; be the polygon obtained by applying the [[w:Pentagram map#The scaling symmetry|rescaling action]] &lt;math&gt;R_\zeta&lt;/math&gt; on &lt;math&gt;P&lt;/math&gt;.  A [[w:Lax matrix|Lax matrix]] &lt;math&gt;\hat{T}(\zeta) \in \mathrm{GL}_3&lt;/math&gt; is a lift of the monodromy of &lt;math&gt;P_\zeta&lt;/math&gt; satisfying a [[w:Lax pair#Zero-curvature equation|zero-curvature equation]].{{Sfn|Weinreich|2022|loc=§5 The Lax representation}} Then, the spectral function is the [[w:Bivariate polynomial|bivariate]] [[w:characteristic polynomial|characteristic polynomial]]
&lt;math display=&quot;block&quot;&gt; Q(\lambda,\zeta) := \det(\lambda\operatorname{Id}-\hat{T}(\zeta)),&lt;/math&gt;or some renormalization of it. The [[w:spectral curve|spectral curve]] is the [[w:Projective variety#projective completion|projective completion]] of the [[w:Algebraic curve|affine curve]] defined by the equation &lt;math&gt;Q(\lambda,\zeta)=0&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=§6. The geometry of the spectral curve}} It is invariant under the pentagram map, and the monodromy invariants appear as the [[w:coefficient|coefficient]]s of &lt;math&gt;Q&lt;/math&gt;.{{Sfn|Soloviev|2013|loc=theorem 6.4}} Its [[w:geometric genus|geometric genus]] is &lt;math&gt;n-1&lt;/math&gt; if &lt;math&gt;n&lt;/math&gt; is odd, and &lt;math&gt;n-2&lt;/math&gt; if &lt;math&gt;n&lt;/math&gt; is even.{{Sfn|Weinreich|2022|p=|loc=theorem 6.4}}

It was first introduced in {{Harv|Soloviev|2013|ps=|p=}} for his proof of [[w:Pentagram map#Algebro-geometric integrability|algebro-geometric integrability]].{{sfn|Soloviev|2013}}

==Complete integrability==
The pentagram map on the moduli space has been proved to be a [[w:completely integrable|completely integrable]] [[w:discrete dynamical system|discrete dynamical system]], both in the [[w:Integrable system#Hamiltonian systems and Liouville integrability|Arnold-Liouville]]{{Efn|Over the [[w:real number|real number]]s.}} and the [[w:Integrable system#Complete integrability over the complex numbers|algebro-geometric]]{{Efn|Over [[w:algebraically closed field|algebraically closed field]]s of [[w:Characteristic (algebra)|characteristic]] different from 2.}} senses. In any case, this means that the moduli space is [[w:almost everywhere|almost everywhere]] [[w:Foliation|foliated]] by [[w:Torus#Flat torus|flat tori]] (or in the algebraic setting, [[w:Abelian variety|Abelian varieties]]), where the motion is a [[w:Translation (geometry)|translation]]. This [[w:Generic property|generically]] induces a [[w:quasiperiodic motion|quasiperiodic motion]] on the corresponding torus.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2009}}

===Arnold–Liouville integrability===

The proof of the integrability of the pentagram map on a real twisted polygon was achieved in {{Harvard citation|Ovsienko|Schwartz|Tabachnikov|2010}}. This is done by noticing that the monodromy invariants &lt;math&gt;O_n&lt;/math&gt; and &lt;math&gt;E_n&lt;/math&gt; are [[w:Casimir invariant|Casimir invariant]]s for the bracket, meaning (in this context) that&lt;math display=&quot;block&quot;&gt; \{O_n,f\}=\{E_n,f\} = 0 &lt;/math&gt;for all functions &lt;math&gt;f&lt;/math&gt;.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}} When &lt;math&gt;n&lt;/math&gt; is even, this is also true for the monodromy invariants &lt;math&gt;O_{\lfloor n/2 \rfloor }&lt;/math&gt; and &lt;math&gt;E_{\lfloor n/2 \rfloor }&lt;/math&gt;.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}}

This allows to consider the Casimir [[w:level set|level set]], where each Casimir has a specified value. Because of [[w:Sard's theorem|Sard's theorem]], any generic level set is a [[w:smooth manifold|smooth manifold]].{{Sfn|Schwartz|2017|p=44}} They form a [[w:foliation|foliation]] in [[w:Poisson manifold#Symplectic leaves|symplectic leaves]], on which the Poisson bracket gives rise to a [[w:symplectic form|symplectic form]].{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=corollary 2.13}}

Each of these symplectic leaves has an iso-monodromy [[w:foliation|foliation]], namely, a decomposition into the common level sets of the remaining monodromy functions. By using again [[w:Sard's theorem|Sard's theorem]], they are generically [[w:Symplectic manifold#Lagrangian submanifolds|Lagrangian manifolds]].{{Sfn|Schwartz|2017|p=45}} Moreover, they are compact.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=§3.3. Compactness of the level sets}} Since the monodromy invariants Poisson-commute and there are enough of them, the discrete [[w:Liouville–Arnold theorem|Liouville–Arnold theorem]] can be applied to prove that the level sets are [[w:Torus#Flat torus|flat tori]] over which the dynamics is a translation.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=412}}
===Algebro-geometric integrability===

In {{Harvard citation|Soloviev|2013}}, it was shown that the pentagram map admits a [[w:Lax representation|Lax representation]] with a spectral parameter, which allows to prove its algebro-geometric integrability. This means that the space of polygons (either twisted or closed) is parametrized by its spectral data, consisting of [[w:Pentagram map#The spectral curve|its spectral curve]], with marked points and a [[w:Divisor (algebraic geometry)|divisor]] given by a [[w:Floquet theory|Floquet]]–[[w:Bloch's theorem|Bloch]] equation. This gives an embedding to the [[w:Jacobian variety|Jacobian variety]] through the [[w:Abel–Jacobi map|Abel–Jacobi map]], where the motion is expressed in terms of translation.{{sfn|Soloviev|2013|loc=theorems A, B and C}} The previously defined Poisson bracket is also retrieved.{{sfn|Soloviev|2013|loc=theorem D}} 

This integrability was generalized in {{Harvard citation|Weinreich|2022}} from the field of [[w:complex number|complex number]]s to any [[w:algebraically closed field|algebraically closed field]] of [[w:Characteristic (algebra)|characteristic]] different from 2. The translation on a torus is replaced by a translation on an [[w:Abelian variety|Abelian variety]] (in fact, a Jacobian variety again).{{sfn|Weinreich|2022|loc=theorem 1.4}}

=== Dimension of the invariant manifold ===

For twisted &lt;math&gt;n&lt;/math&gt;-gons, the [[w:dimension|dimension]] of the invariant tori (or Jacobian varieties) is{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=421}}

: &lt;math&gt;\begin{cases} 
n-1 &amp; \text{when }n \text{ is odd,}\\
n-2 &amp; \text{when }n \text{ is even.}
\end{cases}&lt;/math&gt;

Moreover, when &lt;math&gt;n&lt;/math&gt; is even, there are two isomorphic Jacobians on which the iterates of the pentagram map alternate. But on each of them, the second iterate is a translation.{{Sfn|Weinreich|2022|loc=theorem 1.4}}

=== For closed polygons ===
There is no Poisson structure on the space of closed polygons.{{Sfn|Soloviev|2013|loc=corollary 4.1}} Nevertheless, the one from twisted polygons can be used to prove integrability.{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=2153}}

Algebro-geometric integrability holds for closed polygons in a same manner as for the twisted ones.{{Sfn|Soloviev|2013|loc=theorem C}} However, Arnold-Liouville integrability is proved for real closed polygons only when they are convex. This is done by restricting the [[w:Hamiltonian vector field|Hamiltonian vector field]]s of monodromy functions to smaller dimensional tori, and showing that enough of them are still independent.{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|loc=corollary 1.1}}

In both situation, the dimension of the invariant manifolds decreases by &lt;math&gt;3&lt;/math&gt; for closed &lt;math&gt;n&lt;/math&gt;-gons (compared to the twisted case), and is equal to{{Sfn|Soloviev|2013|loc=theorem C}}{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|loc=theorem 1}}
:&lt;math&gt;\begin{cases} 
n-4 &amp; \text{when }n \text{ is odd,}\\
n-5 &amp; \text{when }n \text{ is even.}
\end{cases}&lt;/math&gt;

==Connections to other topics==

===The Boussinesq equation===

The continuous limit of a convex polygon is a parametrized convex curve in the plane. When the time parameter is suitably chosen, the [[w:Discretization|continuous limit]] of the pentagram map is the classical [[w:Boussinesq approximation (water waves)|Boussinesq equation]]. This equation is a classical example of an [[w:integrable|integrable]] [[w:partial differential equation|partial differential equation]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 5}}

Here is a description of the geometric action of the Boussinesq equation. Given a [[w:locally convex|locally convex]] curve &lt;math&gt; C:\mathbb R\to \mathbb R^2 &lt;/math&gt; and real numbers &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;t&lt;/math&gt;, consider the [[w:chord (geometry)|chord]] connecting &lt;math&gt; C(x-t) &lt;/math&gt; to &lt;math&gt; C(x+t) &lt;/math&gt;. The [[w:Envelope (mathematics)|envelope]] of all these chords is a new curve &lt;math&gt; C_t(x) &lt;/math&gt;. When &lt;math&gt;t&lt;/math&gt; is extremely small, the curve &lt;math&gt; C_t(x) &lt;/math&gt; is a good model for the time &lt;math&gt;t&lt;/math&gt; evolution of the original curve &lt;math&gt; C_0(x) &lt;/math&gt; under the Boussinesq equation. This construction is also similar to the pentagram map. Moreover, the pentagram invariant bracket is a discretization of a well known invariant Poisson bracket associated to the Boussinesq equation.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=§6.4 Discretization}}

===Cluster algebras===

The pentagram map{{Sfn|Glick|2011}} and some of its generalizations{{Sfn|Gekhtman|Shapiro|Tabachnikov|Vainshtein|2012}}{{Sfn|Glick|Pylyavskyy|2016}} are identified as special cases of discrete dynamical systems powered by [[w:cluster algebra|cluster algebra]]. Using the results from {{Harvard citation|Goncharov|Kenyon|2013}}, this provides a link with the [[w:Poisson–Lie group|Poisson–Lie group]]s, [[w:dimer model|dimer model]]s and other so-called cluster-integrable systems.{{Sfn|Fock|Marshakov|2016}}{{Sfn|Izosimov|2022c}} These methods allow to retrieve the Poisson-bracket and Hamiltonians used to prove complete integrability{{Sfn|Affolter|George|Ramassamy|2025|loc=§5 The pentagram map}} and provide [[w:Lax representation|Lax representation]]s.{{Sfn|Izosimov|2022b}}

=== Octahedron recurrence ===
Using a method to compute [[w:Determinant|determinants]] called [[w:Dodgson_condensation|Dodgson condensation]], {{Harvard citation|Schwartz|2008}} proves that the pentagram map satisfies a property called the &quot;octahedron recurrence&quot;.{{Sfn|Schwartz|2008|loc=§5 The Method of Condensation}} This property turns out to be shared by other dynamical systems defined by geometric constructions similar to one of the pentagram map.{{Sfn|Affolter|de Tilière|Melotti|2025|loc=§9 The Pentagram Map}} It is also shared by higher dimensional pentagram maps defined through cluster algebras mutations, referred as &lt;math&gt;T&lt;/math&gt;-systems.{{Sfn|Kedem|Vichitkunakorn|2015||loc=}}

=== Singularity theory ===
The pentagram map exhibit a property called singularity confinement, which is typical from [[w:integrable system|integrable system]]s.{{Sfn|Grammaticos|Ramani|Papageorgiou|1991}} It states that if a polygon &lt;math&gt;P&lt;/math&gt; is [[w:Singular point of an algebraic variety|singular]] for the pentagram map &lt;math&gt;T&lt;/math&gt;, then there exists an integer &lt;math&gt;m&lt;/math&gt; such that &lt;math&gt;P&lt;/math&gt; not singular for the iterate map &lt;math&gt;T^m&lt;/math&gt;.{{Sfn|Glick|2012}}

Moreover, the pentagram map (along with some of its generalizations and other discrete dynamical systems) exhibit the Devron property.{{Efn|The name comes from an episode of [[w:Star Trek|Star Trek]].{{Sfn|Glick|2015|loc=§1 Introduction}}}} This means that if a polygon &lt;math&gt;P&lt;/math&gt; is singular for some iterate of the pentagram map &lt;math&gt;T^m&lt;/math&gt;, then it will also be singular for some iterate of the inverse map &lt;math&gt;T^{-m'}&lt;/math&gt;.{{Sfn|Glick|2015}}

== Generalizations ==
The definition of twisted polygons still makes sense in any [[w:projective space|projective space]] &lt;math&gt;\mathbb P^d&lt;/math&gt;, under the action of the [[w:Projective linear group|projective group]] &lt;math&gt;\mathbb P \mathrm{GL}_{d+1}&lt;/math&gt;. The pentagram map can be generalized in many ways, and some of them are presented here. Not all of them are integrable.{{Sfn|Khesin|Soloviev|2015|}} Some are [[w:discretization|discretization]]s of [[w:PDEs|PDEs]] from the [[w:KdV hierarchy|KdV hierarchy]], seen as higher dimensional version of [[w:Boussinesq approximation (water waves)|Boussinesq]] or [[w:Kadomtsev–Petviashvili equation|KP]] equations.{{Sfn|Marí-Beffa|2012}}{{Sfn|Wang|2023}} The description of all generalized pentagram maps in terms of [[w:cluster algebra|cluster algebra]]s is still an open question.{{Sfn|Gekhtman|Izosimov|2025|p=14}}

=== Polygons in general positions ===
Let &lt;math&gt;d \geq 2&lt;/math&gt; and &lt;math&gt;P&lt;/math&gt; be a twisted polygon of &lt;math&gt;\mathbb P^d&lt;/math&gt; in [[w:general position|general position]].

==== Short diagonal pentagram maps ====
The &lt;math&gt;k&lt;/math&gt;-th ''short diagonal hyperplane'' &lt;math&gt;H_k^{sh}&lt;/math&gt; is uniquely defined by passing through the vertices &lt;math&gt;v_k,v_{k+2},\dots,v_{k+2d-2}&lt;/math&gt;. [[w:Generic property#In algebraic geometry|Generically]], the intersection of &lt;math&gt;d&lt;/math&gt; consecutive hyperplanes uniquely defines a new point

: &lt;math&gt;T_{sh}v_k:=H_k^{sh}\cap H_{k+1}^{sh}\cap \dots \cap H_{k+d-1}^{sh}.&lt;/math&gt;

Doing this for every vertex defines a new twisted polygon. This map, denoted by &lt;math&gt;T_{sh}&lt;/math&gt;, is again projectively equivariant.{{Sfn|Khesin|Soloviev|2013}}

==== Generalized pentagram maps ====
The previous procedure can be generalized. Let &lt;math&gt;I=(i_1,\dots,i_{d-1}),~J=(j_1,\dots,j_{d-1})&lt;/math&gt; be two sets of integers, respectively called the jump tuple and the intersection tuple. Define the &lt;math&gt;k&lt;/math&gt;-th hyperplane &lt;math&gt;H_k^I&lt;/math&gt; to be passing through the vertices &lt;math&gt;v_k,v_{k+i_1},\dots,v_{k+i_1+\dots+i_{d-1}}&lt;/math&gt;. A new point is given by the intersection

: &lt;math&gt;T_{I,J}v_k:=H_k^I \cap H_{k+j_1}^I \cap \dots \cap H_{k+j_1+\dots +j_{d-1}}^I.&lt;/math&gt;

The map &lt;math&gt;T_{I,J}&lt;/math&gt; is called a generalized pentagram map.{{Sfn|Khesin|Soloviev|2015a}} The original pentagram map is recovered by considering&lt;math&gt;d=2,~I=(2),~J=(1)&lt;/math&gt;.

Integrability can be numerically tested by picking a random polygon &lt;math&gt;P&lt;/math&gt; with [[wikipedia:Rational_point|rational coordinates]] and studying the growth rate of the [[wikipedia:Height_function|height]] of its iterates. This is called the [[wikipedia:Integrable_system#Diophantine_integrability|diophantine integrability]] test, and some generalized pentagram maps don't seem to pass it.{{Sfn|Khesin|Soloviev|2015a|loc=§5 and §6}} However, it is conjectured that the maps &lt;math&gt;T_{I,I}&lt;/math&gt; are integrable for any &lt;math&gt;I&lt;/math&gt;.{{Sfn|Bolsinov|Matveev|Miranda|Tabachnikov|2018|loc=conjecture 4.13 (B. Khesin, F. Soloviev)}}

Some of these maps are [[w:discretization|discretization]]s of higher dimensional counterpart of the [[w:Boussinesq approximation (water waves)|Boussinesq equation]] in the [[w:KdV hierarchy|KdV hierarchy]].{{Sfn|Khesin|Soloviev|2015b|loc=theorem 4.1}}{{Sfn|Izosimov|2022b|loc=theorem 4.1}}

==== Dented pentagram maps ====
Fix an integer &lt;math&gt;m\in \{1,\dots ,d-1\}&lt;/math&gt;. Consider the jump tuple &lt;math&gt;I_m:=(1,\dots,1,2,1,\dots,1)&lt;/math&gt;, where the &lt;math&gt;2&lt;/math&gt; is at the &lt;math&gt;m&lt;/math&gt;-th place, and the intersection tuple &lt;math&gt;J:=(1,\dots,1)&lt;/math&gt;. The dented pentagram map is &lt;math&gt;T_m :=T_{I_m,J}&lt;/math&gt;. They are proved to be integrable.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 2.14}}

For an integer &lt;math&gt;p \geq 2&lt;/math&gt;, the deep dented pentagram map (of depth &lt;math&gt;p&lt;/math&gt;) &lt;math&gt;T_m^p&lt;/math&gt; is the same map as before, but the number &lt;math&gt;2&lt;/math&gt; in the definition of &lt;math&gt;I_m&lt;/math&gt; is replaced by &lt;math&gt;p&lt;/math&gt;. This kind of pentagram maps are again integrable.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 6.2}}

=== Corrugated polygons ===
A twisted polygon &lt;math&gt;P&lt;/math&gt; lying in &lt;math&gt;\mathbb P^d&lt;/math&gt; is said to be corrugated if for any &lt;math&gt;k\in \mathbb Z&lt;/math&gt;, the vertices &lt;math&gt;v_k,v_{k+1},v_{k+d},v_{k+d+1}&lt;/math&gt; span a projective two-dimensional plane. Such polygons are not in [[w:general position|general position]]. A new point is defined by

: &lt;math&gt;T_\text{cor}v_k:=\overline{v_k v_{k+d}}\cap \overline{v_{k+1} v_{k+d+1}}.&lt;/math&gt;

The map &lt;math&gt;T_\text{cor}&lt;/math&gt; yields a new corrugated polygon. They are [[w:Integrable system#Hamiltonian systems and Liouville integrability|completely Liouville-integrable]].{{Sfn|Gekhtman|Shapiro|Tabachnikov|Vainshtein|2012|loc=theorem 4.4}}

In fact, they can be retrieved as some dented pentagram map applied on corrugated polygons.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 5.3}}

=== Grassmannian polygons ===
Let &lt;math&gt;d \geq 3, m \geq 1&lt;/math&gt; be integers. The pentagram map can also be generalized to the [[w:Grassmannian|Grassmannian]] space &lt;math&gt;\mathrm{Gr}(m,md)&lt;/math&gt;, which consists of &lt;math&gt;m&lt;/math&gt;-[[w:Dimension (vector space)|dimensional]] [[w:linear subspace|linear subspace]]s of an &lt;math&gt;md&lt;/math&gt;-dimensional [[w:vector space|vector space]]. When &lt;math&gt;m=1&lt;/math&gt;, the linear subspaces are [[w:Vector space#vector line|lines]], which retrieves the definition of [[w:projective space|projective space]]s &lt;math&gt;\mathbb P^d&lt;/math&gt;.{{Sfn|Felipe|Marí-Beffa|2019|loc=§2 definitions and notations}}

A point &lt;math&gt;v\in\operatorname{Gr}(m,md)&lt;/math&gt; is represented by an &lt;math&gt;md \times m&lt;/math&gt; matrix &lt;math&gt;X_v&lt;/math&gt; such that its columns form a [[w:Basis (linear algebra)|basis]] of &lt;math&gt;v&lt;/math&gt;. Consider the [[w:Group action|action]] of the [[w:general linear group|general linear group]] &lt;math&gt;\mathrm{GL}_{md}&lt;/math&gt; by multiplication on the left of &lt;math&gt;X_v&lt;/math&gt;. This defines an action on the Grassmannian, even though it is not [[w:Faithful action|faithful]].{{Efn|Because there can be many lifts for &lt;math&gt;v&lt;/math&gt;, and because some matrices act trivially.}} Hence, the polygons of &lt;math&gt;\mathrm{Gr}(m,md)&lt;/math&gt; and their moduli spaces are defined as before, after the change of underlying group.{{Sfn|Felipe|Marí-Beffa|2019|loc=§2 definitions and notations}}

Depending on the parity of &lt;math&gt;d&lt;/math&gt;, one can define linear subspaces spanned by some &lt;math&gt;X_{v_k}&lt;/math&gt;'s such that taking their intersection generically defines a new point &lt;math&gt;v\in\mathrm{Gr}(m,md)&lt;/math&gt;.{{Sfn|Felipe|Marí-Beffa|2019|loc=sections 4 and 5}} This generalization of the pentagram map is integrable in a [[w:noncommutative|noncommutative]] sense.{{Sfn|Ovenhouse|2020}}

=== Over rings ===
The pentagram map admits a generalization by considering [[w:Projective space#Generalizations|projective planes]] over [[w:stably finite ring|stably finite ring]]s, instead of [[w:Field (mathematics)|field]]s. In particular, this retrieves the pentagram map over Grassmannians. Again, it admits a [[w:Lax representation|Lax representation]].{{Sfn|Hand|Izosimov|2025}}

== References ==
{{reflist|25em}}

===Notes===
{{notelist}}

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*{{Cite journal|ref=harv |last=Schwartz|first=Richard|date=2026-02-14|title=The Flapping Birds in the Pentagram Zoo|url=https://armj.math.stonybrook.edu/Articles/241224-Schwartz/index.html|journal=Arnold Mathematical Journal|volume=011|issue=004|pages=10|doi=10.56994/ARMJ.011.004.002|issn=2199-6792}}
*{{Cite journal |ref=harv |title=Elementary Surprises in Projective Geometry |url=http://link.springer.com/10.1007/s00283-010-9137-8 |journal=The Mathematical Intelligencer |date=2010 |issn=0343-6993 |pages=31–34 |volume=32 |issue=3 |doi=10.1007/s00283-010-9137-8 |language=en |first1=Richard Evan |last1=Schwartz |first2=Serge |last2=Tabachnikov |hdl=21.11116/0000-0004-24EE-8 }}
*{{Cite journal|ref=harv |title=The Pentagram Integrals on Inscribed Polygons|url=https://www.combinatorics.org/ojs/index.php/eljc/article/view/v18i1p171|journal=The Electronic Journal of Combinatorics|date=2011-09-02|issn=1077-8926|volume=18|issue=1|doi=10.37236/658|first1=Richard Evan|last1=Schwartz|first2=Serge|last2=Tabachnikov |article-number=P171 }}
*{{Citation |last=Soloviev |first=Fedor |title=Integrability of the pentagram map |date=1 December 2013 |journal=[[w:Duke Mathematical Journal|Duke Mathematical Journal]] |volume=162 |issue=15 |pages=2815–2853 |doi=10.1215/00127094-2382228 |arxiv=1106.3950 |url=https://doi.org/10.1215/00127094-2382228}}
*{{Cite journal |ref=harv |last=Tabachnikov |first=Serge |date=2019-05-07 |title=Kasner Meets Poncelet |url=https://doi.org/10.1007/s00283-019-09897-5 |journal=The Mathematical Intelligencer |volume=41 |issue=4 |pages=56–59 |doi=10.1007/s00283-019-09897-5 |arxiv=1707.09267 |issn=0343-6993}}
*{{Cite journal |ref=harv |last=Tupan |first=Alexandru |date=2022-07-03 |title=Pentagram Configurations for Pentagons and Hexagons |url=https://www.tandfonline.com/doi/full/10.1080/00029890.2022.2060695 |journal=The American Mathematical Monthly |language=en |volume=129 |issue=6 |pages=554–565 |doi=10.1080/00029890.2022.2060695 |issn=0002-9890}}
*{{Cite journal|ref=harv |title=Pentagram-Type Maps and the Discrete KP Equation|url=https://link.springer.com/10.1007/s00332-023-09961-7|journal=Journal of Nonlinear Science|date=2023|issn=0938-8974|volume=33|issue=6|doi=10.1007/s00332-023-09961-7|language=en|first=Bao|last=Wang |article-number=101 |bibcode=2023JNS....33..101W }}
*{{Cite journal |ref=harv |title=The algebraic dynamics of the pentagram map |journal=Ergodic Theory and Dynamical Systems |date=2022-11-25 |issn=0143-3857 |pages=3460–3505 |volume=43 |issue=10 |doi=10.1017/etds.2022.82 |first=Max H. |last=Weinreich}}</text>
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      <text bytes="61065" sha1="9filxapi1spqj27m2t3x41zyy57b7de" xml:space="preserve">{{Article info
| last1 = Stiegler
| orcid1 = 0009-0001-5789-6923
| first1 = Jean-Baptiste
| affiliation1 = Université Paris-Saclay
| correspondence1 = jean-baptiste.stiegler@universite-paris-saclay.fr
| journal = WikiJournal of Science
| et_al = true
| w1 = Pentagram map
| from w1 = true
| keywords = Pentagram map, Dynamical system, Projective geometry, Moduli space, Integrable systems
| license = CC-BY-SA 4.0
| submitted = 2025-12-08
| abstract = In [[w:mathematics|mathematics]], the '''pentagram map''' is a [[w:Dynamical system#Discrete dynamical system|discrete dynamical system]] acting on [[w:polygons|polygons]] in the [[w:projective plane|projective plane]]. It defines a new polygon whose vertices are obtained as the intersection points of the shortest [[w:Diagonal|diagonals]] of the initial polygon. This is a [[w:Projective linear group|projectively]] [[w:Equivariant map|equivariant]] procedure, hence it [[w:Quotient space (topology)|descends]] to the [[w:moduli space|moduli space]] of polygons and defines another dynamical system (which is also referred to as the pentagram map). It was first introduced by [[w:Richard Schwartz (mathematician)|Richard Schwartz]] in 1992.{{Sfn|Schwartz|1992}}

The pentagram map on the moduli space is famous for its [[w:Completely integrable|complete integrability]] and its link with [[w:cluster algebra|cluster algebras]].{{sfn|Gekhtman|Izosimov|2025|p=14}}

It admits many generalizations in [[w:Projective space|projective spaces]] and other settings.
}}
== Introduction ==

=== Informal definition ===

==== On polygons ====
[[File:Pentagram pentagon nolabel big.svg|alt=|thumb|300x300px|The pentagram map applied on a [[w:Convex set|convex]] [[w:pentagon|pentagon]].]]
Initially, the pentagram map was defined for [[w:convex polygon|convex polygon]]s (with at least five sides) on the [[w:euclidean plane|Euclidean plane]]. Given such a polygon &lt;math&gt;P&lt;/math&gt; with &lt;math&gt;n&lt;/math&gt; sides, one can draw the &quot;shortest [[w:diagonal|diagonal]]s&quot;, meaning the [[w:Line segment|segments]] whose endpoints are a [[w:Vertex (geometry)|vertex]] and one of its second neighbors (as in Figure 1). The intersections of the shortest diagonals are then taken as the vertices of a new &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;T(P)&lt;/math&gt;; this new polygon is the output of the pentagram map.{{Sfn|Berger|2005}}

The same construction can be done on [[w:Concave polygon|non-convex polygons]], but there are several complications. First, some consecutive short diagonals may not intersect, so one must extend the segments to [[w:Line (geometry)|lines]]. Second, the image &lt;math&gt;T(P)&lt;/math&gt; can fail to be a new &lt;math&gt;n&lt;/math&gt;-gon because some consecutive vertices could coincide. However, this [[w:Generic property|generically]] doesn't happen.{{Sfn|Ovsienko|Schwartz|Tabachnikov|p=411|2009}} Finally, it is possible that two diagonals are [[w:Parallel (geometry)|parallel]] and don't intersect on the [[w:euclidean plane|Euclidean plane]]. This is resolved by extending the Euclidean plane to the [[w:real projective plane|real projective plane]] by the addition of a [[w:line at infinity|line at infinity]], where the [[w:Vanishing point|intersection point]] lies (see Figure 3). Hence, the pentagram map is defined for generic polygons in the real projective plane.{{Sfn|Berger|2005|p=25}}

More generally, the construction of the pentagram map is well defined whenever the concepts of lines and their intersections make sense. This is encompassed by the notion of a general [[w:projective plane|projective plane]], of which the real projective plane is one example; but the pentagram map can also be considered over other [[w:Field (mathematics)|fields]], for instance the [[w:complex number|complex number]]s, which give the [[w:complex projective plane|complex projective plane]].{{Sfn|Weinreich|2022|loc=§3.1.1}}

==== On the moduli space of polygons ====
Since the pentagram map is constructed by drawing lines and marking their intersections, it [[w:Commutative property|commutes]] with any transformation that sends lines to lines. Such maps are called [[w:projective transformations|projective transformations]]. Hence, polygons can be identified [[w:up to|up to]] [[w:Perspectivity#Projectivity|projective transformations]]. This identification gives the [[w:Quotient space (topology)|quotient space]] (technically called a [[w:moduli space|moduli space]]) of [[w:Equivalence class|classes]] of polygons. 

The pentagram map on polygons induces another dynamical system on the moduli space,{{Sfn|Schwartz|1992|loc=§1 Projective geometry}} whose behavior differs quite a lot from the initial one.{{Efn|Compare the paragraph about the [[w:Pentagram map#Collapsing of convex polygons|collapsing of convex polygons]] and the one about [[w:Pentagram map#Complete integrability|complete integrability]].}} The dynamic is trivial for the classes of pentagons and hexagons, but this is no longer the case for polygons with more vertices.{{Efn|See the paragraph about [[w:Pentagram_map#Pentagons_and_hexagons|pentagons and hexagons]].}}

=== Historical elements ===
The pentagram map for general polygons was introduced in {{Harvard citation|Schwartz|1992}}, but the simplest case is the one of [[w:pentagons|pentagons]], hence the name &quot;[[w:pentagram|pentagram]]&quot;.{{Sfn|Marí-Beffa|2014|p=1}} Their study goes back to {{Harvard citation|Clebsch|1871}},{{Sfn|Izosimov|2022a|p=1085}} {{Harvard citation|Kasner|1928}}{{Sfn|Tabachnikov|2019}} and {{Harvard citation|Motzkin|1945}}.{{Sfn|Schwartz|2013|p=1}}

The pentagram map interacts with some classical configuration theorems of [[w:projective geometry|projective geometry]]. It provides results analogous to the ones of [[w:Pascal's theorem|Pascal's theorem]] and [[wikipedia:Brianchon's_theorem|Brianchon's theorem]].{{Sfn|Schwartz|Tabachnikov|2010}} Some specific configurations make [[w:Desargues' theorem|Desargues's theorem]] and [[w:Poncelet's porism|Poncelet's porism]] appear.{{Sfn|Berger|2005|loc=§4 and §5}}{{Efn|See the paragraph about [[w:Pentagram map#Poncelet polygons|Poncelet polygons]].}}

==Definitions and first properties==

=== Definition of the map ===
[[File:Pentagram pentagon label big.svg|alt=|thumb|300x300px|The pentagram map on a convex pentagon, with vertices labeled.]]
[[File:Pentagram on nonconvex pentagon.svg|alt=|thumb|300x300px|The pentagram map applied on a [[w:Self-intersecting polygon|self-intersecting]] (in particular, non-convex) pentagon. The vertex &lt;math&gt;w_2&lt;/math&gt; is on the [[w:line at infinity|line at infinity]], because it is the [[w:Vanishing point|intersection of two parallel lines]].]]
Let &lt;math&gt;n\geq 5&lt;/math&gt; be an integer. A polygon &lt;math&gt;P&lt;/math&gt; with &lt;math&gt;n&lt;/math&gt; sides, or &lt;math&gt;n&lt;/math&gt;-gon, is a tuple of [[w:Vertex (geometry)|vertices]] &lt;math&gt;(v_1,\dots,v_n)&lt;/math&gt; lying in some [[w:projective plane|projective plane]] &lt;math&gt;\mathbb P ^2&lt;/math&gt;,{{Efn|In the following, the figures represent polygons on the real plane, where the intuition is easier to grasp.}} where the indices are understood [[w:Modular arithmetic|modulo]] &lt;math&gt;n&lt;/math&gt;. The [[w:Dimension of an algebraic variety|dimension]] of the space of &lt;math&gt;n&lt;/math&gt;-gons is &lt;math&gt;2n&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=definition 1.1}}

Suppose that the vertices are in sufficiently [[w:general position|general position]], meaning that no consecutive triple of points are [[w:Collinearity|collinear]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=1}} Taking the intersection of two consecutive &quot;shortest&quot; [[w:diagonal|diagonal]]s{{Efn|Meaning the line between a vertex &lt;math&gt;v_k&lt;/math&gt; and a &quot;second neighbour&quot; &lt;math&gt;v_{k\pm 2}&lt;/math&gt;.}} defines a new point&lt;math display=&quot;block&quot;&gt; w_k := \overline{v_{k-1} v_{k+1}} \cap \overline{v_{k} v_{k+2}}. &lt;/math&gt;This procedure defines a new &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;T(P)=(w_1,\dots,w_n)&lt;/math&gt;, as in Figure 2.{{Sfn|Schwartz|1992|p=71}}

The labeling of the indices of &lt;math&gt;T(P)&lt;/math&gt; is not [[w:canonical|canonical]]. In most papers, a choice is made at the beginning of the paper and the formulas are tuned accordingly.{{Sfn|Izosimov|2016|loc=remark 1.5}}

The pentagram map on polygons is a [[w:birational map|birational map]] &lt;math&gt;T:(\mathbb P^2)^n&lt;/math&gt;{{nowrap|{{font|size=145%|⇢}}}}&lt;math&gt;(\mathbb P^2)^n&lt;/math&gt;. Indeed, each [[w:Homogeneous coordinates|coordinate]] of &lt;math&gt;w_k&lt;/math&gt; is given as a [[w:rational function|rational function]] of the coordinates of &lt;math&gt;v_{k-1},\dots,v_{k+2}&lt;/math&gt;, since it is defined as the intersection of lines passing by them. Moreover, the [[w:inverse map|inverse map]] is given by taking the intersections &lt;math&gt;\overline{w_{k-2} w_{k-1}} \cap \overline{w_{k} w_{k+1}} &lt;/math&gt;, which is rational for the same reason.{{Sfnp|Weinreich|2022|loc=definition 1.2}}

=== Moduli space ===
The pentagram map is defined by taking [[w:Line (geometry)|lines]] and intersections of them. The biggest [[w:Group (mathematics)|group]] which maps lines to lines is the one of [[w:projective transformations|projective transformations]], denoted by &lt;math&gt;\mathbb P \mathrm{GL}_{3}&lt;/math&gt;. Such a transformation &lt;math&gt;M&lt;/math&gt; [[w:Group action|acts]] on a polygon &lt;math&gt;P&lt;/math&gt; by sending it to &lt;math&gt;M \cdot P:=(Mv_1,\dots,Mv_n)&lt;/math&gt;. The pentagram map [[w:Commutative property|commutes]] with this action, and thereby induces another [[w:dynamical system|dynamical system]] on the [[w:moduli space|moduli space]] of projective [[w:equivalence classes|equivalence classes]] of polygons, whose [[w:Dimension of an algebraic variety|dimension]] is &lt;math&gt;2n-8&lt;/math&gt;.{{Sfn|Schwartz|1992|loc=§1 Projective geometry}}

===Twisted polygons===
[[File:Twisted heptagon.svg|alt=|thumb|300x300px|An example of twisted [[w:heptagon|heptagon]] on the real plane.]]
The pentagram map naturally generalizes to the larger space of twisted polygons (see example in Figure 4). For any integer &lt;math&gt;n\geq5&lt;/math&gt;, a twisted &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;P&lt;/math&gt; is the data of:

* a [[w:Sequence#Indexing|bi-infinite sequence]] of points &lt;math&gt;(v_k)_{k\in\mathbb Z}&lt;/math&gt; in the projective plane (called the vertices),
* a [[w:projective transformation|projective transformation]] &lt;math&gt;M \in \mathbb P \mathrm{GL}_3&lt;/math&gt; (called the [[w:monodromy|monodromy]]),

such that for any &lt;math&gt;k \in \mathbb Z&lt;/math&gt;, the property &lt;math&gt;v_{k+n}=Mv_k&lt;/math&gt; is satisfied. The dimension of the space of twisted &lt;math&gt;n&lt;/math&gt;-gons is &lt;math&gt;2n+8&lt;/math&gt;.{{Sfn|Schwartz|2008}}

When &lt;math&gt;M&lt;/math&gt; is the [[w:Identity_element|identity]], this gives back the initial definition of polygons (which are said to be closed). The space of closed &lt;math&gt;n&lt;/math&gt;-gons is of [[w:codimension|codimension]] &lt;math&gt;8&lt;/math&gt; in the space of twisted ones.{{Sfn|Soloviev|2013|p=2816}}

The action of projective transformations over the space of closed polygons generalizes to the space of twisted ones (the monodromy is changed by [[w:Matrix similarity|conjugation]]). This provides again a moduli space, of dimension &lt;math&gt;2n&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=definition 1.3}}

== Collapsing of convex polygons ==

=== Exponential shrinking ===
[[File:Pentagram map convex heptagon iterate.svg|alt=|thumb|300x300px|The pentagram map iterated on a convex [[w:heptagon|heptagon]], exhibiting the convergence.]]
Let &lt;math&gt;P&lt;/math&gt; be a closed [[w:Convex polygon#Strictly convex polygon|strictly convex polygon]] lying on the real plane. One of the first results proved by Richard Schwartz it that its iterates under the pentagram map shrink [[w:Exponential growth|exponentially fast]] to a point, as illustrated in Figure 5. This follows from two facts.
# The image of a strictly convex polygon is contained in its [[w:Interior (topology)|interior]], and is also strictly convex.{{Sfn|Glick|2020|p=2818}}
# There exists a constant &lt;math&gt;0&lt; \eta_P&lt;1&lt;/math&gt;, depending on &lt;math&gt;P&lt;/math&gt;, such that for any &lt;math&gt;N \in \mathbb N&lt;/math&gt;, the diameters of the iterates verify the inequality &lt;math display=&quot;inline&quot;&gt;\operatorname{diam}(T^N(P))\leq\eta_P^N \operatorname{diam}(P). &lt;/math&gt;{{Sfn|Schwartz|1992|loc=theorem 3.1}}

Hence, by [[w:Cantor's intersection theorem#Variant in complete metric spaces|Cantor's intersection theorem]], the sequence of polygons collapses toward a point.{{Sfn|Schwartz|1992|loc=§3 Convex polygons}}

The behavior on the moduli space is very different, since the dynamics is [[w:Recurrent point|recurrent]].{{Sfn|Schwartz|2001|loc=theorem 1.1}} It is even a [[w:quasiperiodic motion|quasiperiodic motion]],{{Sfn|Ovsienko|Schwartz|Tabachnikov|2009}} as discussed in [[w:Pentagram map#Complete integrability|the section about integrability]].

=== Coordinates of the limit point ===
The limit point coordinates were given in {{Harvard citation|Glick|2020}}. They satisfy some [[w:Degree of a polynomial|degree]] 3 [[w:polynomial equation|polynomial equations]], whose coefficients are [[w:rational function|rational function]]s in the coordinates of the vertices of the starting polygon. The proof relies on the fact that the limit point must be an [[w:eigenline|eigenline]] of a certain [[w:linear operator|linear operator]] of &lt;math&gt;\mathbb R^3&lt;/math&gt;.{{sfn|Glick|2020}}

This operator was reinterpreted in {{Harvard citation|Aboud|Izosimov|2022}} as the infinitesimal monodromy of the polygon. The [[w:Pentagram map#The scaling symmetry|scaling symmetry]] is used to [[w:Deformation (mathematics)|deform]] a closed polygon &lt;math&gt;P&lt;/math&gt; into a family of twisted ones '''&lt;math&gt;(P_z)_{z\in \mathbb C^*}&lt;/math&gt;''' with monodromy &lt;math&gt;M_z&lt;/math&gt;. The infinitesimal monodromy is defined to be:{{sfn|Aboud|Izosimov|2022}}
&lt;math display=&quot;block&quot;&gt;\left.\frac{dM_z}{dz}\right|_{z=1}.&lt;/math&gt;

=== Generalization ===
The collapsing of polygons may also happen in some [[w:Pentagram map#Generalizations|generalization of the pentagram map]], when considering some specific configurations of polygons in the real plane. The coordinates of the collapse point are given by a formula analogous to the one for the original pentagram map.{{Sfn|Schwartz|2026}}

== Periodic orbits on the moduli space ==
For some configurations of closed polygons, the iterate of the pentagram map will send &lt;math&gt;P&lt;/math&gt; to a projectively equivalent polygon (up to some shift of the indices). This means that, on the moduli space, the orbit of the class of &lt;math&gt;P&lt;/math&gt; is [[w:Periodic orbit|periodic]].

===Pentagons and hexagons===
[[File:penta hexagon.svg|300px|thumb|The outward hexagon is projectively equivalent to the inward one, with respect to their labeling.]]The following two facts are proved by checking [[w:cross-ratio|cross-ratio]] equalities, so they are true for polygons in any [[w:projective plane|projective plane]] (not just the [[w:Real projective plane|real one]]).{{Sfn|Schwartz|1992|loc=§2 Pentagons and hexagons}}

The pentagram map &lt;math&gt;T&lt;/math&gt; is the identity on the moduli space of [[w:pentagon|pentagon]]s.{{Sfn|Schwartz|1992|loc=theorem 2.1}}{{Sfn|Clebsch|1871}}{{Sfn|Motzkin|1945}} The second iterate &lt;math&gt;T^2&lt;/math&gt; is the identity on the space of labeled [[w:hexagon|hexagon]]s, up to a shift of labeling (see Figure 6).{{Sfn|Schwartz|1992|loc=theorem 2.3}} This phenomenon doesn't generalize to generic polygons with at least seven sides, for which the motion is [[w:Quasiperiodic motion|quasiperiodic]].{{Sfn|Tupan|2022}}

==== Generalization ====
The result about pentagons and hexagons generalizes to some [[w:Pentagram map#Generalizations|higher pentagram maps]] in &lt;math&gt;\mathbb P ^k&lt;/math&gt;, for polygons with &lt;math&gt;k+3&lt;/math&gt; or &lt;math&gt;2k+2&lt;/math&gt; sides. The proof uses a generalization of the [[w:Gale transform|Gale transform]].{{Sfn|Dirdak|2024}}

=== Poncelet polygons ===
A polygon is said to be Poncelet{{Efn|The name comes from [[w:Jean-Victor Poncelet|Jean-Victor Poncelet]] and [[w:Poncelet porism|his porism]].{{Sfn|Izosimov|2022a|p=1085}}}} if it is [[w:Inscribed figure|inscribed]] in a [[w:Conic section|conic]] and circumscribed about another one.{{Sfn|Schwartz|2015|loc=|p=433}}{{Efn|In particular, pentagons are Poncelet since [[w:five points determine a conic|five points determine a conic]].{{Sfn|Schwartz|2015|loc=|p=433}}}} For a convex Poncelet &lt;math&gt;n&lt;/math&gt;-gon &lt;math&gt;P&lt;/math&gt; lying on the [[w:real projective plane|real projective plane]], the polygon &lt;math&gt;T^2(P)&lt;/math&gt; is projectively equivalent to &lt;math&gt;P&lt;/math&gt;.{{Sfn|Schwartz|2015|loc=theorem 1.1}} In fact, when &lt;math&gt;n&lt;/math&gt; is odd, the converse is also true.{{Sfn|Izosimov|2022a|loc=corollary 1.1}}

However, this converse statement is no longer true when the polygons are considered over the [[w:complex projective plane|complex projective plane]].{{Sfn|Izosimov|2022a|loc=remark 1.3}}

==Coordinates for the moduli space==
The moduli space can be described by different [[w:Coordinate_system|coordinate systems]]. The following ones give simple expressions for the dynamics, as presented in the next section.

=== Corner coordinates ===
[[File:Corner coordinates big.svg|thumb|300x300px|The geometric construction of the points defining the corner invariants.]]
Define the [[w:cross-ratio|cross-ratio]] of four [[w:Collinearity|collinear]] points to be

: &lt;math&gt; [a,b,c,d]=\frac{(a-b)(c-d)}{(a-c)(b-d)}. &lt;/math&gt;

The corner invariants are a system of coordinates on the space of twisted polygons, constructed by taking intersections as in Figure 7.{{Sfn|Schwartz|2001|loc=figure 2}} The left and right invariants are respectively defined{{Efn|The ordering of the vertices in the cross-ratios can differ from a paper to another one, which slightly changes the formulas in the following sections.}} as the following cross-ratios:

: &lt;math&gt;x_k:=[v_{k-2},v_{k-1},\overline{v_{k-2}v_{k-1}}\cap\overline{v_{k}v_{k+1}},\overline{v_{k-2}v_{k-1}}\cap\overline{v_{k+1}v_{k+2}}],&lt;/math&gt;

: &lt;math&gt;y_k:=[\overline{v_{k+1}v_{k+2}}\cap\overline{v_{k-2}v_{k-1}}, \overline{v_{k+1}v_{k+2}}\cap\overline{v_{k-1}v_{k}},v_{k+1},v_{k+2}].&lt;/math&gt;

Since the cross-ratio is [[w:Cross-ratio#Projective geometry|projective invariant]], the sequences &lt;math&gt;(x_k)_{k \in \mathbb Z}&lt;/math&gt; and &lt;math&gt;(y_k)_{k \in \mathbb Z}&lt;/math&gt; associated to a twisted &lt;math&gt;n&lt;/math&gt;-gon are &lt;math&gt;n&lt;/math&gt;-periodic.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=415}}

The corner invariants are elements of &lt;math&gt;\mathbb{P}^1\smallsetminus\{0,1,\infty\}&lt;/math&gt;, and they realize an [[w:Isomorphism_of_varieties|isomorphism of varieties]] between the moduli space of twisted &lt;math&gt;n&lt;/math&gt;-gons and &lt;math&gt;(\mathbb{P}^1\smallsetminus\{0,1,\infty\})^{2n}&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=theorem 3.6}}

===ab-coordinates===

There is a second set of coordinates for the moduli space of twisted &lt;math&gt;n&lt;/math&gt;-gons defined over a [[w:Field (mathematics)|field]] &lt;math&gt;F&lt;/math&gt; satisfying &lt;math&gt;\mathrm{SL}_3(F)\cong \mathbb P\mathrm{GL}_3(F)&lt;/math&gt;,{{Sfn|Weinreich|2022|loc=remark 3.8}} and such that &lt;math&gt;n&lt;/math&gt; is not divisible by &lt;math&gt;3&lt;/math&gt;.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=section 4.1}}

The vertices &lt;math&gt;v_k&lt;/math&gt; in the [[w:projective plane|projective plane]] &lt;math&gt;\mathbb P^2(F)&lt;/math&gt; can be [[w:Lift (mathematics)|lifted]] to [[w:Vector space|vectors]] &lt;math&gt;V_k&lt;/math&gt; in the [[w:affine space|affine space]] &lt;math&gt;F^3&lt;/math&gt; so that each consecutive triple of vectors spans a [[w:parallelepiped|parallelepiped]] having [[w:determinant|determinant]] equal to &lt;math&gt;1&lt;/math&gt;. This leads to the relation defining the &lt;math&gt;ab&lt;/math&gt;-coordinates:{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=equation 4.1}}

: &lt;math&gt;V_{k+3} = a_k V_{k+2} + b_k V_{k+1} + V_k.&lt;/math&gt;

This bring out an analogy between twisted polygons and solutions of third order linear [[w:ordinary differential equations|ordinary differential equations]], normalized to have unit [[w:Wronskian|Wronskian]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=remark 6.6}}

They are linked to the corner coordinates by:{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=lemma 4.5}}
: &lt;math&gt;x_k=\frac{a_{k-2}}{b_{k-2}b_{k-1}},&lt;/math&gt;
: &lt;math&gt;y_k=-\frac{b_{k-1}}{a_{k-2}a_{k-1}}.&lt;/math&gt;

==Formulas on the moduli space==

===As a birational map ===
The pentagram map is a [[w:birational map|birational map]] on the moduli space, because it can be decomposed as the [[w:Function composition|composition]] of two [[w:Birational geometry|birational]] [[w:Involution (mathematics)|involutions]].{{Sfn|Schwartz|2008|loc=§1.2 The Pentagram Map}} The corner invariants change in the following way:{{Sfn|Ovsienko|Schwartz|loc=lemma 2.4|Tabachnikov|2010}}

: &lt;math&gt;x_k'=x_k\frac{1-x_{k-1} y_{k-1}}{1-x_{k+1}y_{k+1}},&lt;/math&gt;

: &lt;math&gt;y_k'=y_{k+1}\frac{1-x_{k+2} y_{k+2}}{1-x_k y_k}.&lt;/math&gt;

=== The scaling symmetry ===
The [[w:multiplicative group|multiplicative group]] &lt;math&gt;F\smallsetminus\{0\}&lt;/math&gt; [[w:One-parameter group|acts]] on the moduli space in the following way:

: &lt;math&gt;R_s\cdot(x_1,\dots,x_n,y_1,\dots,y_n)=(sx_1,\dots,sx_n,s^{-1}y_1,\dots,s^{-1}y_n),&lt;/math&gt;

where &lt;math&gt;R&lt;/math&gt; is called the scaling action and &lt;math&gt;s&lt;/math&gt; is the scaling parameter. This action commutes with the pentagram map on the moduli space (as presented in the previous formulas). This property is called the scaling symmetry, and is instrumental in proving the [[w:Pentagram map#Complete integrability|complete integrability]] of the dynamics.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=corollary 2.5}}

==Invariant structures==

===Monodromy invariants===

The monodromy invariants, introduced in {{Harvard citation|Schwartz|2008}}, are a collection of [[w:Function (mathematics)|functions]] on the [[w:moduli space|moduli space]] that are invariant under the pentagram map.{{Sfn|Schwartz|2008|loc=theorem 1.2}} The simplest example of them are 
:&lt;math&gt; O_n= x_1x_2\cdots x_{n}, \quad E_n = y_1y_2\cdots y_n. &lt;/math&gt;
The other monodromy invariants can be retrieved through different points of view: through the [[w:Pentagram map#The scaling symmetry|scaling symmetry]], as [[w:Combinatorics|combinatorial]] objects, or as some [[w:determinant|determinant]]s.{{Sfn|Schwartz|Tabachnikov|2011|loc=§2 The Monodromy Invariants}} The one involving scaling symmetry is presented here.

Let &lt;math&gt;M\in \mathrm{GL}_3&lt;/math&gt; be a [[w:Lift (mathematics)|lift]] of the monodromy of a twisted &lt;math&gt;n&lt;/math&gt;-gon. The quantities

: &lt;math&gt;\Omega_1=\frac{\operatorname{trace}^3(M)}{\det(M)}, \quad \Omega_2=\frac{\operatorname{trace}^3(M^{-1})}{\det(M^{-1})},&lt;/math&gt;

are independent of the choice of lift and are invariant under [[w:Matrix similarity|conjugation]], so they are well defined for the projective class of the polygon. They are invariant under the pentagram map, since the monodromy matrix doesn't change.{{Sfn|Schwartz|Tabachnikov|2011|loc=|p=5}} Now, the quantities

: &lt;math&gt;\tilde{\Omega}_1=O_n^2E_n\Omega_1, \quad \tilde{\Omega}_2=O_nE_n^2\Omega_2,&lt;/math&gt;

have the same properties, but turn out to be polynomials in the corner invariants.{{Efn|Some papers consider the cube roots of this functions, but it doesn't change the following definitions of the monodromy invariants.}} They can be written as{{Sfn|Schwartz|Tabachnikov|2011|loc=|p=5}}

: &lt;math&gt;
\tilde{\Omega}_1=\biggl(\sum_{k=0}^{\lfloor n/2\rfloor}O_k\biggr)^3, \quad
\tilde{\Omega}_2=\biggl(\sum_{k=0}^{\lfloor n/2\rfloor}E_k\biggr)^3,
&lt;/math&gt;

where each &lt;math&gt;O_k&lt;/math&gt; and &lt;math&gt;E_k&lt;/math&gt; are [[w:homogeneous polynomial|homogeneous polynomial]]s respectively of weight &lt;math&gt;k&lt;/math&gt; and &lt;math&gt;-k&lt;/math&gt;,{{Sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=11}} meaning they change under the [[w:Pentagram map#The scaling symmetry|rescaling action]] on variables by{{Sfn|Schwartz|Tabachnikov|2011|p=5}}

: &lt;math&gt; R_s(O_k)= s^k O_k, \quad R_s(E_k)= s^{-k} E_k. &lt;/math&gt;

The quantities &lt;math&gt;O_1,\dots,O_{\lfloor n/2 \rfloor},O_n, E_1,\dots,E_{\lfloor n/2 \rfloor},E_n,&lt;/math&gt; are unchanged by the dynamics, and are called the monodromy invariants. Moreover, they are [[w:algebraically independent|algebraically independent]].{{Sfn|Schwartz|2008|loc=theorem 1.2}}

==== Polygons on conics ====
Whenever &lt;math&gt;P&lt;/math&gt; is [[w:Inscribed figure|inscribed]] in a [[w:conic section|conic section]], one has &lt;math&gt;O_k(P)=E_k(P)&lt;/math&gt; for all &lt;math&gt;k&lt;/math&gt;.{{Sfn|Schwartz|Tabachnikov|2011|loc=theorem 1.1}} Moreover, if &lt;math&gt;P&lt;/math&gt; is circumscribed about another conic,{{Efn|See the paragraph about [[w:Pentagram map#Poncelet polygons|Poncelet polygons]].}} then its monodromy invariants are characterized by the pair of conics.{{Sfn|Schwartz|2015|loc=theorem 1.2}} For such odd-gons, the translation on the [[w:Jacobian variety|Jacobian variety]]{{Efn|See the paragraph about [[w:Pentagram map#Algebro-geometric integrability|algebraic integrability]].}} is restricted to the [[w:Prym variety|Prym variety]] (which is a half-dimensional torus in the Jacobian).{{Sfn|Izosimov|2016|loc=theorem 1.3}}

===Poisson bracket===

An invariant [[w:Poisson bracket|Poisson bracket]] on the space of twisted polygons was found in {{Harvard citation|Ovsienko|Schwartz|Tabachnikov|2010}}. The monodromy invariants [[w:Poisson bracket#Constants of motion|commute]] with respect to it:
&lt;math display=&quot;block&quot;&gt; \{O_i,O_j\}=\{O_i,E_j\}=\{E_i,E_j\}=0 &lt;/math&gt;for all &lt;math&gt;i,j&lt;/math&gt;.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}}

The Poisson bracket is defined in terms of the corner coordinates by:
&lt;math display=&quot;block&quot;&gt; \begin{align}
 \{x_i,x_{i\pm1}\} &amp;= \mp x_i x_{i+1}, \\
 \{y_i,y_{i\pm 1}\} &amp;= \mp y_i y_{i+1}, \\
\end{align}&lt;/math&gt;and &lt;math display=&quot;block&quot;&gt; \{x_i,x_j\} = \{y_i,y_j\} = \{x_i,y_j\} = 0&lt;/math&gt;for all other &lt;math&gt; i,j.&lt;/math&gt;{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=equation 2.16}}

=== The spectral curve ===

Let &lt;math&gt;\zeta&lt;/math&gt; be an element of the [[w:multiplicative group|multiplicative group]] and &lt;math&gt;P_\zeta&lt;/math&gt; be the polygon obtained by applying the [[w:Pentagram map#The scaling symmetry|rescaling action]] &lt;math&gt;R_\zeta&lt;/math&gt; on &lt;math&gt;P&lt;/math&gt;.  A [[w:Lax matrix|Lax matrix]] &lt;math&gt;\hat{T}(\zeta) \in \mathrm{GL}_3&lt;/math&gt; is a lift of the monodromy of &lt;math&gt;P_\zeta&lt;/math&gt; satisfying a [[w:Lax pair#Zero-curvature equation|zero-curvature equation]].{{Sfn|Weinreich|2022|loc=§5 The Lax representation}} Then, the spectral function is the [[w:Bivariate polynomial|bivariate]] [[w:characteristic polynomial|characteristic polynomial]]
&lt;math display=&quot;block&quot;&gt; Q(\lambda,\zeta) := \det(\lambda\operatorname{Id}-\hat{T}(\zeta)),&lt;/math&gt;or some renormalization of it. The [[w:spectral curve|spectral curve]] is the [[w:Projective variety#projective completion|projective completion]] of the [[w:Algebraic curve|affine curve]] defined by the equation &lt;math&gt;Q(\lambda,\zeta)=0&lt;/math&gt;.{{Sfn|Weinreich|2022|loc=§6. The geometry of the spectral curve}} It is invariant under the pentagram map, and the monodromy invariants appear as the [[w:coefficient|coefficient]]s of &lt;math&gt;Q&lt;/math&gt;.{{Sfn|Soloviev|2013|loc=theorem 6.4}} Its [[w:geometric genus|geometric genus]] is &lt;math&gt;n-1&lt;/math&gt; if &lt;math&gt;n&lt;/math&gt; is odd, and &lt;math&gt;n-2&lt;/math&gt; if &lt;math&gt;n&lt;/math&gt; is even.{{Sfn|Weinreich|2022|p=|loc=theorem 6.4}}

It was first introduced in {{Harv|Soloviev|2013|ps=|p=}} for his proof of [[w:Pentagram map#Algebro-geometric integrability|algebro-geometric integrability]].{{sfn|Soloviev|2013}}

==Complete integrability==
The pentagram map on the moduli space has been proved to be a [[w:completely integrable|completely integrable]] [[w:discrete dynamical system|discrete dynamical system]], both in the [[w:Integrable system#Hamiltonian systems and Liouville integrability|Arnold-Liouville]]{{Efn|Over the [[w:real number|real number]]s.}} and the [[w:Integrable system#Complete integrability over the complex numbers|algebro-geometric]]{{Efn|Over [[w:algebraically closed field|algebraically closed field]]s of [[w:Characteristic (algebra)|characteristic]] different from 2.}} senses. In any case, this means that the moduli space is [[w:almost everywhere|almost everywhere]] [[w:Foliation|foliated]] by [[w:Torus#Flat torus|flat tori]] (or in the algebraic setting, [[w:Abelian variety|Abelian varieties]]), where the motion is a [[w:Translation (geometry)|translation]]. This [[w:Generic property|generically]] induces a [[w:quasiperiodic motion|quasiperiodic motion]] on the corresponding torus.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2009}}

===Arnold–Liouville integrability===

The proof of the integrability of the pentagram map on a real twisted polygon was achieved in {{Harvard citation|Ovsienko|Schwartz|Tabachnikov|2010}}. This is done by noticing that the monodromy invariants &lt;math&gt;O_n&lt;/math&gt; and &lt;math&gt;E_n&lt;/math&gt; are [[w:Casimir invariant|Casimir invariant]]s for the Poisson bracket, meaning (in this context) that&lt;math display=&quot;block&quot;&gt; \{O_n,f\}=\{E_n,f\} = 0 &lt;/math&gt;for every smooth function &lt;math&gt;f&lt;/math&gt;.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}} When &lt;math&gt;n&lt;/math&gt; is even, this is also true for the monodromy invariants &lt;math&gt;O_{\lfloor n/2 \rfloor }&lt;/math&gt; and &lt;math&gt;E_{\lfloor n/2 \rfloor }&lt;/math&gt;.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 1}}

This allows to consider the Casimir [[w:level set|level set]], where each Casimir invariant has a specified value. Because of [[w:Sard's theorem|Sard's theorem]], any generic level set is a [[w:smooth manifold|smooth manifold]].{{Sfn|Schwartz|2017|p=44}} They form a [[w:foliation|foliation]] in [[w:Poisson manifold#Symplectic leaves|symplectic leaves]], on which the Poisson bracket gives rise to a [[w:symplectic form|symplectic form]].{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=corollary 2.13}}

Each of these symplectic leaves has an iso-monodromy [[w:foliation|foliation]], namely, a decomposition into the common level sets of the remaining monodromy functions. By using again [[w:Sard's theorem|Sard's theorem]], they are generically [[w:Symplectic manifold#Lagrangian submanifolds|Lagrangian manifolds]].{{Sfn|Schwartz|2017|p=45}} Moreover, they are compact.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=§3.3. Compactness of the level sets}} Since the monodromy invariants Poisson-commute and there are enough of them, the discrete [[w:Liouville–Arnold theorem|Liouville–Arnold theorem]] can be applied to prove that the level sets are [[w:Torus#Flat torus|flat tori]] over which the dynamics is a translation.{{sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=412}}
===Algebro-geometric integrability===

In {{Harvard citation|Soloviev|2013}}, it was shown that the pentagram map admits a [[w:Lax representation|Lax representation]] with a spectral parameter, which allows to prove its algebro-geometric integrability. This means that the space of polygons (either twisted or closed) is parametrized by its spectral data, consisting of [[w:Pentagram map#The spectral curve|its spectral curve]], with marked points and a [[w:Divisor (algebraic geometry)|divisor]] given by a [[w:Floquet theory|Floquet]]–[[w:Bloch's theorem|Bloch]] equation. This gives an embedding to the [[w:Jacobian variety|Jacobian variety]] through the [[w:Abel–Jacobi map|Abel–Jacobi map]], where the motion is expressed in terms of translation.{{sfn|Soloviev|2013|loc=theorems A, B and C}} The previously defined Poisson bracket is also retrieved.{{sfn|Soloviev|2013|loc=theorem D}} 

This integrability was generalized in {{Harvard citation|Weinreich|2022}} from the field of [[w:complex number|complex number]]s to any [[w:algebraically closed field|algebraically closed field]] of [[w:Characteristic (algebra)|characteristic]] different from 2. The translation on a torus is replaced by a translation on an [[w:Abelian variety|Abelian variety]] (in fact, a Jacobian variety again).{{sfn|Weinreich|2022|loc=theorem 1.4}}

=== Dimension of the invariant manifold ===

For twisted &lt;math&gt;n&lt;/math&gt;-gons, the [[w:dimension|dimension]] of the invariant tori (or Jacobian varieties) is{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|p=421}}

: &lt;math&gt;\begin{cases} 
n-1 &amp; \text{when }n \text{ is odd,}\\
n-2 &amp; \text{when }n \text{ is even.}
\end{cases}&lt;/math&gt;

Moreover, when &lt;math&gt;n&lt;/math&gt; is even, there are two isomorphic Jacobians on which the iterates of the pentagram map alternate. But on each of them, the second iterate is a translation.{{Sfn|Weinreich|2022|loc=theorem 1.4}}

=== For closed polygons ===
There is no Poisson structure on the space of closed polygons.{{Sfn|Soloviev|2013|loc=corollary 4.1}} Nevertheless, the one from twisted polygons can be used to prove integrability.{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|p=2153}}

Algebro-geometric integrability holds for closed polygons in a same manner as for the twisted ones.{{Sfn|Soloviev|2013|loc=theorem C}} However, Arnold-Liouville integrability is proved for real closed polygons only when they are convex. This is done by restricting the [[w:Hamiltonian vector field|Hamiltonian vector field]]s of monodromy functions to smaller dimensional tori, and showing that enough of them are still independent.{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|loc=corollary 1.1}}

In both situation, the dimension of the invariant manifolds decreases by &lt;math&gt;3&lt;/math&gt; for closed &lt;math&gt;n&lt;/math&gt;-gons (compared to the twisted case), and is equal to{{Sfn|Soloviev|2013|loc=theorem C}}{{sfn|Ovsienko|Schwartz|Tabachnikov|2013|loc=theorem 1}}
:&lt;math&gt;\begin{cases} 
n-4 &amp; \text{when }n \text{ is odd,}\\
n-5 &amp; \text{when }n \text{ is even.}
\end{cases}&lt;/math&gt;

==Connections to other topics==

===The Boussinesq equation===

The continuous limit of a convex polygon is a parametrized convex curve in the plane. When the time parameter is suitably chosen, the [[w:Discretization|continuous limit]] of the pentagram map is the classical [[w:Boussinesq approximation (water waves)|Boussinesq equation]]. This equation is a classical example of an [[w:integrable|integrable]] [[w:partial differential equation|partial differential equation]].{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=theorem 5}}

Here is a description of the geometric action of the Boussinesq equation. Given a [[w:locally convex|locally convex]] curve &lt;math&gt; C:\mathbb R\to \mathbb R^2 &lt;/math&gt; and real numbers &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;t&lt;/math&gt;, consider the [[w:chord (geometry)|chord]] connecting &lt;math&gt; C(x-t) &lt;/math&gt; to &lt;math&gt; C(x+t) &lt;/math&gt;. The [[w:Envelope (mathematics)|envelope]] of all these chords is a new curve &lt;math&gt; C_t(x) &lt;/math&gt;. When &lt;math&gt;t&lt;/math&gt; is extremely small, the curve &lt;math&gt; C_t(x) &lt;/math&gt; is a good model for the time &lt;math&gt;t&lt;/math&gt; evolution of the original curve &lt;math&gt; C_0(x) &lt;/math&gt; under the Boussinesq equation. This construction is also similar to the pentagram map. Moreover, the pentagram invariant bracket is a discretization of a well known invariant Poisson bracket associated to the Boussinesq equation.{{Sfn|Ovsienko|Schwartz|Tabachnikov|2010|loc=§6.4 Discretization}}

===Cluster algebras===

The pentagram map{{Sfn|Glick|2011}} and some of its generalizations{{Sfn|Gekhtman|Shapiro|Tabachnikov|Vainshtein|2012}}{{Sfn|Glick|Pylyavskyy|2016}} are identified as special cases of discrete dynamical systems powered by [[w:cluster algebra|cluster algebra]]. Using the results from {{Harvard citation|Goncharov|Kenyon|2013}}, this provides a link with the [[w:Poisson–Lie group|Poisson–Lie group]]s, [[w:dimer model|dimer model]]s and other so-called cluster-integrable systems.{{Sfn|Fock|Marshakov|2016}}{{Sfn|Izosimov|2022c}} These methods allow to retrieve the Poisson bracket and Hamiltonians used to prove complete integrability{{Sfn|Affolter|George|Ramassamy|2025|loc=§5 The pentagram map}} and provide [[w:Lax representation|Lax representation]]s.{{Sfn|Izosimov|2022b}}

=== Octahedron recurrence ===
Using a method to compute [[w:Determinant|determinants]] called [[w:Dodgson_condensation|Dodgson condensation]], {{Harvard citation|Schwartz|2008}} proves that the pentagram map satisfies a property called the &quot;octahedron recurrence&quot;.{{Sfn|Schwartz|2008|loc=§5 The Method of Condensation}} This property turns out to be shared by other dynamical systems defined by geometric constructions similar to one of the pentagram map.{{Sfn|Affolter|de Tilière|Melotti|2025|loc=§9 The Pentagram Map}} It is also shared by higher dimensional pentagram maps defined through cluster algebras mutations, referred as &lt;math&gt;T&lt;/math&gt;-systems.{{Sfn|Kedem|Vichitkunakorn|2015||loc=}}

=== Singularity theory ===
The pentagram map exhibits a property called singularity confinement, which is typical from [[w:integrable system|integrable system]]s.{{Sfn|Grammaticos|Ramani|Papageorgiou|1991}} It states that if a polygon &lt;math&gt;P&lt;/math&gt; is [[w:Singular point of an algebraic variety|singular]] for the pentagram map &lt;math&gt;T&lt;/math&gt;, then there exists an integer &lt;math&gt;m&lt;/math&gt; such that &lt;math&gt;P&lt;/math&gt; not singular for the iterate map &lt;math&gt;T^m&lt;/math&gt;.{{Sfn|Glick|2012}}

Moreover, the pentagram map (along with some of its generalizations and other discrete dynamical systems) exhibit the Devron property.{{Efn|The name comes from an episode of [[w:Star Trek|Star Trek]].{{Sfn|Glick|2015|loc=§1 Introduction}}}} This means that if a polygon &lt;math&gt;P&lt;/math&gt; is singular for some iterate of the pentagram map &lt;math&gt;T^m&lt;/math&gt;, then it will also be singular for some iterate of the inverse map &lt;math&gt;T^{-m'}&lt;/math&gt;.{{Sfn|Glick|2015}}

== Generalizations ==
The definition of twisted polygons still makes sense in any [[w:projective space|projective space]] &lt;math&gt;\mathbb P^d&lt;/math&gt;, under the action of the [[w:Projective linear group|projective group]] &lt;math&gt;\mathbb P \mathrm{GL}_{d+1}&lt;/math&gt;. The pentagram map can be generalized in many ways, and some of them are presented here. Not all of them are integrable.{{Sfn|Khesin|Soloviev|2015|}} Some are [[w:discretization|discretization]]s of [[w:PDEs|PDEs]] from the [[w:KdV hierarchy|KdV hierarchy]], seen as higher dimensional version of [[w:Boussinesq approximation (water waves)|Boussinesq]] or [[w:Kadomtsev–Petviashvili equation|KP]] equations.{{Sfn|Marí-Beffa|2012}}{{Sfn|Wang|2023}} The description of all generalized pentagram maps in terms of [[w:cluster algebra|cluster algebra]]s is still an open question.{{Sfn|Gekhtman|Izosimov|2025|p=14}}

=== Polygons in general positions ===
Let &lt;math&gt;d \geq 2&lt;/math&gt; and &lt;math&gt;P&lt;/math&gt; be a twisted polygon of &lt;math&gt;\mathbb P^d&lt;/math&gt; in [[w:general position|general position]].

==== Short diagonal pentagram maps ====
The &lt;math&gt;k&lt;/math&gt;-th ''short diagonal hyperplane'' &lt;math&gt;H_k^{sh}&lt;/math&gt; is uniquely defined by passing through the vertices &lt;math&gt;v_k,v_{k+2},\dots,v_{k+2d-2}&lt;/math&gt;. [[w:Generic property#In algebraic geometry|Generically]], the intersection of &lt;math&gt;d&lt;/math&gt; consecutive hyperplanes uniquely defines a new point

: &lt;math&gt;T_{sh}v_k:=H_k^{sh}\cap H_{k+1}^{sh}\cap \dots \cap H_{k+d-1}^{sh}.&lt;/math&gt;

Doing this for every vertex defines a new twisted polygon. This map, denoted by &lt;math&gt;T_{sh}&lt;/math&gt;, is again projectively equivariant.{{Sfn|Khesin|Soloviev|2013}}

==== Generalized pentagram maps ====
The previous procedure can be generalized. Let &lt;math&gt;I=(i_1,\dots,i_{d-1}),~J=(j_1,\dots,j_{d-1})&lt;/math&gt; be two sets of integers, respectively called the jump tuple and the intersection tuple. Define the &lt;math&gt;k&lt;/math&gt;-th hyperplane &lt;math&gt;H_k^I&lt;/math&gt; to be passing through the vertices &lt;math&gt;v_k,v_{k+i_1},\dots,v_{k+i_1+\dots+i_{d-1}}&lt;/math&gt;. A new point is given by the intersection

: &lt;math&gt;T_{I,J}v_k:=H_k^I \cap H_{k+j_1}^I \cap \dots \cap H_{k+j_1+\dots +j_{d-1}}^I.&lt;/math&gt;

The map &lt;math&gt;T_{I,J}&lt;/math&gt; is called a generalized pentagram map.{{Sfn|Khesin|Soloviev|2015a}} The original pentagram map is recovered by considering&lt;math&gt;d=2,~I=(2),~J=(1)&lt;/math&gt;.

Integrability can be numerically tested by picking a random polygon &lt;math&gt;P&lt;/math&gt; with [[wikipedia:Rational_point|rational coordinates]] and studying the growth rate of the [[wikipedia:Height_function|height]] of its iterates. This is called the [[wikipedia:Integrable_system#Diophantine_integrability|diophantine integrability]] test, and some generalized pentagram maps don't seem to pass it.{{Sfn|Khesin|Soloviev|2015a|loc=§5 and §6}} However, it is conjectured that the maps &lt;math&gt;T_{I,I}&lt;/math&gt; are integrable for any &lt;math&gt;I&lt;/math&gt;.{{Sfn|Bolsinov|Matveev|Miranda|Tabachnikov|2018|loc=conjecture 4.13 (B. Khesin, F. Soloviev)}}

Some of these maps are [[w:discretization|discretization]]s of higher dimensional counterpart of the [[w:Boussinesq approximation (water waves)|Boussinesq equation]] in the [[w:KdV hierarchy|KdV hierarchy]].{{Sfn|Khesin|Soloviev|2015b|loc=theorem 4.1}}{{Sfn|Izosimov|2022b|loc=theorem 4.1}}

==== Dented pentagram maps ====
Fix an integer &lt;math&gt;m\in \{1,\dots ,d-1\}&lt;/math&gt;. Consider the jump tuple &lt;math&gt;I_m:=(1,\dots,1,2,1,\dots,1)&lt;/math&gt;, where the &lt;math&gt;2&lt;/math&gt; is at the &lt;math&gt;m&lt;/math&gt;-th place, and the intersection tuple &lt;math&gt;J:=(1,\dots,1)&lt;/math&gt;. The dented pentagram map is &lt;math&gt;T_m :=T_{I_m,J}&lt;/math&gt;. They are proved to be integrable.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 2.14}}

For an integer &lt;math&gt;p \geq 2&lt;/math&gt;, the deep dented pentagram map (of depth &lt;math&gt;p&lt;/math&gt;) &lt;math&gt;T_m^p&lt;/math&gt; is the same map as before, but the number &lt;math&gt;2&lt;/math&gt; in the definition of &lt;math&gt;I_m&lt;/math&gt; is replaced by &lt;math&gt;p&lt;/math&gt;. This kind of pentagram maps are again integrable.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 6.2}}

=== Corrugated polygons ===
A twisted polygon &lt;math&gt;P&lt;/math&gt; lying in &lt;math&gt;\mathbb P^d&lt;/math&gt; is said to be corrugated if for any &lt;math&gt;k\in \mathbb Z&lt;/math&gt;, the vertices &lt;math&gt;v_k,v_{k+1},v_{k+d},v_{k+d+1}&lt;/math&gt; span a projective two-dimensional plane. Such polygons are not in [[w:general position|general position]]. A new point is defined by

: &lt;math&gt;T_\text{cor}v_k:=\overline{v_k v_{k+d}}\cap \overline{v_{k+1} v_{k+d+1}}.&lt;/math&gt;

The map &lt;math&gt;T_\text{cor}&lt;/math&gt; yields a new corrugated polygon. They are [[w:Integrable system#Hamiltonian systems and Liouville integrability|completely Liouville-integrable]].{{Sfn|Gekhtman|Shapiro|Tabachnikov|Vainshtein|2012|loc=theorem 4.4}}

In fact, they can be retrieved as some dented pentagram map applied on corrugated polygons.{{Sfn|Khesin|Soloviev|2015b|loc=theorem 5.3}}

=== Grassmannian polygons ===
Let &lt;math&gt;d \geq 3, m \geq 1&lt;/math&gt; be integers. The pentagram map can also be generalized to the [[w:Grassmannian|Grassmannian]] space &lt;math&gt;\mathrm{Gr}(m,md)&lt;/math&gt;, which consists of &lt;math&gt;m&lt;/math&gt;-[[w:Dimension (vector space)|dimensional]] [[w:linear subspace|linear subspace]]s of an &lt;math&gt;md&lt;/math&gt;-dimensional [[w:vector space|vector space]]. When &lt;math&gt;m=1&lt;/math&gt;, the linear subspaces are [[w:Vector space#vector line|lines]], which retrieves the definition of [[w:projective space|projective space]]s &lt;math&gt;\mathbb P^d&lt;/math&gt;.{{Sfn|Felipe|Marí-Beffa|2019|loc=§2 definitions and notations}}

A point &lt;math&gt;v\in\operatorname{Gr}(m,md)&lt;/math&gt; is represented by an &lt;math&gt;md \times m&lt;/math&gt; matrix &lt;math&gt;X_v&lt;/math&gt; such that its columns form a [[w:Basis (linear algebra)|basis]] of &lt;math&gt;v&lt;/math&gt;. Consider the [[w:Group action|action]] of the [[w:general linear group|general linear group]] &lt;math&gt;\mathrm{GL}_{md}&lt;/math&gt; by multiplication on the left of &lt;math&gt;X_v&lt;/math&gt;. This defines an action on the Grassmannian, even though it is not [[w:Faithful action|faithful]].{{Efn|Because there can be many lifts for &lt;math&gt;v&lt;/math&gt;, and because some matrices act trivially.}} Hence, the polygons of &lt;math&gt;\mathrm{Gr}(m,md)&lt;/math&gt; and their moduli spaces are defined as before, after the change of underlying group.{{Sfn|Felipe|Marí-Beffa|2019|loc=§2 definitions and notations}}

Depending on the parity of &lt;math&gt;d&lt;/math&gt;, one can define linear subspaces spanned by some &lt;math&gt;X_{v_k}&lt;/math&gt;'s such that taking their intersection generically defines a new point &lt;math&gt;v\in\mathrm{Gr}(m,md)&lt;/math&gt;.{{Sfn|Felipe|Marí-Beffa|2019|loc=sections 4 and 5}} This generalization of the pentagram map is integrable in a [[w:noncommutative|noncommutative]] sense.{{Sfn|Ovenhouse|2020}}

=== Over rings ===
The pentagram map admits a generalization by considering [[w:Projective space#Generalizations|projective planes]] over [[w:stably finite ring|stably finite ring]]s, instead of [[w:Field (mathematics)|field]]s. In particular, this retrieves the pentagram map over Grassmannians. Again, it admits a [[w:Lax representation|Lax representation]].{{Sfn|Hand|Izosimov|2025}}

== References ==
{{reflist|25em}}

===Notes===
{{notelist}}

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      <text bytes="35088" sha1="skn9wddbqgphpflpgecivywvybw4y70" xml:space="preserve">==Overview==
A. N. Wilson says,&lt;blockquote&gt;At Cimiez, in April 1897, the Queen found herself staying in the same hotel as the great Sarah Bernhardt: as venerated for her acting as she was celebrated for her rackety life of love. (Bertie, as the Queen was no doubt completely aware, had become obsessed by her when she did a London season in 1879, attending her [963–964] performances night after night; though she was only a flirtation, she was invited to his Coronation years later, and placed with Mrs Keppel, Jennie Churchill and the other mistresses in the chancel gallery nicknamed the ‘King’s Loose Box’. ...)&lt;ref&gt;Wilson, A. N. ''Victoria: A Life''. Penguin, 2014. Apple Books:  https://books.apple.com/us/book/victoria/id828766078.&lt;/ref&gt;{{rp|963–964 of 1204}}&lt;/blockquote&gt;

==Also Known As==
* Family name: Bernhard
* Henriette-Rosine Bernard

==Acquaintances, Friends and Enemies==
===Acquaintances===
===Friends===

* Charles de Morny, Duke of Morny (half-brother of Napoleon III)&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|date=2025-12-07|title=Sarah Bernhardt|url=https://en.wikipedia.org/w/index.php?title=Sarah_Bernhardt&amp;oldid=1326131063|journal=Wikipedia|language=en}}&lt;/ref&gt;
* Charles Gounod&lt;ref name=&quot;:0&quot; /&gt;
* Madame Guérard, lived with Bernhardt and Maurice&lt;ref name=&quot;:0&quot; /&gt;
* George Sand

===Lovers===

* Henri, Hereditary Prince de Ligne (1864)&lt;ref name=&quot;:0&quot; /&gt;

===Enemies===
[[File:Harvard_Theatre_Collection_-_Sarah_Bernhardt_TCS_2_(Cleopatra)_(cropped).jpg|thumb|Sarah Bernhardt as Cleopatra, 1891, Sarony, non-cropped version available]]
[[File:Sarah_Bernhardt,_1891_LCCN2016852695.jpg|thumb|Sarah Bernhardt as Cleopatra, 1891]]

==Organizations and Social Networks==
[[File:Sara_Bernhardt_-_Sarony,_N.Y._LCCN90716396.jpg|thumb|Sarah Bernhardt as Cleopatra, 1891, can get better copy from LoC[[File:Sarah_Bernardt.JPG|thumb|Sarah Bernhardt as Cleopatra, 1893]]]]

==Timeline==
'''1857''', Bernhardt found out that her father had died.&lt;ref name=&quot;:0&quot; /&gt;

'''1862''', Bernhardt's debut at the Comédie-Française.&lt;ref name=&quot;:0&quot; /&gt;

'''1862 August 31''', Bernhardt's debut at the Theatre Français.&lt;ref name=&quot;:0&quot; /&gt;

'''1864''', Bernhardt moved to the Gymnase theatre company, from which she was invited to recite 2 poems at a reception at the Tuileries Palace hosted by Empress Eugènie and Napoleon III, but she unwittingly read poetry by Victor Hugo, a critic of the monarchy, and the court walked out.&lt;ref name=&quot;:0&quot; /&gt;

'''1866 early''', Bernhardt read for Felix Duquesnel, director of the Théâtre de L'Odéon, nearly as prestigious to the Comédie-Française but with a less traditional repertoire.&lt;ref name=&quot;:0&quot; /&gt;

'''1884''', Bernhardt created the role of Theodora in the new Sardou play.

'''1896''', Bernhardt&lt;blockquote&gt;used the new technology of lithography to produce vivid color posters, and in 1894, she hired Czech artist Alphonse Mucha to design the first of a series of posters for her play ''Gismonda''. He continued to make posters of her for six years.&lt;ref name=&quot;:102&quot;&gt;{{Cite journal|date=2025-07-30|title=Sarah Bernhardt|url=https://en.wikipedia.org/w/index.php?title=Sarah_Bernhardt&amp;oldid=1303400174|journal=Wikipedia|language=en}}&lt;/ref&gt;&lt;/blockquote&gt;Mucha's lithographs were in the Art Nouveau style. His poster of Salammbô is shown at the very top of the section on [[Social Victorians/People/Bourke#Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball|Gwendolen Bourke's costume for the ball]].
[[File:Henri_de_Toulouse-Lautrec,_Sarah_Bernhardt_in_&quot;Cleopatra&quot;_(Sarah_Bernhardt_dans_&quot;Cléopatre&quot;),_1896,_NGA_42139.jpg|left|thumb|Toulouse-Lautrec's Bernhardt as Cleopatra, 1896]]

== Major Roles ==

=== Cleopatra ===
Sarah Bernhardt performed Victorien Sardou's and Émile Moreau's 1890 ''Cléopâtra'' (with music by Xavier Leroux).&lt;ref&gt;{{Cite journal|date=2025-08-04|title=Cleopatra|url=https://en.wikipedia.org/w/index.php?title=Cleopatra&amp;oldid=1304135144|journal=Wikipedia|language=en}}&lt;/ref&gt;She habitually took a personal interest in her costumes, sometimes doing research in museums and art galleries,&lt;ref name=&quot;:10&quot;&gt;{{Cite journal|date=2025-07-30|title=Sarah Bernhardt|url=https://en.wikipedia.org/w/index.php?title=Sarah_Bernhardt&amp;oldid=1303400174|journal=Wikipedia|language=en}}&lt;/ref&gt; and in ''Cléopâtra'' she used her own pet garter snakes for the asp that kills her. She was photographed in the costume of the 1891 performances of ''Cleopatra'' (first 3 photographs on the right). Henri Toulouse-Lautrec drew her in the same role in 1896 (below left).

==== The Historical Cleopatra ====
Cleopatra lived from 70/69 B.C.E. to 10 or 12 August 30 B.C.E., the last of the Hellenistic pharaohs.&lt;ref&gt;{{Cite journal|date=2025-08-04|title=Cleopatra|url=https://en.wikipedia.org/w/index.php?title=Cleopatra&amp;oldid=1304135144|journal=Wikipedia|language=en}}&lt;/ref&gt; But nonscholarly late 19th-century Britons, Europeans and Americans would have known her less as a historical figure than a cultural one, by her presence in the arts and in popular culture. 

About 6,000–7,000 references to Cleopatra appear per year in British newspapers between 1890 and 1891, so Cleopatra was present as a name referring generally to the powerful queen of antiquity, especially of Egypt, Rome and Greece. She was painted by the major painters of the late 19th century and appeared in plays, novels, operas, ballets and poems. She is rendered white almost universally by Europeans and especially Americans&lt;ref&gt;{{Cite journal|date=2025-04-20|title=Egyptomania in the United States|url=https://en.wikipedia.org/w/index.php?title=Egyptomania_in_the_United_States&amp;oldid=1286505313|journal=Wikipedia|language=en}}&lt;/ref&gt; of whatever century. And beyond her presence as herself, ships were named after her, and she is implicated in depictions of Julius Caesar and Mark Antony as well as the &quot;Egyptomania&quot; of the time, including Giuseppe Verdi's popular 1871 ''Aida'',&lt;ref&gt;{{Cite journal|date=2025-08-13|title=Aida|url=https://en.wikipedia.org/w/index.php?title=Aida&amp;oldid=1305615525|journal=Wikipedia|language=en}}&lt;/ref&gt; which is set in &quot;Old Kingdom&quot; Egypt (that is, some undetermined time in the far past). Egypt was present in the imaginations of the Romantics and kept there by the Victorians, by the deciphering of hieroglyphics beginning with the Rosetta Stone in 1822,&lt;ref&gt;{{Cite journal|date=2025-08-15|title=Rosetta Stone|url=https://en.wikipedia.org/w/index.php?title=Rosetta_Stone&amp;oldid=1305991621|journal=Wikipedia|language=en}}&lt;/ref&gt; by the presence of Egyptian artifacts in the British Museum, and by the widely discussed role in the 1870s of Prime Minister Benjamin Disraeli, the Earl of Beaconsfield in the purchase of British control of the Suez Canal.&lt;ref&gt;{{Cite journal|date=2025-07-26|title=Benjamin Disraeli|url=https://en.wikipedia.org/w/index.php?title=Benjamin_Disraeli&amp;oldid=1302642906|journal=Wikipedia|language=en}}&lt;/ref&gt;
[[File:Lillie Langtry as Cleopatra.jpg|alt=Old photo of a woman with her long hair down, dressed as a queen from the ancient world of Egypt|thumb|Lillie Langtry as Cleopatra, 1891]]
Besides Shakespeare's ''Antony and Cleopatra'', other plays, late-19th-century paintings or novels featuring Cleopatra would have been reviewed and advertised in contemporary periodicals. For example, Émile Moreau and Victorien Sardou's ''Cléopâtre'' was produced in 1890, starring [[Social Victorians/People/Sarah Bernhardt|Sarah Bernhardt]], who took the show on tour to the U.K. and U.S. Lillie Langtry also performed Cleopatra in Shakespeare's play and was photographed by society photographer W. &amp; D. Downey (bottom right).

H. Rider Haggard renamed his 1890 ''Harmachio'' in 1891 to ''Cleopatra: Being an Account of the Fall and Vengeance of '''Harmachis''''' [https://en.wikipedia.org/wiki/Cleopatra_(Haggard_novel)&lt;nowiki&gt;].&lt;/nowiki&gt;

* Lawrence Alma-Tadema (1875 Cleopatra,1883 The Meeting of Antony and Cleopatra)
* Frederick Arthur Bridgman (1896 Cleopatra on the Terraces of Philae [https://commons.wikimedia.org/wiki/File:Frederick_Arthur_Bridgman_-_Cleopatra_on_the_Terraces_of_Philae.JPG&lt;nowiki&gt;])&lt;/nowiki&gt;
* Alexandre Cabanel (1887, ''Cleopatra Testing Poisons on Condemned Prisoners''[https://en.wikipedia.org/wiki/Cleopatra_Testing_Poisons_on_Condemned_Prisoners&lt;nowiki&gt;])&lt;/nowiki&gt;
* John Collier (1890 ''The Death of Cleopatra'')
* John William Waterhouse (1888 Cleopatra)
* Richard Caton Woodville: ''Cleopatra''
** ''The Death of Cleopatra'' (1889) for ''The Illustrated London News''
* and many more

Although it is too late to be an influence on any costume at this ball, in 1898 George Grossmith, Jr., and Paul Rubens created the burlesque ''Great Caesar''.&lt;ref&gt;{{Cite journal|date=2025-08-04|title=Cleopatra|url=https://en.wikipedia.org/w/index.php?title=Cleopatra&amp;oldid=1304135144|journal=Wikipedia|language=en}}&lt;/ref&gt;

In 1893, ''The Queen'' advertised &quot;the Cleopatra,&quot; a &quot;charming evening cloak, in rich Bengaline Silk, lined Silk.&quot;&lt;ref&gt;{{Cite journal|date=2025-08-04|title=Cleopatra|url=https://en.wikipedia.org/w/index.php?title=Cleopatra&amp;oldid=1304135144|journal=Wikipedia|language=en}}&lt;/ref&gt;

The 9th edition of the ''Encylopædia Britannica'', the edition that would have been available at this time, has an article about Cleopatra that runs about one full column. It emphasizes her &quot;remarkable charms of person&quot;: &lt;blockquote&gt;CLEOPATRA (''Κλεπάτρα''), the name of several Egyptian princesses of the house of the Ptolemies. The best known was the daughter of Ptolemy Auletes, born 69 &lt;small&gt;B&lt;/small&gt;.&lt;small&gt;C&lt;/small&gt;. Her father left her, at the age of seventeen, heir to his kingdom jointly with her younger brother Ptolemy, whose wife, in accordance with Egyptian custom, she was to become. A few years afterwards her brother, or rather her guardians, deprived her of all royal authority. She withdrew into Syria, and there made preparation to recover her rights by force of arms. It was at this juncture that Julius Cæsar followed Pompey into Egypt, resolved to settle there, if possible, the existing dispute as to the throne. The personal fascinations of Cleopatra, which she was not slow in bringing to bear upon him, soon won him entirely to her side; and as Ptolemy and his advisers still refused to admit her to a share in the kingdom, Cæsar undertook a war on her behalf, in which Ptolemy lost his life, and she was replaced on the throne in conjunction with a younger brother, to whom she was also contracted in marriage. Her relations with Cæsar were matter of public notoriety, and soon after his return to Rome she joined him there, in company with her boy-husband (of whom, however, she soon rid herself by poison), but living openly with her Roman lover, somewhat to the scandal of his fellow-citizens. After Cæsar’s assassination, aware of her unpopularity, she returned at once to her native country. But subsequently, during the civil troubles at Rome, she took the part of Antony, on whom she is said to have already made some impression in her earlier years, when he was campaigning in Egypt. When he was in Cilicia, she made a purpose journey to visit him, sailing up the Cydnus in a gorgeously-decked galley, arrayed in all the attractive splendour which Eastern magnificence could bring in aid of her personal charms. Antony became from that time forth her infatuated slave, followed her to Egypt, and lived with her there for some time in the most profuse and wanton luxury. They called themselves “Osiris” and “Isis,” and claimed to be regarded as divinities. His marriage with Octavia broke this connection for a while, but it was soon renewed, and Cleopatra assisted him in his future campaigns both with money and supplies. This infatuation of his rival with a personage already so unpopular at Rome as Cleopatra, was taken advantage of by Octavianus Cæsar (Augustus), who declared war against her personally. In the famous seafight at Actium, between the fleets of Octavianus and Antony, Cleopatra, who had accompanied him into action with an Egyptian squadron, took to flight while the issue was yet doubtful, and though hotly pursued by the enemy succeeded in escaping to Alexandria, where she was soon joined by her devoted lover. When the cause of Antony was irretrievably ruined, and all her attempts to strengthen herself against the Roman conqueror by means of foreign alliances had failed, she made overtures of submission. Octavianus suggested to her, as a way to his favour, the assassination of his enemy Antony. She seems to have entertained the base proposal, — enticing him to join her in [Col. 1c-2a] a mausoleum which she had built, in order that “they might die together,” and where he fulfilled his part of the compact by committing suicide, in the belief that she had already done so. The charms which had succeeded so easily with Julius and with Antony failed to move the younger Cæsar, though he at once granted her an interview; and rather than submit to be carried by him as a prisoner to Rome, she put an end to her life — by applying an asp to her bosom, according to the common version of the story — in the thirty-ninth year of her age. With her ended the dynasty of the Ptolemies in Egypt. Besides her remarkable charms of person, she had very considerable abilities, and unusual literary tastes. She is said to have been able to converse in seven languages. She had three children by Antony, and, as some say, a son, called Cæsarion, by Julius Cæsar.&lt;ref&gt;{{Cite journal|date=2025-08-04|title=Cleopatra|url=https://en.wikipedia.org/w/index.php?title=Cleopatra&amp;oldid=1304135144|journal=Wikipedia|language=en}}&lt;/ref&gt;&lt;/blockquote&gt;
[[File:Theodora - Basilica San Vitale (Ravenna, Italy) - croped.jpg|alt=Photograph of an ancient mosaic showing woman with a halo and with attendants|left|thumb|Detail of icon of Theodora and attendants in the Basilica San Vitale, Ravenna, Italy]]
[[File:Sarah Bernhardt as Theodora by Nadar.jpg|alt=Old photograph of a woman barring a door and acting like she's overhearing something|thumb|Sarah Bernhardt, 1884, Theodora]]

=== Theodora ===
[[File:Sarah Bernhardt as the Empress Theodora.jpg|alt=Old photograph of an actor seated on a throne|thumb|Sarah Bernhardt, seated as Theodora, 1884]]
For Sardou's 1884 ''Theodora'', Bernhardt visited Ravenna and based her costumes for the title character on her sketches of the clothing of the icon in the mosaic murals there.&lt;ref&gt;{{Cite journal|date=2026-06-21|title=Sarah Bernhardt: Return to Paris, European tour, Fédora to Theodora (1881–1886)|url=https://en.wikipedia.org/wiki/Sarah_Bernhardt#Return_to_Paris,_European_tour,_Fédora_to_Theodora_(1881–1886)|journal=Wikipedia|language=en}}&lt;/ref&gt; The 6th-century mosaic (left) shows Theodora, who has a halo. Her headdress is very elaborately bejeweled. Her cloak and underdress are not ornate like later the theatrical costumes based on it, but they both have a decorative panel at the hemline. The attendant to Theodora's left is wearing a drape with a pattern of circles. Later designers of costumes may have repeated the motif of circles and the panels at the hemlines.

Two images of Bernhardt in costume show two different highly theatrical costumes from Sardou's play. In the first (above right), Bernhardt is wearing a distinctive belted tabard over an underdress. The tabard is decorated with a motif of appliquéd circles, and the underdress has a panel at the bottom of what may be faces in circles. The second costume (below right) shows Theodora as more regal and formal, but the motifs of appliquéd circles on the tabard and the panel at the hem of the underdress are repeated.

==== The Historical Theodora ====
The 9th edition of the ''Encyclopædia Britannica'' has a substantial entry on her under her own name:&lt;blockquote&gt;THEODORA, the wife of the emperor J&lt;small&gt;USTINIAN&lt;/small&gt; (''q''.''v''.), was born probably in Constantinople, though according to some in Cyprus, in the early years of the 6th century, and died in 547. We shall first give the usually received account of her life and character, and then proceed to inquire how far this account deserves to be accepted. According to Procopius, our chief, but by no means a trustworthy authority for her life, she was the daughter of Acacius, a bear-feeder of the amphitheatre at Constantinople to the Green Faction, and while still a child was sent on to the stage to earn her living in the performances called mimes. She had no gift for either music or dancing, but made herself notorious by the spirit and impudence of her acting in the rough farces, as one may call them, which delighted the crowd of the capital. Becoming a noted courtesan, she accompanied a certain Hecebolus to Pentapolis (in North Africa), of which he had been appointed governor, and, having quarrelled with him, betook herself first to Alexandria, and then back to Constantinople through the cities of Asia Minor. In Constantinople (where, according to a late but apparently not quite groundless story, she now endeavoured to support herself by spinning, and may therefore have been trying to reform her life) she attracted the notice of Justinian, then patrician, and, as the all-powerful nephew of the emperor Justin, practically ruler of the empire. He desired to marry her, but could not overcome the opposition of his aunt, the empress Euphemia. After her death (usually assigned to the year 523) the emperor yielded, and, as a law, dating from the time of Constantine, forbade the marriage of women who had followed the stage with senators, this law was repealed. Thereupon Justinian married Theodora, whom he had already caused to be raised to the patriciate. They were some time after (527) admitted by Justin to a share in the sovereignty; and, on his death four months later, Justinian and Theodora became sole rulers of the Roman world. He was then about forty-four years of age, and she some twenty years younger. Procopius relates in his unpublished history (Άνέκδοπα) many repulsive tales regarding Theodora’s earlier life, but his evident hatred of her, though she had been more than ten years dead when the ''Anecdota'' were written, and the extravagances which the book contains, oblige us to regard him as a very doubtful witness. Some confirmation of the reported opposition of the imperial family to the marriage has been found in the story regarding the conduct of Justinian’s own mother Vigilantia, which Nicholas Alemanni, the first editor of the ''Anecdota'', in his notes to that book, quotes from a certain “Life of Justinian” by Theophilus, to which he frequently refers, without saying where he found it. Since the article J&lt;small&gt;USTINIAN&lt;/small&gt; (''q''.''v''.) was published, the present writer has discovered in Rome what is believed to be the only MS. of this so-called life of Justinian; and his examination of its contents, which he has lately published, makes him think it worthless as an authority. See article T&lt;small&gt;HEOPHILUS&lt;/small&gt;.

Theodora speedily acquired unbounded influence over her husband. He consulted her in everything, and allowed her to interfere directly, as and when she pleased, in the government of the empire. She had a right to interfere, for she was not merely his consort, but empress regnant, and as such entitled equally with himself to the exercise of all prerogatives. In the most terrible crisis of Justinian’s reign, the great Nika insurrection of 532, her courage and firmness in refusing to fly when the rebels were attacking the palace saved her husband’s crown, and no doubt strengthened her command over his mind. Officials took an oath of allegiance to her as well as to the emperor (''Nov''., viii.). She even corresponded with foreign ambassadors, and instructed Belisarius how to deal with the popes. Pro- [253–254] copius describes her as acting with harshness, seizing on trivial pretexts persons who had offended her, stripping some of their property, throwing others into dungeons, where they were cruelly tortured or kept for years without the knowledge of their friends. The city was full of her spies, who reported to her everything said against herself or the administration. She surrounded herself with ceremonious pomp, and required all who approached to abase themselves in a manner new even to that half-Oriental court. She was an incessant and tyrannical match-maker, forcing men to accept wives and women to accept husbands at her caprice. She constituted herself the protectress of faithless wives against outraged husbands, yet professed great zeal for the moral reformation of the city, enforcing severely the laws against vice, and immuring in a “house of repentance” on the Asiatic side of the Bosphorus five hundred courtesans whom she had swept out of the streets of the capital. How much of all this is true we have no means of determining, for it rests on the sole word of Procopius. But there are slight indications in other writers that she had a reputation for severity.

In the religious strife which distracted the empire Theodora took part with the Monophysites, and her coterie usually contained several leading prelates and monks of that party. As Justinian was a warm upholder of the decrees of Chalcedon, this difference of the royal pair excited much remark and indeed much suspicion. Many saw in it a design to penetrate the secrets of both ecelesiastical factions, and so to rule more securely. In other matters also the wife spoke and acted very differently from the husband; but their differences do not seem to have disturbed either his affection or his confidence. The maxim in Constantinople was that the empress was a stronger and a safer friend than the emperor; for, while he abandoned his favourites to her wrath, she stood by her protégés, and never failed to punish any one whose heedless tongue had assailed her character.

Theodora bore to Justinian no son, but one daughter, — at least it would seem that her grandson, who is twice mentioned, was the offspring of a legitimate daughter, whose name, however, is not given. According to Procopius, she had before her marriage become the mother of a son, who when grown up returned from Arabia, revealed himself to her, and forthwith disappeared for ever; but this is a story to be received with distrust. That her behaviour as a wife was irreproachable may be gathered from the fact that Procopius mentions only one scandal affecting it, the case of Areobindus. Even he does not seem to believe this case, for, while referring to it as a mere rumour, the only proof he gives is that, suspecting Areobindus of some offence, she had torture applied to this supposed paramour. Her health was delicate, and, though she took all possible care of it, frequently quitting the capital for the seclusion of her villas on the Asiatic shore, she died comparatively young. Theodora was small in stature and rather pale, but with a graceful figure, beautiful features, and a piercing glance. There remains in the apse of the famous church of St Vitale at Ravenna a contemporaneous mosaic portrait of her, to which the artist, notwithstanding the stiffness of the material, has succeeded in giving some character.

[The next paragraphs are printed in smaller font.]

The above account is in substance that which historians of the last two centuries and a half have accepted and repeated regarding this famous empress. But it must be admitted to be open to serious doubts. Everything relating to the early career of Theodora, the faults of her girlhood, the charges of cruelty and insolence in her government of the empire, rest on the sole authority of the Anecdota of Procopius, — a book whose credit is shaken by its bitterness and extravagance. If we reject it, little is left against her, except of course that action in ecclesiastical affairs which excited the wrath of Baronius, who had denounced her before the ''Anecdota'' were published.

In favour of the picture which Procopius gives of the empress it may be argued (1) that she certainly did interfere constantly and [Col. 1c–2a] arbitrarily in the administration of public affairs, and showed herself therein the kind of person who would be cruel and unscrupulous in her choice of means, and (2) that we gather from other writers an impression that she was harsh and tyrannical, as, for instance, from the references to her in the lives of the popes in the ''Liber Pontificalis'' (which used to pass under the name of Anastasius, the papal librarian). Her threat to the person whom she commanded to bring Vigilius to her was ‘‘nisi hoc feceris, per Viventem in sæcula excoriari te faciam.” Much of what we find in these lives is legendary, but they are some evidence of Theodora’s reputation. Again (3) the statute (''Cod''., v. 4, 23) which repeals the older law so far as relates to ''sceniæ mulieres'' is now generally attributed to Justin, and agrees with the statement of Procopius that an alteration of the law was made to legalize her marriage. There is therefore reason for holding that she was an actress, and, considering what the Byzantine stage was (as appears even by the statute in question), her life cannot have been irreproachable.

Against the evidence of Procopius, with such confirmations as have been indicated, there is to be set the silence of other writers, contemporaries like Agathias and Evagrius, as well as such later historians as Theophanes, none of whom repeat the charges as to Theodora’s life before her marriage. To this consideration no great weight need be attached. It is difficult to establish any view of the controversy without a long and minute examination of the authorities, and in particular of the ''Anecdota''. But the most probable conclusions seem to be — (1) that the odious details which Procopius gives, and which Gibbon did not blush to copy, deserve no more weight than would be given nowadays to the malignant scandal of disappointed courtiers under a despotic government, where scandal is all the blacker because it is propagated in secret (see P&lt;small&gt;ROCOPIUS&lt;/small&gt;); (2) that apparently she was an actress and a courtesan, and not improbably conspicuous in both those charaeters; and (3) that it is impossible to determine how far the specific charges of cruelty and oppression brought against her by Procopius deserve credence. We are not bound to accept them, for they are uncorroborated; yet the accounts of Justinian’s government given in the ''Anecdota'' agree in too many respects with what we know ''aliunde'' to enable us to reject them altogether; and it must be admitted that there is a certain internal consistency in the whole picture which the ''Anecdota'' present of the empress. About the beauty, the intellectual gifts, and the imperious will of Theodora there can be no doubt, for as to these all our authorities agree. She was evidently an extraordinary person, born to shine in any station of life.

Her fortunes have employed many pens. Among the latest serious works dealing with them may be mentioned M. Antonin Débidour’s ''L’Impératrice Theodora: Etude Critique'', Paris, 1885, which endeavours to vindicate her from the aspersions of Procopius; and among more imaginative writings are Sir Henry Pottinger’s interesting romance ''Blue and Green'' (London, Hurst and Blackett, 1879), M. Rhangabé’s tragedy Θεοδωρα (Leipsic, 1884), and M. Sardou’s play ''Theodora'', produced in Paris in 1884. See also Dr F. Dahn’s ''Prokopios von Cäsarea'', 1865. (J. BR.)&lt;ref&gt;J. Br. [James Bryce]. &quot;Theodora.&quot; ''Encyclopædia Britannica: A Dictionary of Arts, Sciences, and General Information''. Ed., Thomas Spencer Baynes, 9th ed. Vol. XXIII (Vol. 23): ''T to UPS''. pp. 253, Col. 2a – 254, Col. 2b. ''Internet Archive'' https://archive.org/details/encyclopaedia-britannica-9ed-1875/Vol%2023%20%28T-UPS%29%20193592732.23/page/254/mode/2up.&lt;/ref&gt;&lt;/blockquote&gt;Edward Gibbons' 1776 ''Decline and Fall of the Roman Empire'' has a passage on Theodora, which would also have been easily available to the people who attended the Duchess of Devonshire's ball. It is part of his chapter on the &quot;Reign of Justinian&quot;:&lt;blockquote&gt;Those who believe that the female mind is totally depraved by the loss of chastity, will eagerly listen to all the invectives of private envy, or popular resentment which have dissembled the virtues of Theodora, exaggerated her vices, and condemned with rigor the venal or voluntary sins of the youthful harlot. From a motive of shame, or contempt, she often declined the servile homage of the multitude, escaped from the odious light of the capital, and passed the greatest part of the year in the palaces and gardens which were pleasantly seated on the sea-coast of the Propontis and the Bosphorus. Her private hours were devoted to the prudent as well as grateful care of her beauty, the luxury of the bath and table, and the long slumber of the evening and the morning. Her secret apartments were occupied by the favorite women and eunuchs, whose interests and passions she indulged at the expense of justice; the most illustrious person ages of the state were crowded into a dark and sultry antechamber, and when at last, after tedious attendance, they were admitted to kiss the feet of Theodora, they experienced, as her humor might suggest, the silent arrogance of an empress, or the capricious levity of a comedian. Her rapacious avarice to accumulate an immense treasure, may be excused by the apprehension of her husband's death, which could leave no alternative between ruin and the throne; and fear as well as ambition might exasperate Theodora against two generals, who, during the malady of the emperor, had rashly declared that they were not disposed to acquiesce in the choice of the capital. But the reproach of cruelty, so repugnant even to her softer vices, has left an indelible stain on the memory of Theodora. Her numerous spies observed, and zealously reported, every action, or word, or look, injurious to their royal mistress. Whomsoever they accused were cast into her peculiar prisons, ... inaccessible to the inquiries of justice; and it was rumored, that the torture of the rack, or scourge, had been inflicted in the presence of the female tyrant, insensible to the voice of prayer or of pity. ... Some of these unhappy victims perished in deep, unwholesome dungeons, while others were permitted, after the loss of their limbs, their reason, or their fortunes, to appear in the world, the living monuments of [38–39] her vengeance, which was commonly extended to the children of those whom she had suspected or injured. The senator or bishop, whose death or exile Theodora had pronounced, was delivered to a trusty messenger, and his diligence was quickened by a menace from her own mouth. &quot;If you fail in the execution of my commands, I swear by Him who liveth forever, that your skin shall be flayed from your body.&quot; ...&lt;/p&gt;

If the creed of Theodora had not been tainted with heresy, her exemplary devotion might have atoned, in the opinion of her contemporaries, for pride, avarice, and cruelty. But, if she employed her influence to assuage the intolerant fury of the emperor, the present age will allow some merit to her religion, and much indulgence to her speculative errors. ... The name of Theodora was introduced, with equal honor, in all the pious and charitable foundations of Justinian; and the most benevolent institution of his reign may be ascribed to the sympathy of the empress for her less fortunate sisters, who had been seduced or compelled to embrace the trade of prostitution. A palace, on the Asiatic side of the Bosphorus, was converted into a stately and spacious monastery, and a liberal maintenance was assigned to five hundred women, who had been collected from the streets and brothels of Constantinople. In this safe and holy retreat, they were devoted to perpetual confinement; and the despair of some, who threw themselves headlong into the sea, was lost in the gratitude of the penitents, who had been delivered from sin and misery by their generous benefactress. ... The prudence of Theodora is celebrated by Justinian himself; and his laws are attributed to the sage counsels of his most reverend wife whom he had received as the gift of the Deity. ... Her courage was displayed amidst the tumult of the people and the terrors of the court. Her chastity, from the moment of her union with Justinian, is founded on the silence of her implacable enemies; and although the daughter of Acacius might be satiated with love, yet some applause is due to the firmness of a mind which could sacrifice pleasure and habit to the stronger sense either of duty or interest. The wishes and prayers of Theodora could never obtain the blessing of a lawful son, and she buried an infant daughter, the sole offspring of her marriage. ... Notwithstanding this disappointment, her dominion was permanent and absolute; she preserved, by art or merit, the affections of Justinian; and their seeming dissensions were always fatal to the courtiers who believed them to be sincere. Perhaps her health had been [39–40] impaired by the licentiousness of her youth; but it was always delicate, and she was directed by her physicians to use the Pythian warm baths. In this journey, the empress was followed by the Praetorian praefect, the great treasurer, several counts and patricians, and a splendid train of four thousand attendants: the highways were repaired at her approach; a palace was erected for her reception; and as she passed through Bithynia, she distributed liberal alms to the churches, the monasteries, and the hospitals, that they might implore Heaven for the restoration of her health. ... At length, in the twenty-fourth year of her marriage, and the twenty-second of her reign, she was consumed by a cancer; ... and the irreparable loss was deplored by her husband, who, in the room of a theatrical prostitute, might have selected the purest and most noble virgin of the East. [ellipsis points mark where footnote numbers were edited out]&lt;ref&gt;Gibbons, Edward. ''The Decline and Fall of the Roman Empire''. Vol. 4. The Ages Digital Library Collections. Albany, OR: Books for the Ages, 1997. Pp. 40–42. ''Internet Archive'' https://archive.org/details/DeclineAndFallOfTheRomanEmpireVol.4ByEdwardGibbons/page/38/mode/2up.&lt;/ref&gt;&lt;/blockquote&gt;

== Demographics ==
* Nationality: French
===Residences===
==Family==

* Julie or Youle (Judith) Bernard&lt;ref name=&quot;:0&quot; /&gt;
* [father]
*# Henriette-Rosine Bernard (22 October 1844 – 26 March 1923)&lt;ref name=&quot;:0&quot; /&gt;


* Sarah Bernhardt (Henriette-Rosine Bernard) (22 October 1844 – 26 March 1923)
* Henri, Hereditary Prince de Ligne (not married, but the father of)
*# Maurice Bernhardt (22 December 1864 – )
* Ambroise Aristide Damala (15 January 1855 – 18 August 1889)&lt;ref&gt;{{Cite journal|date=2025-11-03|title=Jacques Damala|url=https://en.wikipedia.org/w/index.php?title=Jacques_Damala&amp;oldid=1320196285|journal=Wikipedia|language=en}}&lt;/ref&gt;

===Relations===

==Questions and Notes==
==Bibliography==
{{reflist}}</text>
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    <title>Awakening to languages</title>
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{{Education}}{{Course}}
== Starting activity ==

=== Context: ===
You are preparing an activity on sounds and words in different languages around the world. While discussing with your students, you discover that several of them speak languages at home that you do not know at all (e.g., Tigrinya, Albanian, Dari, etc.). You are not quite sure whether or how to integrate them into your activity, nor, if you do want to, where to start.

=== Instructions: ===
Take a few minutes to reflect on the following points:

# What would be your initial reactions or questions as a teacher?  (For example: Should I be interested in this? Is it feasible? Is it beneficial for learning?)
# What simple activity ideas could you imagine based on the presence of these languages in your classroom? (Even if you are unfamiliar with them!) How might you connect them to a corpus that includes several widely taught international languages (English, Spanish, German, etc.)?
# In what ways could you involve the students concerned, as well as their families, in acknowledging and valuing their heritage languages?&quot;

== Objectives ==
By the end of this section, you should be able to...

* Become aware of the importance of linguistic diversity at school;
* Understand what awakening to languages is and what its purpose is;
* Understand the foundations and objectives of awakening to languages;
* Discover the benefits of such an approach for students;
* Understand the importance of metalinguistic reflection in learning;
* Find ideas to involve students and families, seeing them as resources for implementing such an approach.

== Key words ==
pluralistic approaches; teaching plurilingualism; metalinguistic reflection; language awareness; plurilingual and intercultural competence.

== Prerequisites ==

* Basic knowledge of linguistics and language acquisition (optional).
* Basic knowledge of language families

== Table of contents ==

# Introduction
# History
# Definitions
# Take home messages
# Self-assessment

== Introduction ==
Awakening to languages is one of the four pluralistic approaches, and it represents an innovative pedagogical approach. It is a form of learning through problem solving, with specific objectives and the use of various strategies and cognitive processes (Charitonidou &amp; Ioannitou, 2012).

Specifically, it is an original approach used in primary schools, where children discover the diversity of languages and their functions (Candelier, 2003; Dabène, 2003). This approach includes the development of attitudes, skills and knowledge about languages, and is based on several objectives:

* linguistic (discovery of languages and their specific features),
* cognitive (mobilization of mental processes that support learning, but also relating languages to each other),
* sociolinguistic (understanding the roles and social representations of languages),
* psychological (ability to take a detached view of linguistic diversity).

By promoting a better understanding of language, this approach encourages curiosity, metalinguistic reflection, and openness to linguistic and cultural diversity.

== History ==
Awakening to languages became widely recognized in 2003 thanks to the European EvLang program, but it was inspired by earlier initiatives.

In the 1970s in Australia, faced with high levels of immigration, a language awareness approach was introduced in schools, even though English remained dominant (Dompmartin-Normand, 2011).

In the 1980s in the United Kingdom, Eric Hawkins developed the Language Awareness movement because students were struggling with English and lacked motivation to learn other languages (Bullock, 1975; Fidler, 2006). He suggested a bridge course between languages to encourage reflection on language and understanding of how it works (Hawkins, 1984; Bernaus et al., 2008).

In France, the idea was taken up in the late 1980s, especially in primary schools, in order to promote the languages of origin of migrant children and stimulate interest in all languages, even those not taught (Little &amp; Kirwan, 2018; Candelier, 2005).

Gradually, concepts such as metalinguistic reflection and cross-curricular skills were developed (Hawkins, 1999), while pilot projects were carried out in multicultural schools (Billiez, 1992; Dabène, 1991). Other countries followed suit, such as Italy (educazione linguistica) and Switzerland (EOLE program) (Bernaus et al., 2007). This work led to the EvLang and Ja-Ling programs, which disseminated the approach on a large scale (Kervran, 2006).

== Definitions ==
Awakening to languages is a linguistic approach through which students discover the world of languages by exploring their diversity and their functions. To understand this approach, we can refer to the definition given by Michel Candelier, coordinator of the EvLang project:

Awakening to languages is fostered when some activities focus on languages that the school does not aim to teach, whether or not they are spoken at home by certain pupils. This does not mean that only such activities are part of the approach, since it is above all a comprehensive, often comparative process involving these languages, the language of schooling, and the foreign languages taught. (Candelier, 2003, translated by the lesson’s authors)

What makes this approach unique is that no language is excluded. It allows for the full recognition of students' languages, even if they are not the main language of the school, and can also help with language learning throughout schooling.

When students engage in awakening to languages activities, they explore new sounds, observe different writing systems, compare languages reflecting on their similarities and differences, and become aware of the value of their own linguistic repertoire. In this way, they develop their ability to analyze human language (Armand, 2000; 2005).

The purpose of this approach is also to highlight the languages students bring from home (Candelier et al., 2012). Thus, awakening to languages can serve as preparation for language learning, which Zarate (1995) calls ''propédeutique'', laying solid foundations in the early years. It develops skills such as linguistic observation, reflection, positive attitudes towards languages, and openness to intercultural encounters (Dabène, 2003; Bernaus et al., 2008).

In short, it is preparation for language learning, which seeks to arouse students' curiosity, interest, and confidence in languages and cultures, and to strengthen their ability to observe, analyze, and make connections between different languages.

In order to implement this approach, it is useful to refer to the tools/resources offered by the EvLang program. The approach presented below:

# Setting the scene or anchoring. This first step serves to introduce the topic. It links the new material to the students' existing knowledge and their everyday life in the classroom. The aim is to create a “learning contract” with the students and ask an interesting and open-ended question (e.g., “Why don't we speak just one language in the world?”) that motivates the students to participate. This is also when the students' representations of languages emerge. They exchange their points of view, ask questions that interest them, and this encourages their engagement. Students also share their personal experiences, their linguistic histories, and their intuitive knowledge about languages, their form, and their role in everyday life.
# The research situation. During this stage, students become a kind of detectives. They have to solve a problem presented by the teacher. A “linguistic tension” arises, which pushes them to discover new knowledge on their own. They develop strategies: observing, comparing, analyzing, hypothesizing, organizing, and discussing their ideas. They work in groups, negotiate, make tentative generalizations, and question what they knew about the language before.
# Synthesis. This is the final stage, where the students' personal and collective discoveries take shape. They express what they have observed and understood by working together to create a concrete or conceptual solution or tool. This is the moment when everything they have discovered becomes clear and new knowledge is constructed.

== An example of an awakening to languages activity ==

=== “The sounds of my classroom” ===

==== 1. Setting the scene: Discovering languages in the classroom ====
'''Objective''': To make students aware of the linguistic diversity of their group.

'''Procedure''':

* The teacher asks each student to think of a word, a short expression, or a special sound in the language they speak at home.
* Each student shares their word or sound aloud with the rest of the class.
* The teacher writes these words or sounds on the board, specifying the language if possible.
* Simple questions are asked to spark curiosity:
** “Who knows this language?”
** “What do you find interesting or surprising about these words?”
** “Have you heard these sounds anywhere else?”

==== 2. Research situation: Observe and compare sounds ====
'''Objective''': To encourage students to listen carefully and think about the differences and similarities between languages.

'''Procedure''':

* The teacher reads the words or repeats the sounds one by one, slowly.
* The students listen and repeat if possible, trying to imitate the sounds.
* Then, in small groups (3-4 students), they discuss the following questions:
** Which sounds seem similar to you?
** Which sounds seem different to you?
** Are there any sounds you don't know?
* Each group writes or draws their observations on a sheet of paper (for example, a table with columns labeled “similar” and “different”).

==== 3. Synthesis: Share and create a collective tool ====
'''Objective''': Develop a collective understanding and highlight the students' discoveries.

'''Procedure''':

* Each group presents its observations to the whole class.
* The teacher writes down the important ideas on a large poster or board entitled “The sounds of our class.”
* Together, create a small booklet or poster with:
** The words or sounds collected,
** The similarities and differences observed,
** A few drawings or symbols to illustrate the sounds.
* Students are encouraged to use this booklet or poster in future language activities.
* The activity ends with a discussion about what these discoveries have taught us:
** “Why is it important to know the languages spoken in the classroom?”
** “What does this teach us about languages in general?”

==== Teaching comments ====

* This activity develops linguistic curiosity and metalinguistic reflection from the very beginning of learning.
* It values all of the students' languages, even those that are unfamiliar to the teacher.
* Group work promotes cooperation and dialogue.
* The booklet or poster creates a concrete tool that reminds students of the linguistic richness present in the classroom.
* The teacher can adapt the difficulty of the questions according to the age of the students.
* Families can also be invited to participate by sending words or sounds from home, which strengthens the school-family bond.

== Take home messages ==

* Awakening to languages stands out for its openness to all languages, without exception, including those spoken at home.
* The activities encourage language discovery through comparative work and reflection.
* The approach develops observation skills, curiosity, and interest in languages and cultures, as well as a positive attitude toward diversity.
* It is not a language teaching method, but rather an active preparation for learning.

== Self-assessment ==
&lt;quiz display=&quot;simple&quot;&gt;
{What is language awareness?}
-a) A method for teaching foreign languages.
-b) An activity for learning a language at home.
+c) An approach to discovering the diversity of languages without teaching them.
-d) A course for becoming a language teacher.

{What is the main objective of language awareness?}
-a) To correct students' mistakes in all languages.
+b) To develop curiosity and reflection about languages.
-c) To teach English earlier in school.
-d) To learn to read and write in all the languages spoken in the class.

{What do students do during a language awareness activity?}
-a) They learn to speak a new language in a month.
-b) They have to translate all the words into their own language.
+c) They observe, compare languages, listen, and discuss.
-d) They prepare a dictation in a foreign language.

{Why is it useful to involve students and their families in these activities?}
-a) To have more homework to do at home.
-b) To check if families speak French well.
+c) To value the languages spoken at home and create a link with the school.
-d) To do translations during class.

&lt;/quiz&gt;

== Resources to go further ==

* https://elodil.umontreal.ca/
* https://dulala.fr/
* https://bilem.ac-besancon.fr/faire-classe/leveil-aux-langues/ 

== Bibliography ==
Bernaus, M., Andrade, A.I., Kervran, M., Murkowska, A. &amp; Saez, F. T. (2007) Plurilingual and Pluricultural awareness in language teacher education. Strasbourg: ECML, Council of Europe Publishing.

Bernaus, M., Andrade, A-I., Kervran, M., Murkowska, A. &amp; Trujilli Saez, F. (2008). Plurilingual and pluricultural awareness in language teacher education : a training kit. Strasbourg: Council of Europe.

Billiez, J. (1992). L’enseignement précoce des langues vivantes dans un environnement scolaire multilingue: vers une solution alternative. In Bouchard, R. (éd.) ''Acquisition et enseignement/apprentissage des langues'', (pp.257-269). Lidilem.

Candelier, M. (2003). Evlang – l’éveil aux langues à l’école primaire – Bilan d’une innovation européenne. De Boeck – Duclot. 

Candelier, M. (2005). « L'éveil aux langues : une approche plurielle des langues et des cultures au service de l'extension des compétences linguistiques », Prudent, L-.F., Tupin, F. et Wharton S. (rédacteurs), Du plurilinguisme à l'école – Vers une gestion coordonnée des langues en contextes éducatifs sensibles, (pp. 417-436). Peter Lang.

Charitonidou, A. &amp; Ioannitou, G. (2012). « L’autonomie des enseignants : quels éléments caractérisent l’enseignant autonome et comment ils influent sur la décision de la mise en œuvre d’une innovation pédagogique ? ». Synergies France, nº 9, p. 51-59.

Dabène, L. (2003). Préface. In M. Candelier (coord), L’Éveil au langues à l’école primaire: EvLang: bilan d’une innovation européenne,  13-17, Bruxelles: de Boeck. 

Dabène, L. (1991). L’éveil au langage: compte-rendu d’une expérience en cours. In Les langues vivantes à l’école élémentaire, Actes du colloque de juin 1990, (pp. 105-108). INRP.

Dompmartin-Normand, C. (2011). Éveil aux langues et aux cultures à l'école : une démarche intégrée avec un triple objectif cognitif, affectif et social. ''L’ Autre, 12''(2), 162-168.

Fidler, S. (2006). Awakening to languages in primary school. ''ELT Journal, 60''(4), 346–354.

Hawkins, E. (1984). Awareness of language: An introduction. Cambridge: Cambridge University Press.

Hawkins, E. (1999). Foreign Language Study and Language Awareness, 8 (3-4), 124-142. 

Kervran, M. (2006). Pourquoi et comment faire appel à la diversité des langues du monde à l'école primaire? Spirale- Revue de recherches en éducation, 38, 27-35. 

Little, D., &amp; Kirwan, D. (2018). From plurilingual repertoires to language awareness: Developing primary pupils’ proficiency in the language of schooling. In C. Hélot, C. Frijns, K. Gorp &amp; S. Sierens (Eds.), ''Language Awareness in Multilingual Classrooms in Europe: From Theory to Practice'', (pp.169-206). Walter de Gruyter GmbH &amp; Co KG.

Zarate, G., (1995). Questions autour des pratiques interculturelles associées à l'éveil au langage. In D. Moore (coord) L'éveil au langage. Notions en questions. Rencontres en didactique des langues, 129-134, CREDIF- LIDILEM.


==Credits==
This resource has been created by [[User:Projet PEP|Projet PEP]] ([[User talk:Projet PEP|discuss]] • [[Special:Contributions/Projet PEP|contribs]]) (Erasmus+ project, co-financed by the European Commission) :

* Eftychia Damaskou (University of Thessaly) 
* Lisa Brinkmann (Universität Hamburg).

[[Portal: Plurilingual education]]
[[Category:Languages]]</text>
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      <text bytes="86265" sha1="8ke7zbrcbg8ef1esz1ubh7xiek019y1" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon {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  form the ''edge polygon'' of the 16-cell {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 the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

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

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic 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 ''characteristic left rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== 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 tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the characteristic left rotation 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 helix is an 8-rung ladder twisted 3 times, bent into a circle in the fourth dimension. Each rung is a tesseract edge.

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, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in 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;]]
We can rotate the 24-cell isoclinically in Clifford parallel invariant planes containing 16-cell edges, for example in the characteristic left 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° chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once.

We can also rotate the 24-cell isoclinically in Clifford parallel invariant planes containing 24-cell edges. A complete 24-cell edge plane revolution requires 720° like a complete 16-cell 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 central 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.

[[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 a single &lt;math&gt;r_{1}&lt;/math&gt; edge each, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''characteristic left rotation of the 24-cell'', also Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; {12/5} star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. 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 skew&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 ''characteristic right rotation of the 24-cell'', also Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; 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 skew 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 every 180° of isoclinic rotation each vertex circles a skew 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 skew 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 two twisted parallel strands that intersects each 24-cell vertex once. In every 360° of isoclinic rotation each vertex circles a skew great square 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 triangles 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.  

...  

{{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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== Slight modifications of the article ==

Hello,&lt;br&gt;
I imported this page from the Wikipedia article, which I revamped. But since the import, some contributors made helpful comments and edits. I tried to update them all here, but now I stopped and I will just re-import the Wikipedia article when the peer-review process will start. Please notify me when it happens, or re-import it yourself {{=)}}. [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 09:48, 13 January 2026 (UTC)

==Peer review 1==
{{review
 |reviewer   =Sanjay Ramassamy
 |Q          =Q102641962
 |affiliation=Université Paris-Saclay, CNRS, CEA, Institut de Physique Théorique
 |link       =https://www.normalesup.org/~ramassamy/index.html.en
 |date       = 1 June 2026
 |text       =
This review article is very well-written, mathematically sound and accessible to people outside the field. I only have minor comments below, most of them typos. I recommend publishing the article once the comments are taken into account.

General comment: There are several figures next to the text, but the figures don't seem to be cited in the text. I don't know if this is a journal policy, but it looks a bit unusual to me.

Second sentence of the abstract: there is twice &quot;&quot;a new polygon&quot;&quot;. Maybe you could rephrase it in a way to eliminate one of the occurrences. E.g. something like &quot;&quot;It defines a new polygon whose vertices are obtained as the intersection points of the shortest diagonals of the initial polygon.&quot;&quot;

End of first paragraph of the abstract: maybe you could already reference Schwartz's original paper here.

Euclidean plane: please capitalize the first letter of &quot;&quot;Euclidean&quot;&quot; throughout the article

Section &quot;&quot;On polygons&quot;&quot;: &quot;&quot;Finally, it is possible that two diagonals are parallel and not intersect&quot;&quot; -&gt; &quot;&quot;and don't intersect&quot;&quot;

Section &quot;&quot;On the moduli space of polygons&quot;&quot;: it is the first time that I see the term &quot;&quot;projectivity&quot;&quot;. I checked that it was indeed correct, but in all the talks/articles that I have seen on the topic, people rather used &quot;&quot;up to projective transformations&quot;&quot;.

Section &quot;&quot;Historical elements&quot;&quot;, last sentence: it is not too clear what that sentence means. The pentagram map pertains to the field of incidence geometry, like these 3 theorems. What are the further similarities ? Further down in the article, in the section &quot;&quot;Pentagons and hexagons&quot;&quot;, there is a similar sentence: &quot;&quot;The action of the pentagram map on pentagons and hexagons is similar in spirit to classical configuration theorems in projective geometry such as Pascal's theorem, Desargues's theorem and others&quot;&quot;. Is it just the case of pentagons and hexagons that resembles these theorems ?

Section &quot;&quot;Definition of the map&quot;&quot;, first paragraph: it looks strange to cite Weinreich's paper to justify the rather obvious fact that the dimension of the space of n-gons is 2n. More generally, for review articles in WikiJournal, what is the purpose of citations ? Providing a source where something is nicely explained ? Or providing the first source to show some result ? In this article, it seems to be rather the former.

Section &quot;&quot;Definition of the map&quot;&quot;, second paragraph: &quot;&quot;Taking the intersection of the two...&quot;&quot; -&gt; &quot;&quot;Taking the intersection of two...&quot;&quot;

Section &quot;&quot;Twisted polygons&quot;&quot;: &quot;&quot;space of twisted n-gon&quot;&quot; -&gt; &quot;&quot;space of twisted n-gons&quot;&quot;

&quot;&quot;the dynamic&quot;&quot; -&gt; &quot;&quot;the dynamics&quot;&quot; It comes with a final s even though it is singular, e.g. &quot;&quot;the dynamics is integrable&quot;&quot;

Section &quot;&quot;Pentagons and hexagons&quot;&quot;: &quot;&quot;The two following facts&quot;&quot; -&gt; &quot;&quot;The following two facts&quot;&quot;

Section &quot;&quot;Poncelet polygons&quot;&quot;: circumbscribed -&gt; circumscribed

Section &quot;&quot;Poncelet polygons&quot;&quot;: &quot;&quot;For a convex Poncelet n-gons&quot;&quot; -&gt; n-gon

Section &quot;&quot;ab-coordinates&quot;&quot;: I would write &quot;&quot;vertices v_k&quot;&quot; and &quot;&quot;vectors V_k&quot;&quot; rather than &quot;&quot;vertices v_k's&quot;&quot; and &quot;&quot;vectors V_k's&quot;&quot;

Section &quot;&quot;As a birational map&quot;&quot;: you have twice in a row the word pentagram in the first line

Section &quot;&quot;The scaling symmetry&quot;&quot;: &quot;&quot;an s&quot;&quot; -&gt; &quot;&quot;and s&quot;&quot;. 

Section &quot;&quot;The scaling symmetry&quot;&quot;: &quot;&quot;An homogeneous&quot;&quot; -&gt; &quot;&quot;A homogeneous&quot;&quot;. Why do you define the notion of weight in this section ? It looks weird because you don't use it immediately, but only towards the end of the next section. It would suggest moving it much closer to the place where you first use it.

Section &quot;&quot;The spectral curve&quot;&quot;, last sentence: here you write &quot;&quot;algebraic integrability&quot;&quot;. In the next sentence it is called &quot;&quot;algebro-geometric integrability&quot;&quot;. I prefer the latter formulation.

Section &quot;&quot;The spectral curve&quot;&quot;: &quot;&quot;some renormalization it&quot;&quot; -&gt; missing &quot;&quot;of&quot;&quot;

Section &quot;&quot;Algebro-geometric integrability&quot;&quot;: &quot;&quot;in term of&quot;&quot; -&gt; terms

Section &quot;&quot;Dimension of the invariant manifold&quot;&quot;: &quot;&quot;For a twisted n-gons&quot;&quot; -&gt; &quot;&quot;For twisted n-gons&quot;&quot;

Section &quot;&quot;Dimension of the invariant manifold&quot;&quot;: what does it mean that the dimension of the invariant tori drops by 3 for closed n-gons ? That it is always n-3 regardless of the parity of n ? Shouldn't invariant tori always be even-dimensional ? Maybe make a separate sentence discussing the closed n-gons case.

Section &quot;&quot;Cluster algebras&quot;&quot;: rather than &quot;&quot;special cases of cluster algebra&quot;&quot;, I would suggest something like &quot;&quot;special cases of discrete dynamical systems powered by cluster algebras&quot;&quot;. Because the pentagram map itself is not a cluster algebra. Also, the mutations of the underlying cluster algebra induced by the pentagram map are only a subset of all possible mutations.

Section &quot;&quot;Generalizations&quot;&quot;: &quot;&quot;description ... as cluster algebras&quot;&quot; -&gt; maybe &quot;&quot;in terms of cluster algebras&quot;&quot; ?

Section &quot;&quot;Generalized pentagram maps&quot;&quot;: it could be helpful to write that one recovers the original pentagram map by taking d=2, I={2}, J={1}. What surprises me is that for this original pentagram map the set I and J are not equal and yet it is integrable. How is that compatible with the statement that &quot;&quot;the general case is not integrable&quot;&quot; ? Also, just below, the dented pentagram maps provide another class of integrable examples where I and J are not equal. How do you quantify that most cases are not integrable.

Section &quot;&quot;Corrugated polygons&quot;&quot;: &quot;&quot;they can retrieved&quot;&quot; -&gt; &quot;&quot;they can be retrieved&quot;&quot;

&quot;&quot;Grassmannians polygons&quot;&quot; -&gt; &quot;&quot;Grassmannian polygons&quot;&quot;

&quot;&quot;the space of Grassmannians Gr(m,md)&quot;&quot; -&gt; &quot;&quot;the Grassmannian space Gr(m,md)&quot;&quot;

&quot;&quot;A point in v&quot;&quot; -&gt; &quot;&quot;A point v&quot;&quot;

&quot;&quot;general linear group Gl_{md}&quot;&quot; -&gt; &quot;&quot;general linear group GL_{md}&quot;&quot;

&quot;&quot;faithfull&quot;&quot; -&gt; faithful

&quot;&quot;generically define&quot;&quot; -&gt; &quot;&quot;generically defines&quot;&quot;

&quot;&quot;a new point of v&quot;&quot; -&gt; &quot;&quot;a new point v&quot;&quot;

}}

{{response|1 =Hello, and thanks a lot for the thorough review. I am a bit embarrassed by the numerous typos, they are now fixed. I also reformulated many items following your suggestions. There remains two points I need to answer to.
* Indeed, the citation of papers (even for obvious facts) is more frequent than in classical papers. This is because Wikipedia aims to have every statement linked to a reference (see [[w:Wikipedia:Verifiability]]). Some editors take this very seriously (see [https://en.wikipedia.org/wiki/Wikipedia%20talk:WikiProject%20Mathematics/Archive/2025/Dec this discussion]), so I added citations to almost every paragraphs. I guess it could be mitigated for publication.
* I clarified the statement about the dimension of invariant manifolds for closed polygons, with one more citation. According to it, they will always be odd-dimensional.
Thanks again, [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 15:44, 2 June 2026 (UTC)}}

== Peer review 2 ==

{{review
 |reviewer   =Paul Melotti
 |Q          = Q103240269
 |affiliation=Université Paris-Saclay
 |link       =https://www.imo.universite-paris-saclay.fr/~paul.melotti/
 |date       = 11 June 2026
 |text       =
This is a very well-written summary of results on the pentagram map, a fascinating topic that deserved a good presentation in the wikipedia universe. The paper is presented in a clear and coherent way, and I believe it is accessible to non-specialists, provided some minimal background in projective geometry. As far as I could check, the claims are supported by the plentiful references, and they give a good overview of the topic, its history, connections to various topics in mathematics, and modern perspectives.

As a general remark, I think the special property of the map T on the spaces of pentagons and hexagons, stated in Section &quot;Periodic orbits on the moduli space&quot;, could be stated earlier in the paper, possibly in an informal way. They are quite striking and, in my opinion, motivate the study of the generic transformation.

Here are a few minor remarks:
- several references to pictures use the phrase &quot;on Figure...&quot;, I believe &quot;in Figure...&quot; is more common.
- &quot;its interpretation as a cluster algebra&quot; -&gt; maybe &quot;in terms of a cluster algebra&quot;, or something similar, would be more precise.
- On reference [2] by Gekhtman and Izosimov, &quot;Integrable Systems and Cluster Algebras&quot;, the link to sciencedirect in &quot;Works cited&quot; doesn't seem to work when I click it. This might be on my side, but please check the URL.
- &quot;for generic polygons on the real projective plane&quot; -&gt; &quot;in&quot; the projective plane seems more common?
- &quot;by taking lines and intersections of them&quot; sound a bit weird to me (but I'm not a native speaker so maybe it's okay)
- maybe at the beginning of Section &quot;Coordinates for the moduli space&quot;, announce that these will allow for nice expressions of the map T in those coordinates (as it is done in the following section).
- &quot;This generically makes a quasiperiodic motion.&quot; -&gt; &quot;makes&quot; sounds a bit vague to me, maybe &quot;induces a quasiperiodic motion on the corresponding torus&quot; or something.
- In the subsection &quot;Grassmannian polygon&quot;, second paragraph, I am a bit confused with notations and conventions. If we represent the vector space $v$ by a basis, and put the vectors in columns, we get a matrix of size $md \times m$ and not $m \times md$ right? And then, the action of $GL_{md}$ you are mentioning is simply multiplication on the left?
}} [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 02:30, 15 June 2026 (UTC)

{{response|1 = Hello and many thanks for the review. I implemented the changes following your remarks. There was indeed a confusion in the &quot;Grassmannian polygon&quot; section, which is now fixed. Thank you for your vigilance. [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 13:01, 16 June 2026 (UTC)}}

== Peer review 3 ==

{{review
 |reviewer   =Richard Evan Schwartz
 |Q          =Q3893370
 |affiliation=Brown University
 |link       =
 |date       = 15 June 2026
 |text       =
This article is an update of the wikipedia page for the pentagram map, which I largely wrote myself. (I wrote almost the entire thing because what had been there initially was not very good.)  I think that JB did an excellent job updating the pentagram map page.  The article hits the main points : classical geometric results, Arnold-Liouville integrability, algebro-geometric integrability, Lax Pairs, connections to cluster algebras, Glick's result about the collapse point, and various generalizations.  
}} [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 20:37, 15 June 2026 (UTC)

{{response|1 = Hello and thank you for the review. Indeed, you made a massive contribution to the Wikipedia article, as it can be seen by comparing this two versions: [https://en.wikipedia.org/w/index.php?title=Pentagram%20map&amp;oldid=436156794 before] and [https://en.wikipedia.org/w/index.php?title=Pentagram%20map&amp;oldid=438579263 after]. Of course, as stated in the [[WikiJournal User Group/Publishing|guidelines of the journal]], [https://xtools.wmcloud.org/articleinfo/en.wikipedia.org/Pentagram%20map Wikipedia contributors] are also credited. My contribution to reshape it to the standards of the Wikijournal of Science can be seen [https://en.wikipedia.org/w/index.php?title=Pentagram_map&amp;diff=1359956775&amp;oldid=1317663617 here]. [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 14:17, 16 June 2026 (UTC)}}

== Peer review 4 ==

{{review
 |credentials=I have co-authored several papers on this subject
 |date       = 21 June 2026
 |text       =
It is a very good review that covers various aspects of the pentagram map. 

I have noticed one error/typo: in the sentence &quot;The dynamic is trivial for the classes of pentagons and heptagons, but this stops to be the case for polygons with more vertices.&quot; heptagons should be replaced by hexagons. 

I'd suggest to mention the work by Goncharov and Kenyon that, in particular, implies the integrability of the pentagram map: see MR3675462, and a related paper by Izosimov: MR4472585. Perhaps one should also mention the relation to T-systems, see MR3282370.
}} [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 14:06, 30 June 2026 (UTC)

{{response|1 = Hello and thank you very much for the review. I corrected the typo and added the papers you mentioned. I did a little section about the octahedron recurrence, however it's a bit superficial since defining it properly could make a little Wikipedia article on its own. I hope what I wrote is fine. [[User:Regliste|Regliste]] ([[User talk:Regliste|discuss]] • [[Special:Contributions/Regliste|contribs]]) 20:40, 2 July 2026 (UTC)}}</text>
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{{Education}}{{Course}}
== Starting Activity ==
Imagine you are participating in an online language exchange with students from different countries. You and your peers speak different first languages, but you also share some knowledge of additional languages (e.g., English, Spanish, French, Italian). During your video meetings, group chats, and collaborative projects, you are encouraged to use all your linguistic resources to communicate, clarify, and co-construct meaning.

'''Reflect and respond:'''

# Which languages might you choose to use during the exchange? Why?
# How would you draw on your knowledge of different languages to make yourself understood—and to understand others?
# Have you ever used more than one language in a single interaction? What did you notice about how communication worked?
# What strategies might help you deal with misunderstandings or unfamiliar words?
# In your opinion, what are the potential benefits of using multiple languages in a collaborative learning setting?

== Objectives ==
By the end of this module, you will be able to:

* Define key concepts such as plurilingualism, telecollaboration, tandem learning, and linguistic mediation.
* Explain how telecollaborative practices support plurilingual education and intercultural communication.
* Identify digital tools and communicative strategies that facilitate meaningful multilingual interaction.
* Analyze examples of telecollaborative projects through a plurilingual lens.
* Reflect on your own linguistic repertoire and how it can be activated in virtual exchanges.

== Keywords ==
Plurilingualism, telecollaboration, intercultural competence, digital language exchange, linguistic mediation, online interaction, tandem learning, virtual exchange.

== Introduction ==
Telecollaboration is increasingly understood as an action-oriented and collaborative educational approach, rooted in authentic tasks that mirror real-world communicative, academic, and professional situations. In such contexts, learners are not merely exchanging linguistic forms; they co-construct meaning, negotiate across linguistic and cultural repertoires, and develop transversal competences through collaborative problem-solving. Typical tasks include the reciprocal explanation of culturally embedded concepts, the co-creation of multilingual digital artefacts (e.g., digital menus, promotional videos, brochures), and collaborative projects that require learners to navigate semantic ambiguity, cultural diversity, and multiple perspectives. 

At the same time, the growing presence of multilingualism in digital environments has transformed telecollaboration into an important space for developing plurilingual competences. As multilingual communication increasingly takes place through multimodal online platforms, learners naturally draw on their linguistic and semiotic repertoires when interacting across geographical and cultural boundaries. Digital environments therefore provide authentic opportunities to experience multilingualism not only as an object of study but also as a communicative practice. 

When informed by a plurilingual perspective, telecollaboration becomes a powerful means of valuing learners' entire linguistic repertoires. Rather than enforcing a shared lingua franca, educators may encourage, for instance, intercomprehension strategies, whereby participants communicate using different languages and rely on partial understanding, contextual cues, mediation, and mutual scaffolding. This approach, supported by projects such as ComunicaRT and IOTT (Paone, 2024; Leone &amp; Garbarino, 2019), emphasizes that successful communication depends less on native-like proficiency than on the ability to adapt, mediate, and collaborate across linguistic and cultural boundaries. 

Recent research further suggests that multilingual telecollaborative projects can significantly enhance learners' plurilingual and pluricultural competence, crosslinguistic awareness, and understanding of multilingualism as a dynamic social practice. In teacher education, telecollaboration has also been shown to foster multilingual awareness, encourage reflection on linguistic identities, and prepare future teachers for linguistically and culturally diverse classrooms. Such plurilingual telecollaborative practices are closely aligned with inclusive, learner-centred pedagogies. They foster strategic competence, metalinguistic awareness, intercultural mediation, and reflective language awareness, all of which are essential for navigating increasingly multilingual societies. Learners thus develop as plurilingual social agents who are equipped not only to communicate, but also to mediate understanding between diverse linguistic and cultural communities.

== History of the concept ==
Telecollaboration, often referred to as virtual exchange, originated in the early 1990s as a pedagogical response to the increasing accessibility of internet technologies. Its core idea was to connect language learners from different geographical and cultural backgrounds through online communication tools, thereby fostering both linguistic competence and intercultural awareness. Early implementations—typically based on email exchanges, discussion forums, and asynchronous writing tasks—were shaped by the emerging field of Computer-Mediated Communication (CMC) and influenced by theories of network-based language learning (Warschauer, 1997; Kern, 1996).

As digital technologies evolved, so too did the modalities of telecollaboration. Synchronous tools like video conferencing, collaborative platforms, wikis, blogs, and social media enabled more interactive, multimodal, and project-based exchanges. This led to the development of richer tasks that emphasized collaboration, co-construction of meaning, and intercultural reflection. Telecollaboration thus expanded from a tool for language practice into a comprehensive pedagogical model embedded in task-based and action-oriented learning.

In the 2000s and 2010s, scholars such as Robert O’Dowd played a key role in conceptualizing and consolidating the field. O’Dowd (2018) distinguishes between telecollaboration as a more open-ended educational practice and virtual exchange, which often refers to institutionally supported and structured programmes, such as those sponsored by the European Union (e.g., Erasmus+ Virtual Exchange). These initiatives highlight the potential of online intercultural encounters not only for language development, but also for advancing global citizenship, digital literacy, and inclusivity in education.

Telecollaboration’s evolution has also been supported by interdisciplinary insights from CALL (Computer-Assisted Language Learning), intercultural communication, and plurilingual education. With the growing focus on learner agency, plurilingual competence, and social constructivist approaches, telecollaboration is now widely regarded as a powerful action-oriented, learner-centered practice that promotes authentic communication across languages and cultures.

== Definitions ==
Plurilingualism refers to an individual's capacity to flexibly and dynamically use multiple languages, integrating linguistic knowledge across various languages rather than viewing them as isolated systems. This holistic approach emphasizes the interconnectedness of languages within a person's repertoire, allowing for fluid movement between languages depending on the context and communicative needs.

Telecollaboration involves the use of digital communication tools to connect learners from diverse linguistic and cultural backgrounds for educational purposes. This approach facilitates intercultural exchanges and language learning through online platforms, enabling participants to engage in collaborative projects, discussions, and activities that enhance both linguistic proficiency and cultural understanding.

Tandem Learning is a (telecollaborative) method where two individuals with different native languages partner to assist each other in learning their respective languages. This reciprocal arrangement allows each learner to benefit from the native proficiency of their partner, promoting authentic language use and cultural exchange. Sessions typically involve equal time dedicated to each language, ensuring balanced learning opportunities.

Linguistic Mediation encompasses the ability to interpret, facilitate, and relay information between speakers of different languages. It is a crucial component in plurilingual telecollaboration, as it enables effective communication and understanding in multilingual contexts. Mediators employ their linguistic skills to bridge language gaps, ensuring that meaning is accurately conveyed and comprehended across language barriers.

Collectively, these concepts underscore the importance of integrating multiple languages and cultures in educational settings, leveraging digital tools and collaborative methods to enrich language learning and intercultural competence.

== Practical examples ==

=== '''Example 1: Intercomprehension in Telecollaborative Mentoring''' ===
In a transatlantic telecollaborative project, heritage Spanish speakers in California engaged in Italian language learning through mentoring sessions with Italian university students. These sessions encouraged the use of Spanish as a ''pivot language'' to support the acquisition of Italian, promoting '''intercomprehension''' based on the typological proximity of Romance languages. Learners were encouraged to draw on their plurilingual repertoires to negotiate meaning and build metalinguistic awareness during real-time Zoom interactions​ (Cortés Velásquez, Donato, &amp; Ricciardelli, 2023).

=== '''Example 2: Intercomprehension and Teletandem in the IOTT Project''' ===
The IOTT project, a collaboration between the University of Lyon 2 and the University of Salento, implemented a telecollaborative learning scenario combining intercomprehension and teletandem methodologies. Students from different linguistic backgrounds engaged in synchronous oral sessions via VoIP technologies, utilizing their respective Romance languages to communicate. This approach emphasized the development of receptive skills in intercomprehension, allowing learners to understand related languages without prior formal instruction. The project also incorporated reflective practices, such as learning diaries and self-assessment tools, to enhance metalinguistic awareness and foster autonomous learning strategies. Findings indicated that this integrative model effectively promoted plurilingual competencies and intercultural understanding among participants (Garbarino &amp; Leone, 2020).

=== '''Example 3: The Trans-Atlantic and Pacific Project (TAPP)''' ===
TAPP connects classes from Europe and the US in joint professional and linguistic projects. In multilingual group tasks, students collaborate on writing, usability testing, and translation. For example, US engineering students produced presentations peer-reviewed by European students. These exchanges promote '''co-writing and multilingual mediation''', with English often used as a lingua franca, but with growing attention to local languages and translation practices as tools for intercultural understanding (O’Dowd, 2018).

=== '''Example 4: Digital Storytelling in Multilingual Settings''' ===
In a university course, students participated in a virtual intercultural exchange via Google+ and Google Drive. One of the key tasks involved creating a '''digital story''' in teams. Learners used multiple languages for narration and subtitling, leveraging their full linguistic repertoires. These digital products were shared and peer-reviewed across institutions, showcasing the integration of '''multimodal literacy, plurilingual resources, and intercultural storytelling'''​ (Nicolaou &amp; Sevilla-Pavón, 2016).

=== '''Example 5: Tandem Feedback and Plurilingual Awareness through Focus on Form''' ===
In a virtual exchange between heritage Spanish-speaking American students learning Italian and Italian university students studying foreign languages, participants were paired in plurilingual dyads to engage in peer feedback on written texts in their respective target languages. Each student revised their partner’s writing and then participated in oral discussions to explain and negotiate language use—doing so in their own L1 or stronger language.

The exchange was designed to encourage indirect written corrective feedback combined with oral prompting, a strategy that proved especially effective in stimulating active negotiation of form. Crucially, the learners relied on their plurilingual repertoires—including Spanish, English, and Italian—to reflect on and compare linguistic structures. This interaction enabled not only grammatical development, but also the activation of metalinguistic awareness and cross-linguistic transfer, as participants explored differences and similarities among the languages they knew. The experience offered a concrete example of how telecollaboration can leverage multilingual identities to promote both language development and intercultural competence (Cortés Velásquez &amp; Nuzzo, 2021).

== Take-Home Messages ==

* Plurilingualism and telecollaboration enhance language learning by integrating real-world communication and intercultural experiences.
* Digital tools (e.g., Zoom, Google Docs, Padlet, Flipgrid) facilitate authentic multilingual interactions.
* Linguistic mediation and translanguaging strategies help learners navigate and negotiate meaning across languages.

== Self-Assessment ==

=== Multiple choice ===
&lt;quiz display=simple&gt;

{What is the main advantage of using telecollaboration in plurilingual education?}

-A) It ensures students only use their strongest language.

+B) It promotes interaction with speakers of different languages and cultures.

-C) It replaces traditional face-to-face language learning.

+D) It eliminates the need for language teachers.

{Which of the following is an example of linguistic mediation in telecollaboration?}

-A) Memorizing vocabulary lists before an online exchange.

+B) Helping a partner understand a complex idea by rephrasing it in simpler terms.

-C) Using only one language to avoid confusion in communication.

-D) Writing a summary of a conversation without using any digital tools.
&lt;/quiz&gt;

=== Reflection ===
Think about your own language repertoire. How could you use your different languages strategically in a virtual exchange to support communication and collaboration?

== Resources to Go Further ==

* Erasmus+ Virtual Exchange: https://europa.eu/youth/erasmus-virtual
* Council of Europe – Platform of resources and references for plurilingual nd intercultural education: https://www.coe.int/en/web/platform-plurilingual-intercultural-language-education
* Tandem Language Learning Platforms: https://www.tandem.net/
* Series: Telecollaborative learning and Virtual Exchange in Education, edited by Melinda Ann Dooly Owen and Robert O'Dowd: https://www.peterlang.com/series/te

== Bibliography ==
Cortés Velásquez, D., &amp; Nuzzo, E. (2021). Minding the gap: A small-scale study on negotiation of form in telecollaborative tasks. ''Instructed Second Language Acquisition, 5''(2), 232–257. https://doi.org/10.1558/isla.19812

Cortés Velásquez, D., Donato, R., &amp; Ricciardelli, R. (2023). ''Mentoring and intercomprehension in telecollaboration: A plurilingual approach to teaching Italian to heritage Spanish speakers'' [Manuscript in preparation].

Garbarino, S. (2019). Plurilingual practices and telecollaboration: Towards inclusive and learner-centered pedagogies. In P. Leone (Ed.), ''Plurilingual approaches to language learning and teaching'' (pp. 75–91). FrancoAngeli.

Garbarino, S., &amp; Leone, P. (2020). Innovation dans un projet de télécollaboration orale en intercompréhension : bilan et perspectives du projet IOTT. ''Alsic, 23''(2). https://www.researchgate.net/publication/349002904

Kern, R. (1996). Computer-mediated communication: Using e-mail exchanges to explore personal histories in two cultures. In M. Warschauer (Ed.), ''Telecollaboration in foreign language learning'' (pp. 105–119). University of Hawai’i Press.

Leone, P. (2023). Plurilingual telecollaboration: Mediation and intercomprehension in virtual exchanges. In P. Leone &amp; S. Garbarino (Eds.), ''Plurilingualism in language education and content learning'' (pp. 143–158). Cambridge Scholars Publishing.

Nicolaou, A., &amp; Sevilla-Pavón, A. (2016). Exploring telecollaboration through the lens of university students: A Spanish-Cypriot telecollaborative exchange. In S. Jager, M. Kurek, &amp; B. O’Rourke (Eds.), ''New directions in telecollaborative research and practice: Selected papers from the second conference on telecollaboration in higher education'' (pp. 113–119). Research-publishing.net. https://doi.org/10.14705/rpnet.2016.telecollab2016.497

O’Dowd, R. (2018). From telecollaboration to virtual exchange: State-of-the-art and the role of UNICollaboration in moving forward. ''Journal of Virtual Exchange, 1''(1), 1–23. https://doi.org/10.14705/rpnet.2018.jve.1

Paone, L. (2024). ''Plurilingual approaches and real-world tasks in language education: A pedagogical framework'' [Unpublished manuscript].

Warschauer, M. (1997). Computer-mediated collaborative learning: Theory and practice. ''The Modern Language Journal, 81''(4), 470–481. https://doi.org/10.2307/328890

== Credits ==
This resource has been created by [[User:Projet PEP|Projet PEP]] ([[User talk:Projet PEP|discuss]] • [[Special:Contributions/Projet PEP|contribs]]) (Erasmus+ project, co-financed by the European Commission) :

* Diego Cortes Velasquez (Universitá Roma Tre).</text>
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      <minor />
      <comment>Expanded and revised the introduction and definitions. Clarified key concepts related to telecollaboration, plurilingualism, tandem learning, and linguistic mediation, and incorporated recent research on multilingual telecollaboration.</comment>
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{{Education}}{{Course}}
== Starting Activity ==
Imagine you are participating in an online language exchange with students from different countries. You and your peers speak different first languages, but you also share some knowledge of additional languages (e.g., English, Spanish, French, Italian). During your video meetings, group chats, and collaborative projects, you are encouraged to use all your linguistic resources to communicate, clarify, and co-construct meaning.

'''Reflect and respond:'''

# Which languages might you choose to use during the exchange? Why?
# How would you draw on your knowledge of different languages to make yourself understood—and to understand others?
# Have you ever used more than one language in a single interaction? What did you notice about how communication worked?
# What strategies might help you deal with misunderstandings or unfamiliar words?
# In your opinion, what are the potential benefits of using multiple languages in a collaborative learning setting?

== Objectives ==
By the end of this module, you will be able to:

* Define key concepts such as plurilingualism, telecollaboration, tandem learning, and linguistic mediation.
* Explain how telecollaborative practices support plurilingual education and intercultural communication.
* Identify digital tools and communicative strategies that facilitate meaningful multilingual interaction.
* Analyze examples of telecollaborative projects through a plurilingual lens.
* Reflect on your own linguistic repertoire and how it can be activated in virtual exchanges.

== Keywords ==
Plurilingualism, telecollaboration, intercultural competence, digital language exchange, linguistic mediation, online interaction, tandem learning, virtual exchange.

== Introduction ==
Telecollaboration is increasingly understood as an action-oriented and collaborative educational approach, rooted in authentic tasks that mirror real-world communicative, academic, and professional situations. In such contexts, learners are not merely exchanging linguistic forms; they co-construct meaning, negotiate across linguistic and cultural repertoires, and develop transversal competences through collaborative problem-solving. Typical tasks include the reciprocal explanation of culturally embedded concepts, the co-creation of multilingual digital artefacts (e.g., digital menus, promotional videos, brochures), and collaborative projects that require learners to navigate semantic ambiguity, cultural diversity, and multiple perspectives. 

At the same time, the growing presence of multilingualism in digital environments has transformed telecollaboration into an important space for developing plurilingual competences. As multilingual communication increasingly takes place through multimodal online platforms, learners naturally draw on their linguistic and semiotic repertoires when interacting across geographical and cultural boundaries. Digital environments therefore provide authentic opportunities to experience multilingualism not only as an object of study but also as a communicative practice. 

When informed by a plurilingual perspective, telecollaboration becomes a powerful means of valuing learners' entire linguistic repertoires. Rather than enforcing a shared lingua franca, educators may encourage, for instance, intercomprehension strategies, whereby participants communicate using different languages and rely on partial understanding, contextual cues, mediation, and mutual scaffolding. This approach, supported by projects such as ComunicaRT and IOTT (Paone, 2024; Leone &amp; Garbarino, 2019), emphasizes that successful communication depends less on native-like proficiency than on the ability to adapt, mediate, and collaborate across linguistic and cultural boundaries. 

Recent research further suggests that multilingual telecollaborative projects can significantly enhance learners' plurilingual and pluricultural competence, crosslinguistic awareness, and understanding of multilingualism as a dynamic social practice. In teacher education, telecollaboration has also been shown to foster multilingual awareness, encourage reflection on linguistic identities, and prepare future teachers for linguistically and culturally diverse classrooms. Such plurilingual telecollaborative practices are closely aligned with inclusive, learner-centred pedagogies. They foster strategic competence, metalinguistic awareness, intercultural mediation, and reflective language awareness, all of which are essential for navigating increasingly multilingual societies. Learners thus develop as plurilingual social agents who are equipped not only to communicate, but also to mediate understanding between diverse linguistic and cultural communities.

== History of the concept ==
Telecollaboration, often referred to as virtual exchange, originated in the early 1990s as a pedagogical response to the increasing accessibility of internet technologies. Its core idea was to connect language learners from different geographical and cultural backgrounds through online communication tools, thereby fostering both linguistic competence and intercultural awareness. Early implementations—typically based on email exchanges, discussion forums, and asynchronous writing tasks—were shaped by the emerging field of Computer-Mediated Communication (CMC) and influenced by theories of network-based language learning (Warschauer, 1997; Kern, 1996).

As digital technologies evolved, so too did the modalities of telecollaboration. Synchronous tools like video conferencing, collaborative platforms, wikis, blogs, and social media enabled more interactive, multimodal, and project-based exchanges. This led to the development of richer tasks that emphasized collaboration, co-construction of meaning, and intercultural reflection. Telecollaboration thus expanded from a tool for language practice into a comprehensive pedagogical model embedded in task-based and action-oriented learning.

In the 2000s and 2010s, scholars such as Robert O’Dowd played a key role in conceptualizing and consolidating the field. O’Dowd (2018) distinguishes between telecollaboration as a more open-ended educational practice and virtual exchange, which often refers to institutionally supported and structured programmes, such as those sponsored by the European Union (e.g., Erasmus+ Virtual Exchange). These initiatives highlight the potential of online intercultural encounters not only for language development, but also for advancing global citizenship, digital literacy, and inclusivity in education.

Telecollaboration’s evolution has also been supported by interdisciplinary insights from CALL (Computer-Assisted Language Learning), intercultural communication, and plurilingual education. With the growing focus on learner agency, plurilingual competence, and social constructivist approaches, telecollaboration is now widely regarded as a powerful action-oriented, learner-centered practice that promotes authentic communication across languages and cultures.

== Definitions ==
Plurilingualism refers to an individual's capacity to flexibly and dynamically use multiple languages, integrating linguistic knowledge across various languages rather than viewing them as isolated systems. This holistic approach emphasizes the interconnectedness of languages within a person's repertoire, allowing for fluid movement between languages depending on the context and communicative needs.

Tandem learning is a language learning approach in which two speakers with different first languages work together to support each other's language learning. Based on the principles of reciprocity and learner autonomy, participants alternate between their respective target languages through authentic interaction while developing linguistic and intercultural competences. Tandem learning may take place face-to-face or online (eTandem), the latter representing one form of telecollaboration.

Telecollaboration involves the use of digital communication tools to connect learners from diverse linguistic and cultural backgrounds for educational purposes. This approach facilitates intercultural exchanges and language learning through online platforms, enabling participants to engage in collaborative projects, discussions, and activities that enhance both linguistic proficiency and cultural understanding.

Linguistic mediation refers to the ability to facilitate communication and co-construct meaning between individuals or groups who may not share the same linguistic or cultural background. According to the Common European Framework of Reference for Languages (Council of Europe, 2020), mediation includes activities such as explaining, summarizing, interpreting, adapting information, and facilitating understanding. Within plurilingual telecollaboration, mediation enables learners to draw on their full linguistic repertoires to support communication, collaboration, and mutual understanding.

Together, these concepts illustrate how telecollaborative learning environments can foster plurilingual competence by encouraging learners to mobilize their linguistic and cultural resources, collaborate across languages, and develop mediation, intercultural, and digital competences through authentic online interaction.

== Practical examples ==

=== '''Example 1: Intercomprehension in Telecollaborative Mentoring''' ===
In a transatlantic telecollaborative project, heritage Spanish speakers in California engaged in Italian language learning through mentoring sessions with Italian university students. These sessions encouraged the use of Spanish as a ''pivot language'' to support the acquisition of Italian, promoting intercomprehension based on the typological proximity of Romance languages. Learners were encouraged to draw on their plurilingual repertoires to negotiate meaning and build metalinguistic awareness during real-time Zoom interactions​ (Cortés Velásquez, Donato, &amp; Ricciardelli, 2023).

=== '''Example 2: Intercomprehension and Teletandem in the IOTT Project''' ===
The IOTT project, a collaboration between the University of Lyon 2 and the University of Salento, implemented a telecollaborative learning scenario combining intercomprehension and teletandem methodologies. Students from different linguistic backgrounds engaged in synchronous oral sessions via VoIP technologies, utilizing their respective Romance languages to communicate. This approach emphasized the development of receptive skills in intercomprehension, allowing learners to understand related languages without prior formal instruction. The project also incorporated reflective practices, such as learning diaries and self-assessment tools, to enhance metalinguistic awareness and foster autonomous learning strategies. Findings indicated that this integrative model effectively promoted plurilingual competencies and intercultural understanding among participants (Garbarino &amp; Leone, 2020).

=== '''Example 3: The Trans-Atlantic and Pacific Project (TAPP)''' ===
TAPP connects classes from Europe and the US in joint professional and linguistic projects. In multilingual group tasks, students collaborate on writing, usability testing, and translation. For example, US engineering students produced presentations peer-reviewed by European students. These exchanges promote co-writing and multilingual mediation, with English often used as a lingua franca, but with growing attention to local languages and translation practices as tools for intercultural understanding (O’Dowd, 2018).

=== '''Example 4: Digital Storytelling in Multilingual Settings''' ===
In a university course, students participated in a virtual intercultural exchange via Google+ and Google Drive. One of the key tasks involved creating a digital story in teams. Learners used multiple languages for narration and subtitling, leveraging their full linguistic repertoires. These digital products were shared and peer-reviewed across institutions, showcasing the integration of multimodal literacy, plurilingual resources, and intercultural storytelling​ (Nicolaou &amp; Sevilla-Pavón, 2016).

=== '''Example 5: Tandem Feedback and Plurilingual Awareness through Focus on Form''' ===
In a virtual exchange between heritage Spanish-speaking American students learning Italian and Italian university students studying foreign languages, participants were paired in plurilingual dyads to engage in peer feedback on written texts in their respective target languages. Each student revised their partner’s writing and then participated in oral discussions to explain and negotiate language use—doing so in their own L1 or stronger language.

The exchange was designed to encourage indirect written corrective feedback combined with oral prompting, a strategy that proved especially effective in stimulating active negotiation of form. Crucially, the learners relied on their plurilingual repertoires—including Spanish, English, and Italian—to reflect on and compare linguistic structures. This interaction enabled not only grammatical development, but also the activation of metalinguistic awareness and cross-linguistic transfer, as participants explored differences and similarities among the languages they knew. The experience offered a concrete example of how telecollaboration can leverage multilingual identities to promote both language development and intercultural competence (Cortés Velásquez &amp; Nuzzo, 2021).

== Take-Home Messages ==

* Plurilingualism and telecollaboration enhance language learning by integrating real-world communication and intercultural experiences.
* Digital tools (e.g., Zoom, Google Docs, Padlet, Flipgrid) facilitate authentic multilingual interactions.
* Linguistic mediation and translanguaging strategies help learners navigate and negotiate meaning across languages.

== Self-Assessment ==

=== Multiple choice ===
&lt;quiz display=simple&gt;

{What is the main advantage of using telecollaboration in plurilingual education?}

-A) It ensures students only use their strongest language.

+B) It promotes interaction with speakers of different languages and cultures.

-C) It replaces traditional face-to-face language learning.

+D) It eliminates the need for language teachers.

{Which of the following is an example of linguistic mediation in telecollaboration?}

-A) Memorizing vocabulary lists before an online exchange.

+B) Helping a partner understand a complex idea by rephrasing it in simpler terms.

-C) Using only one language to avoid confusion in communication.

-D) Writing a summary of a conversation without using any digital tools.
&lt;/quiz&gt;

=== Reflection ===
Think about your own language repertoire. How could you use your different languages strategically in a virtual exchange to support communication and collaboration?

== Resources to Go Further ==

* Erasmus+ Virtual Exchange: https://europa.eu/youth/erasmus-virtual
* Council of Europe – Platform of resources and references for plurilingual nd intercultural education: https://www.coe.int/en/web/platform-plurilingual-intercultural-language-education
* Tandem Language Learning Platforms: https://www.tandem.net/
* Series: Telecollaborative learning and Virtual Exchange in Education, edited by Melinda Ann Dooly Owen and Robert O'Dowd: https://www.peterlang.com/series/te

== Bibliography ==
Cortés Velásquez, D., &amp; Nuzzo, E. (2021). Minding the gap: A small-scale study on negotiation of form in telecollaborative tasks. ''Instructed Second Language Acquisition, 5''(2), 232–257. https://doi.org/10.1558/isla.19812

Cortés Velásquez, D., Donato, R., &amp; Ricciardelli, R. (2023). ''Mentoring and intercomprehension in telecollaboration: A plurilingual approach to teaching Italian to heritage Spanish speakers'' [Manuscript in preparation].

Garbarino, S. (2019). Plurilingual practices and telecollaboration: Towards inclusive and learner-centered pedagogies. In P. Leone (Ed.), ''Plurilingual approaches to language learning and teaching'' (pp. 75–91). FrancoAngeli.

Garbarino, S., &amp; Leone, P. (2020). Innovation dans un projet de télécollaboration orale en intercompréhension : bilan et perspectives du projet IOTT. ''Alsic, 23''(2). https://www.researchgate.net/publication/349002904

Kern, R. (1996). Computer-mediated communication: Using e-mail exchanges to explore personal histories in two cultures. In M. Warschauer (Ed.), ''Telecollaboration in foreign language learning'' (pp. 105–119). University of Hawai’i Press.

Leone, P. (2023). Plurilingual telecollaboration: Mediation and intercomprehension in virtual exchanges. In P. Leone &amp; S. Garbarino (Eds.), ''Plurilingualism in language education and content learning'' (pp. 143–158). Cambridge Scholars Publishing.

Nicolaou, A., &amp; Sevilla-Pavón, A. (2016). Exploring telecollaboration through the lens of university students: A Spanish-Cypriot telecollaborative exchange. In S. Jager, M. Kurek, &amp; B. O’Rourke (Eds.), ''New directions in telecollaborative research and practice: Selected papers from the second conference on telecollaboration in higher education'' (pp. 113–119). Research-publishing.net. https://doi.org/10.14705/rpnet.2016.telecollab2016.497

O’Dowd, R. (2018). From telecollaboration to virtual exchange: State-of-the-art and the role of UNICollaboration in moving forward. ''Journal of Virtual Exchange, 1''(1), 1–23. https://doi.org/10.14705/rpnet.2018.jve.1

Paone, L. (2024). ''Plurilingual approaches and real-world tasks in language education: A pedagogical framework'' [Unpublished manuscript].

Warschauer, M. (1997). Computer-mediated collaborative learning: Theory and practice. ''The Modern Language Journal, 81''(4), 470–481. https://doi.org/10.2307/328890

== Credits ==
This resource has been created by [[User:Projet PEP|Projet PEP]] ([[User talk:Projet PEP|discuss]] • [[Special:Contributions/Projet PEP|contribs]]) (Erasmus+ project, co-financed by the European Commission) :

* Diego Cortes Velasquez (Universitá Roma Tre).</text>
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    <title>User:ThinkingScience/April 20th Experiment Notes</title>
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      <text bytes="1661" sha1="0z0jx229bfjur3arwrl6d1c9vgju9os" xml:space="preserve">Idea is not to &quot;spam&quot; Wikiversity but instead focus on the output that the &quot;AI&quot; program got in its output that made an impression on me. For simplicity I could add anything that I read.

I will not include output that encourages &quot;megalomania&quot;, this &quot;AI Mode&quot; just giving me compliments and me just 'eating it all up without a care' unless it's related to an action I'm interested in taking.

* input: Let's take it into action. Can you suggest some wikiversity page for me to edit?
** I found the output uninteresting
* input: Any page on Wikiversity regarding rapport or neurodiversity?
** output: shows me Google search results instead
* input: I'm thinking https://en.wikiversity.org/wiki/Neurodiversity_Movement but I don't wanna modify it because I want to preserve the original knowing I may be &quot;crazy&quot; today and thus I suggest a subpage where I'll &quot;interpret&quot; the content and try to improve what the project is about
** output: it suggests I do that on my user page instead...so I'll do that.

RESULT: 
* Editing my userpage and adding among other things:
** [[User:ThinkingScience/Neurodiversity_Movement_Interpretation]]

'''AI Prompt History referenced to from other projects on Wikiversity/elsewhere'''

Here I am &quot;Making a new subpage that will host so far, unless I archive it with ie. Wayback Machine, then ask for it to be wiped here&quot;:

[[User:ThinkingScience/All_General_AI_Prompt_History_Archive|Archive of prompt history which I interpret the AI Policy is all about]]

== (empty heading)  ==

* input: I've created a 100% human post by myself at https://en.wikiversity.org/w/index.php?title=Wikiversity_talk:Artificial_intelligence&amp;action=edit</text>
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      <comment>/* Inactive curator template */ reply to Juandev: I see, thank you for explaining. (-) ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
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      <text bytes="3927" sha1="gujcpu8uy91s9o801polzz5xc8efe8l" xml:space="preserve">== Notice to colloquium ==

What is the sence of noticing community about that? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:10, 8 June 2026 (UTC)
: A notice would be posted at the inactive SSM's user talk page, and a separate notification at the village pump listing the inactive support staff member(s). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:46, 25 June 2026 (UTC)
::And the reasoning behind why whole community should know, there is inactive staff? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:08, 2 July 2026 (UTC)

== Inactive curator template ==

Just a note if this policy is agreet the template should be fixed. No it counts with 2 years. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:12, 8 June 2026 (UTC)
: I'm not sure what you are trying to explain. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:47, 25 June 2026 (UTC)
::I am saying that if the inactive period is changed, the {{tl|Inactive curator}} template text ''&quot;no edits or no logged actions for 2 years&quot;'' should be changed to the appropriate one. This is just a notice, not to forget to do so. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:10, 2 July 2026 (UTC)
::: I see, thank you for explaining. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:05, 2 July 2026 (UTC)

== Communication with the SSM and deadlines ==

A notification on the user's user page is a decent way to communicate with support staff. If they don't respond, it's clear that there's no point in waiting any longer and their rights have been revoked. On the contrary, if they respond, they suspect that they should start working on Wikiversity, but it may happen that they won't, i.e. SSM will respond, but they will continue to be inactive, so they will have another year of &quot;peace&quot;.

I would probably reduce the inactivity time to '''8 months''' (i.e. 6 months + 2 months, which may take to creat a custodian), but I would leave the response time at a '''month or more'''. I assume that sometimes the reason for inactivity is health problems or personal problems, and in such situations a person is usually not very reactive - i.e. they don't manage to respond quickly to all the requests that come to them. Another reason may be the busy work schedule of university teachers, who, for example, are on the job for 4 months during exams. This means, yes, you have been inactive for a while for some reason and then someone invites you to return to activity, but you are sick, or you are writing a scientific article, grant report, etc. and you don't have much time right now.

Here, it would perhaps require standardized posts for all SSM roles, where a notice would be written that according to the policy, a SSM cannot be inactive for a given period. ''Then a question whether they will resume activity within 2 months.'' Yes - rights retained, no/no answer - rights removed within a month. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:29, 8 June 2026 (UTC)
: I still think we should leave the timeframe as one year to maintain consistency with some other projects. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:40, 25 June 2026 (UTC)
::Well, why not. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:25, 25 June 2026 (UTC)</text>
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        <id>2969951</id>
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      <comment>/* Notice to colloquium */ reply to Juandev ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
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      <text bytes="4204" sha1="1s5mc78fz8bso0gqh2a44o3ci7fhf1c" xml:space="preserve">== Notice to colloquium ==

What is the sence of noticing community about that? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:10, 8 June 2026 (UTC)
: A notice would be posted at the inactive SSM's user talk page, and a separate notification at the village pump listing the inactive support staff member(s). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:46, 25 June 2026 (UTC)
::And the reasoning behind why whole community should know, there is inactive staff? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:08, 2 July 2026 (UTC)
::: This is standard practice as the stewards have done this similar procedure per [[m:Admin activity review]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:06, 2 July 2026 (UTC)

== Inactive curator template ==

Just a note if this policy is agreet the template should be fixed. No it counts with 2 years. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:12, 8 June 2026 (UTC)
: I'm not sure what you are trying to explain. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:47, 25 June 2026 (UTC)
::I am saying that if the inactive period is changed, the {{tl|Inactive curator}} template text ''&quot;no edits or no logged actions for 2 years&quot;'' should be changed to the appropriate one. This is just a notice, not to forget to do so. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:10, 2 July 2026 (UTC)
::: I see, thank you for explaining. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:05, 2 July 2026 (UTC)

== Communication with the SSM and deadlines ==

A notification on the user's user page is a decent way to communicate with support staff. If they don't respond, it's clear that there's no point in waiting any longer and their rights have been revoked. On the contrary, if they respond, they suspect that they should start working on Wikiversity, but it may happen that they won't, i.e. SSM will respond, but they will continue to be inactive, so they will have another year of &quot;peace&quot;.

I would probably reduce the inactivity time to '''8 months''' (i.e. 6 months + 2 months, which may take to creat a custodian), but I would leave the response time at a '''month or more'''. I assume that sometimes the reason for inactivity is health problems or personal problems, and in such situations a person is usually not very reactive - i.e. they don't manage to respond quickly to all the requests that come to them. Another reason may be the busy work schedule of university teachers, who, for example, are on the job for 4 months during exams. This means, yes, you have been inactive for a while for some reason and then someone invites you to return to activity, but you are sick, or you are writing a scientific article, grant report, etc. and you don't have much time right now.

Here, it would perhaps require standardized posts for all SSM roles, where a notice would be written that according to the policy, a SSM cannot be inactive for a given period. ''Then a question whether they will resume activity within 2 months.'' Yes - rights retained, no/no answer - rights removed within a month. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:29, 8 June 2026 (UTC)
: I still think we should leave the timeframe as one year to maintain consistency with some other projects. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:40, 25 June 2026 (UTC)
::Well, why not. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:25, 25 June 2026 (UTC)</text>
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    <title>Peace Economy Project</title>
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      <text bytes="26085" sha1="5tlki12pxogpvohrup0blu578dgbfou" xml:space="preserve">:''This discusses a 2026-06-20 interview with Katerina Canyon&lt;ref name=Canyon&gt;&lt;!--Katerina Canyon--&gt;{{cite Q|Q140290658}}&lt;/ref&gt; about the Peace Economy Project,&lt;ref name=PEP&gt;&lt;!--Peace Economy Project--&gt;{{cite Q|Q140290765}}&lt;/ref&gt; including a video and 29:00 mm:ss podcast excerpted from the interview. The podcast is released 2026-06-27 to the fortnightly &quot;Media &amp; Democracy&quot; show&lt;ref name=M&amp;D&gt;&lt;!--Media &amp; Democracy--&gt;{{cite Q|Q127839818}}&lt;/ref&gt; syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]&lt;ref&gt;&lt;!--Pacifica Radio Network--&gt;{{cite Q|Q2045587}}&lt;/ref&gt; Network of [[w:List of Pacifica Radio stations and affiliates|over 200 community radio stations]].''&lt;ref&gt;&lt;!--list of Pacifica Radio stations and affiliates--&gt;{{cite Q|Q6593294}}&lt;/ref&gt;

:''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]&lt;ref name=NPOV&gt;The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs.&lt;/ref&gt; and treating others with respect.''&lt;ref name=AGF&gt;[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] was different: Contributors there were asked to [[Wikinews:Never assume|&quot;Don't assume things; be skeptical about everything.&quot;]] That's wise. However, we should still treat others with respect while being skeptical.&lt;/ref&gt;

[[File:Interview with Katerina Canyon about the Peace Economy Project.webm|thumb|2026-06-20 interview with Katerina Canyon about the Peace Economy Project]]
[[File:Interview with Katerina Canyon about the Peace Economy Project.ogg|thumb|29:00 mm:ss excerpts from a 2026-06-20 interview with Katerina Canyon about the Peace Economy Project]]

Katerina Canyon&lt;ref name=Canyon/&gt; describes the Peace Economy Project (PEP),&lt;ref name=PEP/&gt; their vision and activities. Canyon is PEP's Executive Director. She holds a BA from [[w:Saint Louis University|Saint Louis University]] and a Master of Arts in Law and Diplomacy (MALD) from the [[w:Fletcher School of Law and Diplomacy|Fletcher School]].&lt;ref&gt;&lt;!--Katerina Canyon: Poet, Spoken Word Artist--&gt;{{cite Q|Q140291133}}&lt;/ref&gt; Her publications include two recent books of poetry,&lt;ref&gt;Canyon (2021); Canyon and Canyon (2017).&lt;/ref&gt; a novel,&lt;ref&gt;Canyon (in preparation).&lt;/ref&gt; and numerous shorter pieces of political commentary disseminated in a variety of outlets. 

PEP works to &quot;Cut Military Spending [and] Fund Human Needs.&quot; They are &quot;guided by the belief that true security is built through care, dignity, and a shared well-being—not punishment, militarization, or fear.&quot; Their work is grounded in 8 core values:&lt;ref&gt;&lt;!--Peace Economy Project Core Values--&gt;{{cite Q|Q140291167}}&lt;/ref&gt; 
# Human Dignity.
# Nonviolence.
# Peaceful and Civic Engagement.
# Community Safety Over Militarization.
# Accountability and Transparency.
# Care-Centered Policy.
# Solidarity Across Difference.
# Thoughtful, Strategic Action.

Canyon is interviewed by Spencer Graves.&lt;ref&gt;&lt;!--Spencer Graves--&gt;{{cite Q|Q56452480}}&lt;/ref&gt;

== Highlights ==

:''These excerpts are rushed, lightly edited for readability, and may not be in final form. The ultimate authority on what was said is the accompanying video.''

When asked for the most important things she would like to communicate to this audience, Canyon said, {{quote|

I would like to communicate to our audience the importance of media literacy. Especially I'd like to talk to poets and other creative influencers in ways of participating in making sure that events in this world are properly and truthfully documented. ... 

I'm much influenced by the poet [[w:Muriel Rukeyser|Muriel Rukeyser]], who was a documentary poet in the early to mid 20th century. She would use her poetics to talk about events that she witnessed, particularly when it had to do with minors and the health effects that they were facing when going into the mines. Many of them were dying. So Muriel Rockeyser wrote beautiful poems about the experience in order to open up dialog and to reveal what was going on with these people, who would not have been represented otherwise. 

They were being overshadowed by corporate structures that also controlled the media, and at times poetry is the way to get through to people. When you look at what happens during wars, it's often poets who stand out and express themselves either through poetry or through independent media.}}

Graves asked if the best known poets might partner with musicians? 

Canyon agreed. &quot;That is true. A lot of poems are musical. And ... a lot of songs are musical. From soft alternative music to heavy metal to rap music, it's all poetics. And they all bear a certain level of truth. You can also say that song makers are the most known truth tellers that we have right now.&quot; 

Graves mentioned [[w:Bob Dylan|Bob Dylan]]'s &quot;[[w:The Lonesome Death of Hattie Carroll|Hattie Carroll was a maid in the kitchen]]&quot;.

=== Peace Economy Project ===

Graves then asked for an overview of the &quot;Peace Economy Project&quot;,&lt;ref name=PEP/&gt; Canyon said, {{quote|

The Peace Economy Project is an organization that's been around since 1977. It started in [[w:St. Louis|St. Louis]]. One of the founders is still on our board today, Mary Ann McGivern.&lt;ref&gt;&lt;!--Mary Ann McGivern--&gt;{{cite Q|Q140309851}}&lt;/ref&gt; She and a group of citizens of St. Louis were having issues with what was going on with [[w:McDonnell Douglas|McDonnell Douglas]], now [[w:Boeing|Boeing]]. They were against weapons manufacturing in their city of St. Louis. So they protested that. From there, the Peace Economy Project grew. A lot of our focus is primarily on St. Louis in Missouri, but we decided that what we face locally, every state in this country faces ... . 

We decided that in order to impact our cities that we needed to not only focus locally but we needed to also focus nationally. We do this through our fellowship programs. We train fellows to do research. Like I said, truth and documentary poetics. But any way of researching and documenting the truth is very important to our mission, which is to educate the public on the effects of an unchecked military industrial complex. We take a lot of college fellows. Sometimes we will take on a high school fellow, but most are college and grad school. 

They will research different areas in order to look into the impact that the military has on different parts of our life. A lot of our fellows investigate environmental impacts that the military has on their areas. We have fellows from St. Louis, obviously, but we also have fellows that live in New York. We have some in Italy. We have fellows in France. 

They're all over the world, and they are investigating the impact that the United States has on their countries and on their communities.}} 

=== ''Warheads to Windmills'' ===

Graves mentioned [[w:Timmon Wallis|Timmon Wallis]] and his work on ''Warheads to Windmills''.&lt;ref&gt;Wallis (2018, 2023)&lt;/ref&gt; &quot;For example, with the [[w:Kansas City National Security Campus|nuclear bomb plant in Kansas City]], the United States and the public would benefit more if the people doing that work were instead facilitating the transformation to a post-fossil fuel economy.&quot; 

Canyon agreed. &quot;If they were investing in environmental sustainability instead of putting together warheads, ... that would be a better investment ... . I have numerous fellows that work on this. A lot of our fellows are environmental students. They understand the concept of a peace economy that if we just give people a means of meeting their basic human needs, a good portion of our problems would just go away.&quot; 

=== Biggest successes === 

Graves asked about their biggest successes so far. Canyon replied, {{quote|

I would say that our major successes have to do with educating people, especially our fellows. They are the ones that are going to be making decisions 20 years from now. We get fellows from [[w:Yale University|Yale]], from [[w:Rutgers University|Rutgers]], from major universities around the world, [[w:London School of Economics|London School of Economics]]. ...

I'm not saying that the military is completely unnecessary. I hope that it will be one day, but I'm not saying that's the case today. But we are saying that the way that they're going about it right now is extreme, we should not be investing as much money as we do in the military.}}

=== &quot;Record Pentagon Spending, Diminished Security: A $1.5 Trillion Warning&quot; ===

Graves noted that a lead article on their website describes, &quot;Record Pentagon spending, diminished security, a $1.5 trillion warning&quot;,&lt;ref&gt;Canyon (2026-05).&lt;/ref&gt; and asked about that. Canyon replied, {{quote|

&quot;This presidency has exceeded any other presidency in military spending. Each time they go to Congress, they say it's for more security, but we are just getting more entrenched in battles. 

[[w:United States withdrawal from the Iran nuclear deal|President Obama had an agreement with Iran that President Trump tore up]]. We are battling them today because of the ignorance of that agreement. Now they're negotiating, and they're probably going to end up with close to the same agreement that was dismissed by the early Trump presidency.}}

=== &quot;When Defense Endangers Civilians&quot; ===

Graves mentioned another article on their website that mentioned the Patriot missiles.&lt;ref&gt;Canyon (2026-03) &quot;When Defense Endangers Civilians: Urgent Questions After the Bahrain Explosion&quot;.&lt;/ref&gt; Graves noted that one of the books by Robin Andersen&lt;ref&gt;[[Media and war|Andersen was interviewed]] for this [[:Category:Media reform to improve democracy|Media &amp; Democracy]] series 2026-03-27.&lt;/ref&gt; says that the introduction of the Patriot missiles, so-called anti-missile missiles, during the [[w:Gulf War|First Persian Gulf War]] actually did more damage than good, because the debris from the Scud missiles was scattered over a wider area with debris from the Patriots actually destroying more property and killing more people.&lt;ref&gt;Andersen (2006, pp. 178-179) quotes MIT prof [[w:Theodore Postol|Postol]] (1991-92).&lt;/ref&gt;

Canyon replied, &quot;I actually did a report on that back in 2012, when I was a fellow for the Peace Economy Project. It's true that we invest in these weapon systems, and a lot of the time we put all of this money into these weapons, and not only are they ineffective, most of the time, they impact our environment in significant ways.&quot; 

Graves affirmed, &quot;Yeah, worse than worse than useless: They're counterproductive.&quot; 

=== Media sell changes in audience behaviors ===

Graves then asked, &quot;To what extent is it fair to say that every media organization sells changes in audience behavior to the people who give them money?&quot;&lt;ref&gt;It's likely more accurate to say that, &quot;Every media organization sells changes in audience behavior to the people who control most of the money for the media.&quot; This is discussed further in [[Media &amp; Democracy lessons for the future]].&lt;/ref&gt;

Canyon replied, &quot;A part of me wants to believe in the media. As a communications major and a poet who just finished her MFA in creative writing, I want to believe that the media wants to give the public the truth. But at the end of the day, media empires are businesses. They are going to go where the money is. They're going to want to do what they need to survive. I believe that there are very few organizations that aren't pandering to those who give them money. The only way to avoid that is to have a media that does not depend on major corporate support or on special interests.&quot; 

Graves continued, &quot;I've interviewed [[Information is a public good per communications prof Pickard|Victor Picard]], who recommends local news nonprofits, maybe multi local multimedia centers funded locally with firewalls that prevent political interference. Boards of directors may be selected at random, like jury duty, and with regular, maybe monthly, meetings, where they invite public input to help journalists select better topics, and maybe recruit volunteer researchers.&quot;&lt;ref&gt;Pickard (2023).&lt;/ref&gt; 

Canyon replied, &quot;I think there does need to be a better investment in peer review and a better investment into local economies, such that the media isn't controlled solely by one ruling class or one ruling entity. I'm truly in favor of local investment. I think that is one significant way to support true documentary media. On top of that, I think that peer review is another way to make sure that happens.&quot; 

=== Primary drivers of every major conflict include differences between the media that the different parties find credible.===

Graves continued, &quot;My research in this area suggests that primary drivers of every major conflict include differences between the media that the different parties find credible.&lt;ref&gt;This is a &quot;[[Media Literacy and You#Key claims|Key claim]]&quot; of the book-in-progress on ''[[Media Literacy and You]]'', which provides substantial documentation to support this claim.&lt;/ref&gt; Your comment.&quot;

Canyon replied, {{quote|

I think that is also true. You want to believe things that are closer to your own experience, so it is very challenging for people who don't have that experience to believe that another experience is true. 

It happens a lot with my poetry when I travel and I read about the experiences that happened to me in my childhood. Quite often, people will come to me and say, &quot;Did that really happen to you?&quot; 

While, if I am in an audience where either we're mostly women or women of color, they'll come up to me and say, &quot;Thank you for writing that. It is very similar to my own experience.&quot; 

It is hard for people to believe things that are outside their sphere. That's why things like what's happening in the [[w:Middle East|Middle East]] is so hard for people to grasp. Here in the United States, we do not understand what it means to have your homes bombed. Or to be chased out of your own home. We don't understand the idea that you need to leave your country. So that is why immigration is such a hot button issue. 

That is why investing in military support is such a hot button issue, because we don't understand what it is to be in those situations.}} 

=== ''Black Like Me'' ===

Graves noted that his grandmother was deeply moved by the book ''[[w:Black Like Me|Black Like Me]]'', which came out in the early part of the civil rights movement.&lt;ref&gt;Griffin (1961).&lt;/ref&gt; 

Canyon said, &quot;There's a book called ''Empathy and the Novel'', which talks about how fiction and literature move people, and how much can a novel or how much can a poem move someone? Can it move them toward action?&lt;ref&gt;Keen (2007).&lt;/ref&gt; 

Graves continued, &quot;That book was written by apparently a white guy who took some kind of medication to turn his skin dark and traveled throughout the [[w:Southern United States|South]], and wrote about his experiences -- a white guy being treated like an African American. That book had, apparently, a pretty decent impact on a lot of folks. 

Canyon replied, &quot;I can see how that would, because he's the same person, except his skin is different. So he was able to say, &quot;I went through this situation as a white guy, and it was fine. But when I changed the color of my skin, it was a different situation.&quot;

=== MALD and MFA === 

Graves noted that Canyon has a Master of Arts in Law and Diplomacy&lt;ref&gt;&lt;!--Master of Arts in Law and Diplomacy (MALD)--&gt;{{cite Q|Q100606027}} (MALD).&lt;/ref&gt; from the [[w:Fletcher School of Law and Diplomacy|Fletcher School of Law and Diplomacy]]. He asked her to talk about that. 

Canyon said, {{quote|

I do have a Master of Art in Law and Diplomacy from the Fletcher School, and I also have a [[w:Master of Fine Arts|Master of Fine Arts]], which I just earned about a week ago from [[w:Mississippi University for Women|Mississippi University for Women]] in creative writing. 

It was challenging for me when I decided to pursue international studies with Fletcher, because I have this need to be a journalist. Truth is very important to me. Ever since I was a child, and I used to watch ''[[w:Lou Grant (TV series)|Lou Grant]]'', I wanted to be a journalist that investigated stories and reported the truth to people and help people understand what's going on in the world. But I'm also a poet, so I had those competing drivers. There is one school that would have done that for me, where I could have studied both journalism and poetry: It was [[w:Dartmouth College|Dartmouth]]. But I didn't go to Dartmouth. I decided to go to Fletcher, and that was a tremendous experience for me. It was hard. It was very challenging. 

You learned a lot about how the mechanisms work --  not only about the US government but [[w:United Nations System|UN systems]], other international organizations around the world. You were put into scenarios of world events, and you had to figure out how you would solve those problems if you had to make those decisions. 

Fletcher trains the next decision makers of this world. I talk about the United States military being in every country in the world. I believe that we could also say that about Fletcher graduates, that we are all over the world trying to do the best we can to make this a better place.}}

=== Deterrence theory === 

Graves said, &quot;I've interviewed [[w:Richard Ned Lebow|Richard Ned Le Beau]], who's a leading expert on [[w:Deterrence theory|deterrence theory]]. He knows that the foundations of deterrence theory assume that your leaders and the leaders of your opposition are rational. But that's contradicted by the available evidence of all kinds of political and military leaders that prove that they're often not terribly rational.&lt;ref&gt;Lebow (2024); Lebow et al. (2023).&lt;/ref&gt; Your comment.&quot;

Canyon agreed, &quot;That is true. The term [[w:Mutually assured destruction|mutually assured destruction]] comes to mind, where we assume that the United States and China and the [[w:Soviet Union|USSR]] are going to be rational and not enter into conflicts that impact the planet. But I would say, are they being rational in the tiny impacts that happen? You are hard pressed to find any government that tries to act in the best interest of this planet, of the people, when you're bombing countries and leaving millions homeless. When you are leaving children starving and rationalizing it under the guise of security, you are not being a rational human being.&quot; 

Graves said, &quot;You can only play chicken so many times before everybody loses.&quot; 

Canyon agreed, &quot;Exactly.&quot; 

=== Department of War ===

Graves noted that the Peace Economy Project website has a recent blog discussing the US &quot;Department of War&quot; by their founder, Mary Ann McGivern.&lt;ref&gt;McGivern (2026).&lt;/ref&gt; 

Canyon replied, {{quote|

Yes, Mary Ann McGivern. I asked her to write that article because before it was dubbed the &quot;Department of War&quot;, Mary Ann didn't like that we were calling it the &quot;[[w:United States Department of Defense|Department of Defense]]&quot;. She said it was completely inaccurate, that they were not concerned with defense. They were not concerned with protecting the United States. They were about creating war. So when it happened,&lt;ref&gt;President Trump signed [[w:Executive Order 14347|Executive Order 14347]] titled, &quot;Restoring the United States Department of War&quot; on 2025-09-05. However, the official name is still the &quot;Department of Defense&quot;, because it was created by the &lt;!--National Security Act Amendments of 1949--&gt;{{cite Q|Q122052416}} and cannot be changed by an executive order. Its predecessor was the &quot;[[w:United States Department of War|Deparment of War]]&quot;, established by Congress in 1789, the first year of the US under its current constitution.&lt;/ref&gt; I said, &quot;Mary Ann, you're the one who needs to talk about this, because a lot of people in the country were upset that they changed the name because they felt that with the name change there would be a focus change.&quot; Mary Ann was saying, &quot;That happened long ago. Now they're being truthful.&quot; 

That was something that Mary Ann was definitely qualified to write about, because she was our founder. She is very passionate about peace. She's a nun. I do have an appreciation for nuns and their ability to stand out against the common narrative, and stand up and say, &quot;No! This is what the truth is,&quot; and not be afraid of that. 

I wouldn't call myself a person of faith, but I admire people who are and use that faith in order to help others to see the greater truth of preserving this planet.}}

Graves countered, &quot;I would say that our Secretary of War, Hegseth, would consider himself a person of faith. He has monthly prayer meetings in the Pentagon during normal working hours.&lt;ref&gt;See the section on &quot;[[w:Pete Hegseth#Pentagon Christian worship services and &quot;biblically sanctioned war|Pentagon Christian worship services and &quot;biblically sanctioned war]]&quot; in the Wikipedia article on &quot;[[w:Pete Hegseth|Pete Hegseth]]&quot;, accessed 2026-06-22.&lt;/ref&gt; The [[w:Military Religious Freedom Foundation|Military Religious Freedom Foundation]] has reported they'd received over 200 complaints from over 50 different US military installations that said that they were told to spread the good word that Jesus has anointed President Trump to light the signal fires in Iran leading to Armageddon.&lt;ref&gt;Mordowanec (2026).&lt;/ref&gt; 

Canyon replied, &quot;That is frightening to say the least. I don't think that there isn't a place for faith in people's life, but there definitely needs to be a [[w:Separation of church and state|separation of church and state]].

=== Close with a poem ===

When Graves noted that we were out of time and asked for final words, Canyon wanted to read a poem she had written a few years ago during the early stages of the war against Palestinians. It's called 

::'''Waiting in [[w:Al-Shifa Hospital|Al-Shifa Hospital]]'''&lt;ref&gt;Canyon agreed to release this poem under the [[w:Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License|Creative Commons Attribution-ShareAlike (CC BY-SA) 4.0 International License]].&lt;/ref&gt;

It was based on a photo that she saw in ''[[w:The New York Times|The New York Times]]'': 

:Waiting in [[w:Al-Shifa Hospital|Al-Shifa Hospital]].
:Privacy requires no space at all. 
:Fluorescent lights glow white along the ceiling, 
:except for the one that is out. 
:Bombs erupt, sirens wail, or children cry. 
:[[wiktionary:Decubitus|Decubitus]] people fill the gurneys that line the hallway. 
:A long white river of linoleum runs between the rows. 
:Rippling walls of blue plastic curtains embarrass the sky. 
:One gray man stands in the middle of it all, 
:like I imagine my granddaddy stood next to my mama during the [[w:Polio#Epidemiology|polio epidemic]]. 
:That was another century. 
:This is a whole new tragedy for people. 
:I want to ask the [[w:monarch butterfly|monarch butterfly]], the puppet, the sun 
:to stop to put down their bread and circuses, 
:But the man wearing a baseball cap 
:picks up his cellphone and turns his back to the camera. 
:I see a woman in a black hijab toward the back, 
:but I cannot make out her face. 
:The man in black lying in the front gurney 
:is missing at least two fingers.
 
== The need for media reform to improve democracy == 

This article is part of [[:category:Media reform to improve democracy]]. A summary of episodes to 2025-11-15 is available in [[Media &amp; Democracy lessons for the future]]. 

==Discussion ==

:''[Interested readers are invited to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]&lt;ref name=NPOV/&gt; and treating others with respect.&lt;ref name=AGF/&gt;]''

== Notes ==
{{reflist}}

== Bibliography ==

* &lt;!--Robin Andersen (2006) A century of media, a century of war--&gt;{{cite Q|Q138795568}}
* &lt;!--Katerina Canyon (2021-10-28) Surviving Home--&gt;{{cite Q|Q140291339}}
* &lt;!--Katerina Canyon (2026-03-23) &quot;When Defense Endangers Civilians: Urgent Questions After the Bahrain Explosion&quot;--&gt;{{cite Q|Q140310269}}
* &lt;!--Katerina Canyon (2026-05-12) &quot;Record Pentagon Spending, Diminished Security: A $1.5 Trillion Warning&quot;--&gt;{{cite Q|Q140310176}}
* &lt;!--Katerina Canyon (in preparation) Los Angeles Nomad--&gt;{{cite Q|Q140291360|date=in preparation}}
* &lt;!--Katerina Canyon and Aja Canyon (2017-09-16) Changing the Lines--&gt;{{cite Q|Q140291385}}
* &lt;!--John Howard Griffin (1961) Black Like Me--&gt;{{cite Q|Q126453723|author=John Howard Griffin}}
* &lt;!--Suzanne Keen (2007-05-03) Empathy and the Novel--&gt;{{cite Q|Q140236008}}
* &lt;!--Richard Ned Lebow (2024-03-26) Deterrence, its failures, and relations between the US and China--&gt;{{cite Q|Q140315485}}
* &lt;!--Richard Ned Lebow, Douglas A. Samuelson, and Spencer Graves (2023-11-29) Richard Ned Lebow on national defense including deterrence--&gt;{{cite Q|Q124351846}}
* &lt;!--Mary Ann McGivern (2026-05-27) The War Department: What’s in a Name?--&gt;{{cite Q|Q140314425}}
* &lt;!--Nick Mordowanec (2026-03-03) &quot;Commanders Accused of Framing Iran War as Biblical Mandate, Jesus' 'Return'&quot;--&gt;{{cite Q|Q138840951}}
* &lt;!--Victor Pickard (2023) Another Media System is Possible: Ripping Open the Overton Window, from Platforms to Public Broadcasting, Janost--&gt;{{cite Q|Q131398460}}
* &lt;!--Theodore Postol (1991-92) &quot;Lessons of the Gulf War experience with Patriot&quot;--&gt;{{cite Q|Q140310330|date=1991-92}}
* &lt;!--Timmon Wallis (2019) Warheads to Windmills: How to pay for a Green New Deal--&gt;{{cite Q|Q104969895}}
* &lt;!--Timmon Wallis (2023-12-15) Warheads to Windmills: Preventing Climate Catastrophe and Nuclear War--&gt;{{cite Q|Q140310033}}

[[Category:Media]]
[[Category:News]]
[[Category:Democracy]]
[[Category:Politics]]
[[Category:Media literacy]]
[[Category:Media reform to improve democracy]]
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      <comment>Projet PEP moved page [[Theories and models of plurilingualism]] to [[Unitary Translanguaging Theory (UTT) and Crosslinguistic Translanguaging Theory (CTT)]]: Better adequacy to the content</comment>
      <origin>2817603</origin>
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      <text bytes="98" sha1="kmb3ni1z7i72cbnbxyub98q75v0de1e" xml:space="preserve">#REDIRECT [[Nitary Translanguaging Theory (UTT) and Crosslinguistic Translanguaging Theory (CTT)]]</text>
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  </page>
  <page>
    <title>Theories and models of plurilingualism</title>
    <ns>0</ns>
    <id>330440</id>
    <redirect title="Unitary Translanguaging Theory (UTT) and Crosslinguistic Translanguaging Theory (CTT)" />
    <revision>
      <id>2817605</id>
      <timestamp>2026-07-02T14:59:10Z</timestamp>
      <contributor>
        <username>Projet PEP</username>
        <id>3002502</id>
      </contributor>
      <comment>Projet PEP moved page [[Theories and models of plurilingualism]] to [[Unitary Translanguaging Theory (UTT) and Crosslinguistic Translanguaging Theory (CTT)]]: Better adequacy to the content</comment>
      <origin>2817605</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="99" sha1="sumwyl92o52p5750vp8xunxhgby401c" xml:space="preserve">#REDIRECT [[Unitary Translanguaging Theory (UTT) and Crosslinguistic Translanguaging Theory (CTT)]]</text>
      <sha1>sumwyl92o52p5750vp8xunxhgby401c</sha1>
    </revision>
  </page>
  <page>
    <title>Talk:Nitary Translanguaging Theory (UTT) and Crosslinguistic Translanguaging Theory (CTT)</title>
    <ns>1</ns>
    <id>330441</id>
    <revision>
      <id>2817607</id>
      <timestamp>2026-07-02T15:59:57Z</timestamp>
      <contributor>
        <username>Projet PEP</username>
        <id>3002502</id>
      </contributor>
      <comment>/* Fix writing */ new section</comment>
      <origin>2817607</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="372" sha1="iricllrg98muiymf7jz99eyaeqags0z" xml:space="preserve">== Fix writing ==

Hi, 
Can somebody help fix the typo in the title. A U is missing at the beginning. It should be 
Unitary Translanguaging Theory (UTT) and Crosslinguistic Translanguaging Theory (CTT)
Thanks in advance
Best wishes [[User:Projet PEP|Projet PEP]] ([[User talk:Projet PEP|discuss]] • [[Special:Contributions/Projet PEP|contribs]]) 15:59, 2 July 2026 (UTC)</text>
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  <page>
    <title>User:Figter87'</title>
    <ns>2</ns>
    <id>330442</id>
    <revision>
      <id>2817624</id>
      <timestamp>2026-07-03T01:00:21Z</timestamp>
      <contributor>
        <username>Figter87'</username>
        <id>3098518</id>
      </contributor>
      <comment>/* */ for emergency borrow money and I will pay up to 1 year</comment>
      <origin>2817624</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="35" sha1="7odob7gsnnttv737ng9chf86j6aufsr" xml:space="preserve">Paano naka utang ng pera sa katipun</text>
      <sha1>7odob7gsnnttv737ng9chf86j6aufsr</sha1>
    </revision>
    <revision>
      <id>2817625</id>
      <parentid>2817624</parentid>
      <timestamp>2026-07-03T01:01:35Z</timestamp>
      <contributor>
        <username>Figter87'</username>
        <id>3098518</id>
      </contributor>
      <minor />
      <comment>/* */ paano maka utang ng pera sa katipun</comment>
      <origin>2817625</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="35" sha1="0dlay2vc404pvfwyqy3oyk67dvdc20v" xml:space="preserve">Paano maka utang ng pera sa katipun</text>
      <sha1>0dlay2vc404pvfwyqy3oyk67dvdc20v</sha1>
    </revision>
  </page>
  <page>
    <title>User:Juandev/R/Wikicode</title>
    <ns>2</ns>
    <id>330443</id>
    <revision>
      <id>2817633</id>
      <timestamp>2026-07-03T07:42:06Z</timestamp>
      <contributor>
        <username>Juandev</username>
        <id>2651</id>
      </contributor>
      <comment>init</comment>
      <origin>2817633</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="422" sha1="bqtue9uj802ko61c9yxb3c2jdv8u6l4" xml:space="preserve">== What is the name of &lt;nowiki&gt;{{{obec|}}}&lt;/nowiki&gt;? ==
This is a template parameter. The string &quot;obec&quot; is the parameter name and the space after the vertical bar is the empty default value.
*Note. The default parameter is only printed if the user does not enter it in the template. This means that &lt;nowiki&gt;{{{obec|unfilled}}}&lt;/nowiki&gt; will insert &quot;unfilled&quot; into the page after saving, unless this parameter is specified.</text>
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