Wikiversity enwikiversity https://en.wikiversity.org/wiki/Wikiversity:Main_Page MediaWiki 1.47.0-wmf.6 first-letter Media Special Talk User User talk Wikiversity Wikiversity talk File File talk MediaWiki MediaWiki talk Template Template talk Help Help talk Category Category talk School School talk Portal Portal talk Topic Topic talk Collection Collection talk Draft Draft talk TimedText TimedText talk Module Module talk Event Event talk 4 28 2814890 2814840 2026-06-09T18:12:07Z STei (WMF) 3091599 /* Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki */ new section 2814890 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == [[MediaWiki:Protectedpagetext#Protected edit request on 11 December 2025]] == I posted an edit request there 5 months ago, so I’ll be taking it to this page. [[Special:Contributions/&#126;2026-28640-56|&#126;2026-28640-56]] ([[User talk:&#126;2026-28640-56|talk]]) 23:33, 12 May 2026 (UTC) :What exactly is the problem? I don't understand what needs to change and why. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:35, 12 May 2026 (UTC) : Pinging @[[User:Atcovi|Atcovi]], @[[User:Jtneill|Jtneill]] and @[[User:Juandev|Juandev]] for further input. Someone is requesting a modification to [[MediaWiki:Protectedpagetext]] to use {{tlx|Protected page text}}, but we might need to discuss whether to use the template. In the meantime, I'll start a sandbox version of the protected page text template. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:19, 14 May 2026 (UTC) ::Sounds good -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:13, 15 May 2026 (UTC) :::+1 Jtneill. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:59, 19 May 2026 (UTC) == Proposal to rehost Wikinews here == As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance. I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:30, 14 May 2026 (UTC) :I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) ::A few days shy of 30, it seems obvious that this is not going to pass. So I '''withdraw''' as presumptively '''failed'''. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:14, 9 June 2026 (UTC) ===Votes=== *{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) *{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC) *{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of "practicing journalism" a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC) *{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:48, 15 May 2026 (UTC) *{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC) *{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 20:29, 30 May 2026 (UTC) * {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC) *{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC) * {{oppose}} per above. Wikiversity<math>\not=</math> Wikinews - not a good idea to mix the scope of projects. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:03, 8 June 2026 (UTC) * {{abstain}} I will abstain since I'm not an active Wikiversity contributor. But I just feel like Wikinews had a very clear and specific goal of providing news, and Wikiversity is just a different project with different goals. For me, it would be odd to rehost Wikinews here. But please do not count my vote, this is only a comment. --[[User:Antimundo|Antimundo]] ([[User talk:Antimundo|discuss]] • [[Special:Contributions/Antimundo|contribs]]) 13:19, 6 June 2026 (UTC) * {{oppose}} Although I think it's a pity that Wikinews is closed. --[[User:Dick Bos|Dick Bos]] ([[User talk:Dick Bos|discuss]] • [[Special:Contributions/Dick Bos|contribs]]) 19:06, 8 June 2026 (UTC) ===Comments and questions=== :Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice. :Initial questions: :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? :* What are "active editions"? :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? :* Are any changes to the scope of Wikinews proposed? :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. :-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:52, 14 May 2026 (UTC) :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? ::*No, not at this time. :* What are "active editions"? ::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04). :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? ::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure). :* Are any changes to the scope of Wikinews proposed? ::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]]. :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. ::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning <em>about</em> what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like. ::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:13, 14 May 2026 (UTC) :::Thanks, Justin — it is food for thought. :::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]]. :::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:49, 14 May 2026 (UTC) :::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]]. :::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose. :::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable. :::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity. :::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects. :::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]]. :::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles. :::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research. :::-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:39, 15 May 2026 (UTC) My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the "Wikinews refugees" to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote & adopt this proposal independently. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 13:47, 15 May 2026 (UTC) :Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:54, 15 May 2026 (UTC) *Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|<span style="background:#cedff2;color:#11e">''' Blue Rasberry '''</span>]][[User talk:Bluerasberry|<span style="cursor:help"><span style="background:#cedff2;color:#11e">(talk)</span></span>]] 19:17, 22 May 2026 (UTC) *:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 21:15, 22 May 2026 (UTC) *::@[[User:Bluerasberry|Bluerasberry]] WikiJournal is not interested in taking on news journalism. WikiJournal is publishing conference proceedings at the request of some Wikimedian educators, and conference proceedings is what a "regular" journal publishes. News journalism is quite different from this, and if WikiJournal starts to deviate towards publishing news journalism, it will create barrier towards future initiatives like being indexed in Medline or Web of Science, and may risk being delisted from Scopus. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 22:43, 5 June 2026 (UTC) *:::Thats a good point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:09, 9 June 2026 (UTC) == [[Wikiversity:Deletion policy]] proposed as policy == {{archive top|Consensus to promote to an official policy. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:30, 1 June 2026 (UTC)}} [[Wikiversity:Deletions]] has been operating as a [[Wikiversity:Guidelines|guideline]]. It has been revised and moved to [[Wikiversity:Deletion policy]], consistent with naming conventions used across sister projects such as Wikipedia, Wikibooks, and Wikiquote. The speedy deletion criteria have also been updated for consistency with [[MediaWiki:Deletereason-dropdown]]. This proposal is for the page to be formally adopted as [[Wikiversity:Policies|Wikiversity policy]]. Community feedback is invited, including suggestions for further improvements that may strengthen the proposed policy. === Voting === *{{support}} Seems reasonable. If there's somehow something missed here, we can just amend it later. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:33, 18 May 2026 (UTC) *{{support}} I don't see any issues with the policy. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:07, 18 May 2026 (UTC) === Comments === {{archive bottom}} == May 2026 Wikimedia Café meetups regarding the Wikimedia Foundation Annual Plan == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 15px; padding-right: 15px;">[[File:Wikimedia Café logo in plain SVG format.svg|75px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of May. Both sessions will focus on the [https://meta.wikimedia.org/wiki/Wikimedia_Foundation_Annual_Plan/2026-2027 the 2026-2027 Wikimedia Foundation Annual Plan]. Participants may attend either or both sessions. #'''Saturday, 30 May 2026 at 15:00 UTC''' ([https://zonestamp.toolforge.org/1780153200 timestamp converter]), at a time friendly to the Americas, Africa, and Europe #'''Sunday, 31 May 2026 at 05:00 UTC''' ([https://zonestamp.toolforge.org/1780203600 timestamp converter]), at a time friendly to Asia and the Pacific Café participants are highly encouraged to read in advance [https://en.wikipedia.org/wiki/User:Sohom_Datta/annual_plan_guide at least this summary of the plan]. Optionally, Café participants are encouraged to read portions of the plan that interest them and [https://meta.wikimedia.org/wiki/Talk:Wikimedia_Foundation_Annual_Plan/2026-2027 ask questions or provide feedback on the Annual Plan talk page]. Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#May_2026_meetings_with_a_focus_on_Wikimedia_Foundation_Annual_Plan/2026-2027 tables of timestamp conversions for both sessions], [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#Agenda._This_will_be_an_approximately_1_hour_Caf%C3%A9_session,_and_is_extendible_for_an_additional_30_minutes_if_needed. the agenda], and [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">↠Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 19:46, 21 May 2026 (UTC) == Vote now in the 2026 U4C election == <section begin="announcement-content" /> Eligible voters are asked to participate in the 2026 [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee|Universal Code of Conduct Coordinating Committee]] election. More information–including an eligibility check, voting process information, candidate information, and a link to the vote–are available on Meta at the [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2026|2026 Election information page]]. The vote closes on 2 June 2026 at [https://zonestamp.toolforge.org/1780358400 00:00 UTC]. Please vote if your account is eligible. Results will be available by 14 June 2026. -- In cooperation with the U4C,<section end="announcement-content" /> [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 17:15, 27 May 2026 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=30513860 --> == Create an autopatrolled user group? == {{tracked|T428269|resolved}} I would like to propose creating the user group <code>autopatrolled</code> (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling. On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC) :'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:45, 29 May 2026 (UTC) :: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC) :::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:42, 29 May 2026 (UTC) :::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC) :::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:16, 30 May 2026 (UTC) : '''Support''' - sounds like a good idea :* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: "Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors." :* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects? : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:28, 30 May 2026 (UTC) ::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]]. ::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only. :: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC) : @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]]: the autopatroller user group has been implemented here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 8 June 2026 (UTC) ::Thanks. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:13, 9 June 2026 (UTC) == How much of Wikiversity’s content is LLM slop? == Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC) :We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:50, 4 June 2026 (UTC) :Recently agreed [[Wikiversity:Artificial intelligence|policy]] welcome users to tag AI generated pages. Me personally I am not against the use of AI. What is the difference in abstract schematic image created by a human and the same by an AI. If the users does not have finances to pay digital artest and you dont want to let them use AI, would you pay the artest for them? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:07, 8 June 2026 (UTC) ::Wikimedia has a lot of ''volunteer'' artists who can illustrate if asked. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:11, 9 June 2026 (UTC) == Draft inactivity policy == I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]]. However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC) :I support those suggestions. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:55, 4 June 2026 (UTC) == Proposed user group and/or possible policy changes == I want to discuss about user group and possible policy changes. # First, interface administrators. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]]. # Second, curators. Given that curators have some sensitive custodian rights (such as <code>delete</code> [but not <code>undelete</code> or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and <code>protect</code>), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions. # Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC) :#Yes, I agree. :#Thats a good point, but I dont know. At least I dont think its a good idea that both groups i.e. crats and custodiants can do that, it may create chaos. :#Another good point. It seems to me that the current situation is somewhat unclear and should be clarified. I understand the original status of [[Wikiversity:Probationary custodians|Probationary custodians]] as a historicall and invalid, but at the same time I consider myself a probationary custodian, because on the Wikiversity:Custodianship page in the ''[[Wikiversity:Custodianship#How does one become a custodian?|How does one become a custodian?]]'' section it says, I quote, ''"II ...then you will be approved as a probationary custodian for a period of at least four weeks"''. :::Mentors should definitely be kept, but for certain applicants the probation and mentorship should be abolished. For example, if someone was an active custodian for 5 years, then loses their rights or gives them up for a year and then wants to resume their custodial activities, there is no reason for them to undergo a training period. It burdens both the mentors and the community with double voting. The only exception could be a situation where policies or tools for custodians change significantly during that year, or the candidate wants to. :[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:08, 9 June 2026 (UTC) == New user what do I do here == I love wikipedia and the wikiversity project seems super interesting. However I know very little about wikiversity and would like to know how i can best contribute to the project. Also if there are forums or discord or reddit that would be very helpful. (One last thing is it normal that my userboxes don't work here) {{unsigned|AUBSTRAWBS}} :Hey {{ping|AUBSTRAWBS}} Welcome to Wikiversity! I've left a welcome message on your talk page so that should provide you a plethora of useful links for you to look at so you can familiarize yourself with the project. Also, feel free to create the userboxes you need. Wikiversity doesn't have as many userboxes as Wikipedia. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:45, 8 June 2026 (UTC) :Thank you very much :) hope to contribute a lot. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 21:50, 8 June 2026 (UTC) == Towards an Ethics policy == In connection with the [[Wikiversity:Community Review/Removal of Wikidebates|discussion of Wikidebates]], I said that it would be good to establish a policy on ethics, or rather a boundary between ethical and unethical content, so that we don't have to discuss individual cases. In addition, today we also have some global policies that prohibit, for example, attacks on members of the Wikimedia movement or undermining other projects. However, at the very beginning, I would start by collecting your opinions. What content or what research should not be allowed on Wikiversity? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 05:52, 9 June 2026 (UTC) :One ethical issue that I think should be non-controversial is related to good faith in the learning modules. So, learning materials should not be hoaxes or encourage behavior or methods that don't work or that misrepresent the facts or the likelihood of something occurring, etc. and authors should also not plagiarize or misrepresent authorship, etc. That was quite a run-on, but I hope that others can tease out what I mean here. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:39, 9 June 2026 (UTC) ::I look at it from a practical perspective. We can give that to the policy, but I see the problem in that we are not able to check it except plagiarism. ::Plagiarism can be partially detected during patrolling. I see a new text, I put part of it in Google and I check if it is copied from the web. It is a problem with copying from books or other offline sources, but sometimes it happens that someone finds out that something is copied from somewhere and it can be deleted. ::The biggest issue we have here is that we are missing Wikipedia's control mechanism: references. Only some types of resources on Wikiversity require references. In-line references are not often used in courses, exercises, lectures, etc. We are thus deprived of one of the excellent control mechanisms and the only option is for the increase in the number of members with various qualifications to check it for their colleagues. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:59, 9 June 2026 (UTC) :::Having a policy and enforcing that policy are indeed two different things. If we are only concerned with issues that we can definitively enforce, then that will definitely change this conversation. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:06, 9 June 2026 (UTC) :AI generated content should not be allowed as it is inherently plagiarism. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:14, 9 June 2026 (UTC) ::And if the user mention it was generated by an AI? Note that there is something called as public domain, that is the author wave its rights. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:53, 9 June 2026 (UTC) :::Plagiarism isn’t copyright violation. Crediting the AI is not crediting the authors the AI stole from without credit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 10:18, 9 June 2026 (UTC) == Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki == Hello community, The Wikimedia Foundation has provided [[foundation:Legal:Wikimedia Foundation Legal and Safety Contact Information|a single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement. We have already rolled out links to English, German, Italian, Spanish Wikipedias and other wikis and we will deploy to your wiki soon. Please [[m:Wikimedia Foundation Legal and Safety Contacts FAQ|read more on the project page]] and leave any comments in this thread or on [[m:Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|the talk page]]. –– [[User:STei (WMF)|STei (WMF)]] ([[User talk:STei (WMF)|discuss]] • [[Special:Contributions/STei (WMF)|contribs]]) 18:12, 9 June 2026 (UTC) kxx4ejohu592dlvcbiilfo1wjfq1bdo 2814891 2814890 2026-06-09T18:24:07Z Koavf 147 /* Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki */ Reply 2814891 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == [[MediaWiki:Protectedpagetext#Protected edit request on 11 December 2025]] == I posted an edit request there 5 months ago, so I’ll be taking it to this page. [[Special:Contributions/&#126;2026-28640-56|&#126;2026-28640-56]] ([[User talk:&#126;2026-28640-56|talk]]) 23:33, 12 May 2026 (UTC) :What exactly is the problem? I don't understand what needs to change and why. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:35, 12 May 2026 (UTC) : Pinging @[[User:Atcovi|Atcovi]], @[[User:Jtneill|Jtneill]] and @[[User:Juandev|Juandev]] for further input. Someone is requesting a modification to [[MediaWiki:Protectedpagetext]] to use {{tlx|Protected page text}}, but we might need to discuss whether to use the template. In the meantime, I'll start a sandbox version of the protected page text template. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:19, 14 May 2026 (UTC) ::Sounds good -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:13, 15 May 2026 (UTC) :::+1 Jtneill. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:59, 19 May 2026 (UTC) == Proposal to rehost Wikinews here == As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance. I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:30, 14 May 2026 (UTC) :I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) ::A few days shy of 30, it seems obvious that this is not going to pass. So I '''withdraw''' as presumptively '''failed'''. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:14, 9 June 2026 (UTC) ===Votes=== *{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) *{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC) *{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of "practicing journalism" a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC) *{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:48, 15 May 2026 (UTC) *{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC) *{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 20:29, 30 May 2026 (UTC) * {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC) *{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC) * {{oppose}} per above. Wikiversity<math>\not=</math> Wikinews - not a good idea to mix the scope of projects. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:03, 8 June 2026 (UTC) * {{abstain}} I will abstain since I'm not an active Wikiversity contributor. But I just feel like Wikinews had a very clear and specific goal of providing news, and Wikiversity is just a different project with different goals. For me, it would be odd to rehost Wikinews here. But please do not count my vote, this is only a comment. --[[User:Antimundo|Antimundo]] ([[User talk:Antimundo|discuss]] • [[Special:Contributions/Antimundo|contribs]]) 13:19, 6 June 2026 (UTC) * {{oppose}} Although I think it's a pity that Wikinews is closed. --[[User:Dick Bos|Dick Bos]] ([[User talk:Dick Bos|discuss]] • [[Special:Contributions/Dick Bos|contribs]]) 19:06, 8 June 2026 (UTC) ===Comments and questions=== :Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice. :Initial questions: :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? :* What are "active editions"? :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? :* Are any changes to the scope of Wikinews proposed? :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. :-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:52, 14 May 2026 (UTC) :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? ::*No, not at this time. :* What are "active editions"? ::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04). :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? ::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure). :* Are any changes to the scope of Wikinews proposed? ::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]]. :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. ::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning <em>about</em> what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like. ::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:13, 14 May 2026 (UTC) :::Thanks, Justin — it is food for thought. :::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]]. :::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:49, 14 May 2026 (UTC) :::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]]. :::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose. :::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable. :::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity. :::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects. :::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]]. :::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles. :::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research. :::-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:39, 15 May 2026 (UTC) My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the "Wikinews refugees" to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote & adopt this proposal independently. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 13:47, 15 May 2026 (UTC) :Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:54, 15 May 2026 (UTC) *Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|<span style="background:#cedff2;color:#11e">''' Blue Rasberry '''</span>]][[User talk:Bluerasberry|<span style="cursor:help"><span style="background:#cedff2;color:#11e">(talk)</span></span>]] 19:17, 22 May 2026 (UTC) *:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 21:15, 22 May 2026 (UTC) *::@[[User:Bluerasberry|Bluerasberry]] WikiJournal is not interested in taking on news journalism. WikiJournal is publishing conference proceedings at the request of some Wikimedian educators, and conference proceedings is what a "regular" journal publishes. News journalism is quite different from this, and if WikiJournal starts to deviate towards publishing news journalism, it will create barrier towards future initiatives like being indexed in Medline or Web of Science, and may risk being delisted from Scopus. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 22:43, 5 June 2026 (UTC) *:::Thats a good point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:09, 9 June 2026 (UTC) == [[Wikiversity:Deletion policy]] proposed as policy == {{archive top|Consensus to promote to an official policy. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:30, 1 June 2026 (UTC)}} [[Wikiversity:Deletions]] has been operating as a [[Wikiversity:Guidelines|guideline]]. It has been revised and moved to [[Wikiversity:Deletion policy]], consistent with naming conventions used across sister projects such as Wikipedia, Wikibooks, and Wikiquote. The speedy deletion criteria have also been updated for consistency with [[MediaWiki:Deletereason-dropdown]]. This proposal is for the page to be formally adopted as [[Wikiversity:Policies|Wikiversity policy]]. Community feedback is invited, including suggestions for further improvements that may strengthen the proposed policy. === Voting === *{{support}} Seems reasonable. If there's somehow something missed here, we can just amend it later. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:33, 18 May 2026 (UTC) *{{support}} I don't see any issues with the policy. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:07, 18 May 2026 (UTC) === Comments === {{archive bottom}} == May 2026 Wikimedia Café meetups regarding the Wikimedia Foundation Annual Plan == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 15px; padding-right: 15px;">[[File:Wikimedia Café logo in plain SVG format.svg|75px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of May. Both sessions will focus on the [https://meta.wikimedia.org/wiki/Wikimedia_Foundation_Annual_Plan/2026-2027 the 2026-2027 Wikimedia Foundation Annual Plan]. Participants may attend either or both sessions. #'''Saturday, 30 May 2026 at 15:00 UTC''' ([https://zonestamp.toolforge.org/1780153200 timestamp converter]), at a time friendly to the Americas, Africa, and Europe #'''Sunday, 31 May 2026 at 05:00 UTC''' ([https://zonestamp.toolforge.org/1780203600 timestamp converter]), at a time friendly to Asia and the Pacific Café participants are highly encouraged to read in advance [https://en.wikipedia.org/wiki/User:Sohom_Datta/annual_plan_guide at least this summary of the plan]. Optionally, Café participants are encouraged to read portions of the plan that interest them and [https://meta.wikimedia.org/wiki/Talk:Wikimedia_Foundation_Annual_Plan/2026-2027 ask questions or provide feedback on the Annual Plan talk page]. Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#May_2026_meetings_with_a_focus_on_Wikimedia_Foundation_Annual_Plan/2026-2027 tables of timestamp conversions for both sessions], [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#Agenda._This_will_be_an_approximately_1_hour_Caf%C3%A9_session,_and_is_extendible_for_an_additional_30_minutes_if_needed. the agenda], and [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">↠Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 19:46, 21 May 2026 (UTC) == Vote now in the 2026 U4C election == <section begin="announcement-content" /> Eligible voters are asked to participate in the 2026 [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee|Universal Code of Conduct Coordinating Committee]] election. More information–including an eligibility check, voting process information, candidate information, and a link to the vote–are available on Meta at the [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2026|2026 Election information page]]. The vote closes on 2 June 2026 at [https://zonestamp.toolforge.org/1780358400 00:00 UTC]. Please vote if your account is eligible. Results will be available by 14 June 2026. -- In cooperation with the U4C,<section end="announcement-content" /> [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 17:15, 27 May 2026 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=30513860 --> == Create an autopatrolled user group? == {{tracked|T428269|resolved}} I would like to propose creating the user group <code>autopatrolled</code> (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling. On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC) :'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:45, 29 May 2026 (UTC) :: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC) :::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:42, 29 May 2026 (UTC) :::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC) :::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:16, 30 May 2026 (UTC) : '''Support''' - sounds like a good idea :* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: "Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors." :* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects? : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:28, 30 May 2026 (UTC) ::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]]. ::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only. :: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC) : @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]]: the autopatroller user group has been implemented here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 8 June 2026 (UTC) ::Thanks. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:13, 9 June 2026 (UTC) == How much of Wikiversity’s content is LLM slop? == Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC) :We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:50, 4 June 2026 (UTC) :Recently agreed [[Wikiversity:Artificial intelligence|policy]] welcome users to tag AI generated pages. Me personally I am not against the use of AI. What is the difference in abstract schematic image created by a human and the same by an AI. If the users does not have finances to pay digital artest and you dont want to let them use AI, would you pay the artest for them? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:07, 8 June 2026 (UTC) ::Wikimedia has a lot of ''volunteer'' artists who can illustrate if asked. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:11, 9 June 2026 (UTC) == Draft inactivity policy == I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]]. However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC) :I support those suggestions. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:55, 4 June 2026 (UTC) == Proposed user group and/or possible policy changes == I want to discuss about user group and possible policy changes. # First, interface administrators. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]]. # Second, curators. Given that curators have some sensitive custodian rights (such as <code>delete</code> [but not <code>undelete</code> or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and <code>protect</code>), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions. # Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC) :#Yes, I agree. :#Thats a good point, but I dont know. At least I dont think its a good idea that both groups i.e. crats and custodiants can do that, it may create chaos. :#Another good point. It seems to me that the current situation is somewhat unclear and should be clarified. I understand the original status of [[Wikiversity:Probationary custodians|Probationary custodians]] as a historicall and invalid, but at the same time I consider myself a probationary custodian, because on the Wikiversity:Custodianship page in the ''[[Wikiversity:Custodianship#How does one become a custodian?|How does one become a custodian?]]'' section it says, I quote, ''"II ...then you will be approved as a probationary custodian for a period of at least four weeks"''. :::Mentors should definitely be kept, but for certain applicants the probation and mentorship should be abolished. For example, if someone was an active custodian for 5 years, then loses their rights or gives them up for a year and then wants to resume their custodial activities, there is no reason for them to undergo a training period. It burdens both the mentors and the community with double voting. The only exception could be a situation where policies or tools for custodians change significantly during that year, or the candidate wants to. :[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:08, 9 June 2026 (UTC) == New user what do I do here == I love wikipedia and the wikiversity project seems super interesting. However I know very little about wikiversity and would like to know how i can best contribute to the project. Also if there are forums or discord or reddit that would be very helpful. (One last thing is it normal that my userboxes don't work here) {{unsigned|AUBSTRAWBS}} :Hey {{ping|AUBSTRAWBS}} Welcome to Wikiversity! I've left a welcome message on your talk page so that should provide you a plethora of useful links for you to look at so you can familiarize yourself with the project. Also, feel free to create the userboxes you need. Wikiversity doesn't have as many userboxes as Wikipedia. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:45, 8 June 2026 (UTC) :Thank you very much :) hope to contribute a lot. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 21:50, 8 June 2026 (UTC) == Towards an Ethics policy == In connection with the [[Wikiversity:Community Review/Removal of Wikidebates|discussion of Wikidebates]], I said that it would be good to establish a policy on ethics, or rather a boundary between ethical and unethical content, so that we don't have to discuss individual cases. In addition, today we also have some global policies that prohibit, for example, attacks on members of the Wikimedia movement or undermining other projects. However, at the very beginning, I would start by collecting your opinions. What content or what research should not be allowed on Wikiversity? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 05:52, 9 June 2026 (UTC) :One ethical issue that I think should be non-controversial is related to good faith in the learning modules. So, learning materials should not be hoaxes or encourage behavior or methods that don't work or that misrepresent the facts or the likelihood of something occurring, etc. and authors should also not plagiarize or misrepresent authorship, etc. That was quite a run-on, but I hope that others can tease out what I mean here. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:39, 9 June 2026 (UTC) ::I look at it from a practical perspective. We can give that to the policy, but I see the problem in that we are not able to check it except plagiarism. ::Plagiarism can be partially detected during patrolling. I see a new text, I put part of it in Google and I check if it is copied from the web. It is a problem with copying from books or other offline sources, but sometimes it happens that someone finds out that something is copied from somewhere and it can be deleted. ::The biggest issue we have here is that we are missing Wikipedia's control mechanism: references. Only some types of resources on Wikiversity require references. In-line references are not often used in courses, exercises, lectures, etc. We are thus deprived of one of the excellent control mechanisms and the only option is for the increase in the number of members with various qualifications to check it for their colleagues. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:59, 9 June 2026 (UTC) :::Having a policy and enforcing that policy are indeed two different things. If we are only concerned with issues that we can definitively enforce, then that will definitely change this conversation. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:06, 9 June 2026 (UTC) :AI generated content should not be allowed as it is inherently plagiarism. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:14, 9 June 2026 (UTC) ::And if the user mention it was generated by an AI? Note that there is something called as public domain, that is the author wave its rights. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:53, 9 June 2026 (UTC) :::Plagiarism isn’t copyright violation. Crediting the AI is not crediting the authors the AI stole from without credit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 10:18, 9 June 2026 (UTC) == Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki == Hello community, The Wikimedia Foundation has provided [[foundation:Legal:Wikimedia Foundation Legal and Safety Contact Information|a single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement. We have already rolled out links to English, German, Italian, Spanish Wikipedias and other wikis and we will deploy to your wiki soon. Please [[m:Wikimedia Foundation Legal and Safety Contacts FAQ|read more on the project page]] and leave any comments in this thread or on [[m:Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|the talk page]]. –– [[User:STei (WMF)|STei (WMF)]] ([[User talk:STei (WMF)|discuss]] • [[Special:Contributions/STei (WMF)|contribs]]) 18:12, 9 June 2026 (UTC) :Thanks for the notice. In case anyone is not clear, we cannot locally change the text at the footer, as it [[:mw:Manual:Footer|requires access to the server settings]]. If we locally needed to change it, we would have to file a ticket at [[:phab:]]. Since the above was sent by someone from the WMF, I think they are on it and it will be updated without any action from anyone here. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:24, 9 June 2026 (UTC) 6ipk9msp1kc5kcj8apbax8dv218e391 2814892 2814891 2026-06-09T18:26:39Z DavidMCEddy 218607 /* Votes */ support 2814892 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == [[MediaWiki:Protectedpagetext#Protected edit request on 11 December 2025]] == I posted an edit request there 5 months ago, so I’ll be taking it to this page. [[Special:Contributions/&#126;2026-28640-56|&#126;2026-28640-56]] ([[User talk:&#126;2026-28640-56|talk]]) 23:33, 12 May 2026 (UTC) :What exactly is the problem? I don't understand what needs to change and why. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:35, 12 May 2026 (UTC) : Pinging @[[User:Atcovi|Atcovi]], @[[User:Jtneill|Jtneill]] and @[[User:Juandev|Juandev]] for further input. Someone is requesting a modification to [[MediaWiki:Protectedpagetext]] to use {{tlx|Protected page text}}, but we might need to discuss whether to use the template. In the meantime, I'll start a sandbox version of the protected page text template. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:19, 14 May 2026 (UTC) ::Sounds good -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:13, 15 May 2026 (UTC) :::+1 Jtneill. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:59, 19 May 2026 (UTC) == Proposal to rehost Wikinews here == As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance. I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:30, 14 May 2026 (UTC) :I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) ::A few days shy of 30, it seems obvious that this is not going to pass. So I '''withdraw''' as presumptively '''failed'''. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:14, 9 June 2026 (UTC) ===Votes=== *{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) *{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC) *{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of "practicing journalism" a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC) *{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:48, 15 May 2026 (UTC) *{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC) *{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 20:29, 30 May 2026 (UTC) * {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC) *{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC) * {{oppose}} per above. Wikiversity<math>\not=</math> Wikinews - not a good idea to mix the scope of projects. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:03, 8 June 2026 (UTC) * {{abstain}} I will abstain since I'm not an active Wikiversity contributor. But I just feel like Wikinews had a very clear and specific goal of providing news, and Wikiversity is just a different project with different goals. For me, it would be odd to rehost Wikinews here. But please do not count my vote, this is only a comment. --[[User:Antimundo|Antimundo]] ([[User talk:Antimundo|discuss]] • [[Special:Contributions/Antimundo|contribs]]) 13:19, 6 June 2026 (UTC) * {{oppose}} Although I think it's a pity that Wikinews is closed. --[[User:Dick Bos|Dick Bos]] ([[User talk:Dick Bos|discuss]] • [[Special:Contributions/Dick Bos|contribs]]) 19:06, 8 June 2026 (UTC) *{{support}} In 2018 I initiated [[:Category:Videoconferences on media and democracy]] as a platform for disseminating public affairs events. In 2021 I officially initiated a podcast series on "Media & Democracy" syndicated for the [[w:List of Pacifica Radio stations and affiliates|Pacifica radio network]]. In 2024 I converted it from irregular to fortnightly. I think this is all educational and supports the Wikiversity education mission, and I think that "rehost Wikinews here" would be appropriate. (I had some experience with Wikinews a few years ago. I felt it was too tightly controlled: Article submissions went stale, because I could not get official permission to publish and I could not get the information needed to understand what I was supposed to do to obtain the official permission. I would be opposed to rehosting Wikinews here if the policy similarly made it unreasonably difficult for volunteer contributor to get the information needed to meet the journalistic standards imposed by the overworked editors.) ===Comments and questions=== :Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice. :Initial questions: :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? :* What are "active editions"? :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? :* Are any changes to the scope of Wikinews proposed? :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. :-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:52, 14 May 2026 (UTC) :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? ::*No, not at this time. :* What are "active editions"? ::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04). :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? ::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure). :* Are any changes to the scope of Wikinews proposed? ::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]]. :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. ::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning <em>about</em> what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like. ::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:13, 14 May 2026 (UTC) :::Thanks, Justin — it is food for thought. :::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]]. :::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:49, 14 May 2026 (UTC) :::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]]. :::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose. :::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable. :::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity. :::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects. :::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]]. :::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles. :::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research. :::-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:39, 15 May 2026 (UTC) My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the "Wikinews refugees" to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote & adopt this proposal independently. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 13:47, 15 May 2026 (UTC) :Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:54, 15 May 2026 (UTC) *Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|<span style="background:#cedff2;color:#11e">''' Blue Rasberry '''</span>]][[User talk:Bluerasberry|<span style="cursor:help"><span style="background:#cedff2;color:#11e">(talk)</span></span>]] 19:17, 22 May 2026 (UTC) *:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 21:15, 22 May 2026 (UTC) *::@[[User:Bluerasberry|Bluerasberry]] WikiJournal is not interested in taking on news journalism. WikiJournal is publishing conference proceedings at the request of some Wikimedian educators, and conference proceedings is what a "regular" journal publishes. News journalism is quite different from this, and if WikiJournal starts to deviate towards publishing news journalism, it will create barrier towards future initiatives like being indexed in Medline or Web of Science, and may risk being delisted from Scopus. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 22:43, 5 June 2026 (UTC) *:::Thats a good point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:09, 9 June 2026 (UTC) == [[Wikiversity:Deletion policy]] proposed as policy == {{archive top|Consensus to promote to an official policy. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:30, 1 June 2026 (UTC)}} [[Wikiversity:Deletions]] has been operating as a [[Wikiversity:Guidelines|guideline]]. It has been revised and moved to [[Wikiversity:Deletion policy]], consistent with naming conventions used across sister projects such as Wikipedia, Wikibooks, and Wikiquote. The speedy deletion criteria have also been updated for consistency with [[MediaWiki:Deletereason-dropdown]]. This proposal is for the page to be formally adopted as [[Wikiversity:Policies|Wikiversity policy]]. Community feedback is invited, including suggestions for further improvements that may strengthen the proposed policy. === Voting === *{{support}} Seems reasonable. If there's somehow something missed here, we can just amend it later. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:33, 18 May 2026 (UTC) *{{support}} I don't see any issues with the policy. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:07, 18 May 2026 (UTC) === Comments === {{archive bottom}} == May 2026 Wikimedia Café meetups regarding the Wikimedia Foundation Annual Plan == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 15px; padding-right: 15px;">[[File:Wikimedia Café logo in plain SVG format.svg|75px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of May. Both sessions will focus on the [https://meta.wikimedia.org/wiki/Wikimedia_Foundation_Annual_Plan/2026-2027 the 2026-2027 Wikimedia Foundation Annual Plan]. Participants may attend either or both sessions. #'''Saturday, 30 May 2026 at 15:00 UTC''' ([https://zonestamp.toolforge.org/1780153200 timestamp converter]), at a time friendly to the Americas, Africa, and Europe #'''Sunday, 31 May 2026 at 05:00 UTC''' ([https://zonestamp.toolforge.org/1780203600 timestamp converter]), at a time friendly to Asia and the Pacific Café participants are highly encouraged to read in advance [https://en.wikipedia.org/wiki/User:Sohom_Datta/annual_plan_guide at least this summary of the plan]. Optionally, Café participants are encouraged to read portions of the plan that interest them and [https://meta.wikimedia.org/wiki/Talk:Wikimedia_Foundation_Annual_Plan/2026-2027 ask questions or provide feedback on the Annual Plan talk page]. Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#May_2026_meetings_with_a_focus_on_Wikimedia_Foundation_Annual_Plan/2026-2027 tables of timestamp conversions for both sessions], [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#Agenda._This_will_be_an_approximately_1_hour_Caf%C3%A9_session,_and_is_extendible_for_an_additional_30_minutes_if_needed. the agenda], and [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">↠Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 19:46, 21 May 2026 (UTC) == Vote now in the 2026 U4C election == <section begin="announcement-content" /> Eligible voters are asked to participate in the 2026 [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee|Universal Code of Conduct Coordinating Committee]] election. More information–including an eligibility check, voting process information, candidate information, and a link to the vote–are available on Meta at the [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2026|2026 Election information page]]. The vote closes on 2 June 2026 at [https://zonestamp.toolforge.org/1780358400 00:00 UTC]. Please vote if your account is eligible. Results will be available by 14 June 2026. -- In cooperation with the U4C,<section end="announcement-content" /> [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 17:15, 27 May 2026 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=30513860 --> == Create an autopatrolled user group? == {{tracked|T428269|resolved}} I would like to propose creating the user group <code>autopatrolled</code> (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling. On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC) :'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:45, 29 May 2026 (UTC) :: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC) :::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:42, 29 May 2026 (UTC) :::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC) :::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:16, 30 May 2026 (UTC) : '''Support''' - sounds like a good idea :* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: "Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors." :* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects? : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:28, 30 May 2026 (UTC) ::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]]. ::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only. :: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC) : @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]]: the autopatroller user group has been implemented here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 8 June 2026 (UTC) ::Thanks. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:13, 9 June 2026 (UTC) == How much of Wikiversity’s content is LLM slop? == Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC) :We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:50, 4 June 2026 (UTC) :Recently agreed [[Wikiversity:Artificial intelligence|policy]] welcome users to tag AI generated pages. Me personally I am not against the use of AI. What is the difference in abstract schematic image created by a human and the same by an AI. If the users does not have finances to pay digital artest and you dont want to let them use AI, would you pay the artest for them? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:07, 8 June 2026 (UTC) ::Wikimedia has a lot of ''volunteer'' artists who can illustrate if asked. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:11, 9 June 2026 (UTC) == Draft inactivity policy == I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]]. However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC) :I support those suggestions. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:55, 4 June 2026 (UTC) == Proposed user group and/or possible policy changes == I want to discuss about user group and possible policy changes. # First, interface administrators. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]]. # Second, curators. Given that curators have some sensitive custodian rights (such as <code>delete</code> [but not <code>undelete</code> or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and <code>protect</code>), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions. # Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC) :#Yes, I agree. :#Thats a good point, but I dont know. At least I dont think its a good idea that both groups i.e. crats and custodiants can do that, it may create chaos. :#Another good point. It seems to me that the current situation is somewhat unclear and should be clarified. I understand the original status of [[Wikiversity:Probationary custodians|Probationary custodians]] as a historicall and invalid, but at the same time I consider myself a probationary custodian, because on the Wikiversity:Custodianship page in the ''[[Wikiversity:Custodianship#How does one become a custodian?|How does one become a custodian?]]'' section it says, I quote, ''"II ...then you will be approved as a probationary custodian for a period of at least four weeks"''. :::Mentors should definitely be kept, but for certain applicants the probation and mentorship should be abolished. For example, if someone was an active custodian for 5 years, then loses their rights or gives them up for a year and then wants to resume their custodial activities, there is no reason for them to undergo a training period. It burdens both the mentors and the community with double voting. The only exception could be a situation where policies or tools for custodians change significantly during that year, or the candidate wants to. :[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:08, 9 June 2026 (UTC) == New user what do I do here == I love wikipedia and the wikiversity project seems super interesting. However I know very little about wikiversity and would like to know how i can best contribute to the project. Also if there are forums or discord or reddit that would be very helpful. (One last thing is it normal that my userboxes don't work here) {{unsigned|AUBSTRAWBS}} :Hey {{ping|AUBSTRAWBS}} Welcome to Wikiversity! I've left a welcome message on your talk page so that should provide you a plethora of useful links for you to look at so you can familiarize yourself with the project. Also, feel free to create the userboxes you need. Wikiversity doesn't have as many userboxes as Wikipedia. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:45, 8 June 2026 (UTC) :Thank you very much :) hope to contribute a lot. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 21:50, 8 June 2026 (UTC) == Towards an Ethics policy == In connection with the [[Wikiversity:Community Review/Removal of Wikidebates|discussion of Wikidebates]], I said that it would be good to establish a policy on ethics, or rather a boundary between ethical and unethical content, so that we don't have to discuss individual cases. In addition, today we also have some global policies that prohibit, for example, attacks on members of the Wikimedia movement or undermining other projects. However, at the very beginning, I would start by collecting your opinions. What content or what research should not be allowed on Wikiversity? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 05:52, 9 June 2026 (UTC) :One ethical issue that I think should be non-controversial is related to good faith in the learning modules. So, learning materials should not be hoaxes or encourage behavior or methods that don't work or that misrepresent the facts or the likelihood of something occurring, etc. and authors should also not plagiarize or misrepresent authorship, etc. That was quite a run-on, but I hope that others can tease out what I mean here. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:39, 9 June 2026 (UTC) ::I look at it from a practical perspective. We can give that to the policy, but I see the problem in that we are not able to check it except plagiarism. ::Plagiarism can be partially detected during patrolling. I see a new text, I put part of it in Google and I check if it is copied from the web. It is a problem with copying from books or other offline sources, but sometimes it happens that someone finds out that something is copied from somewhere and it can be deleted. ::The biggest issue we have here is that we are missing Wikipedia's control mechanism: references. Only some types of resources on Wikiversity require references. In-line references are not often used in courses, exercises, lectures, etc. We are thus deprived of one of the excellent control mechanisms and the only option is for the increase in the number of members with various qualifications to check it for their colleagues. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:59, 9 June 2026 (UTC) :::Having a policy and enforcing that policy are indeed two different things. If we are only concerned with issues that we can definitively enforce, then that will definitely change this conversation. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:06, 9 June 2026 (UTC) :AI generated content should not be allowed as it is inherently plagiarism. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:14, 9 June 2026 (UTC) ::And if the user mention it was generated by an AI? Note that there is something called as public domain, that is the author wave its rights. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:53, 9 June 2026 (UTC) :::Plagiarism isn’t copyright violation. Crediting the AI is not crediting the authors the AI stole from without credit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 10:18, 9 June 2026 (UTC) == Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki == Hello community, The Wikimedia Foundation has provided [[foundation:Legal:Wikimedia Foundation Legal and Safety Contact Information|a single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement. We have already rolled out links to English, German, Italian, Spanish Wikipedias and other wikis and we will deploy to your wiki soon. Please [[m:Wikimedia Foundation Legal and Safety Contacts FAQ|read more on the project page]] and leave any comments in this thread or on [[m:Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|the talk page]]. –– [[User:STei (WMF)|STei (WMF)]] ([[User talk:STei (WMF)|discuss]] • [[Special:Contributions/STei (WMF)|contribs]]) 18:12, 9 June 2026 (UTC) :Thanks for the notice. In case anyone is not clear, we cannot locally change the text at the footer, as it [[:mw:Manual:Footer|requires access to the server settings]]. If we locally needed to change it, we would have to file a ticket at [[:phab:]]. Since the above was sent by someone from the WMF, I think they are on it and it will be updated without any action from anyone here. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:24, 9 June 2026 (UTC) 8v0nl4qt3x0pf9e5br51z5gm0z4qzcv 2814893 2814892 2026-06-09T18:32:44Z Koavf 147 /* Votes */ 2814893 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == [[MediaWiki:Protectedpagetext#Protected edit request on 11 December 2025]] == I posted an edit request there 5 months ago, so I’ll be taking it to this page. [[Special:Contributions/&#126;2026-28640-56|&#126;2026-28640-56]] ([[User talk:&#126;2026-28640-56|talk]]) 23:33, 12 May 2026 (UTC) :What exactly is the problem? I don't understand what needs to change and why. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:35, 12 May 2026 (UTC) : Pinging @[[User:Atcovi|Atcovi]], @[[User:Jtneill|Jtneill]] and @[[User:Juandev|Juandev]] for further input. Someone is requesting a modification to [[MediaWiki:Protectedpagetext]] to use {{tlx|Protected page text}}, but we might need to discuss whether to use the template. In the meantime, I'll start a sandbox version of the protected page text template. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:19, 14 May 2026 (UTC) ::Sounds good -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:13, 15 May 2026 (UTC) :::+1 Jtneill. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:59, 19 May 2026 (UTC) == Proposal to rehost Wikinews here == As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance. I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:30, 14 May 2026 (UTC) :I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) ::A few days shy of 30, it seems obvious that this is not going to pass. So I '''withdraw''' as presumptively '''failed'''. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:14, 9 June 2026 (UTC) ===Votes=== *{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) *{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC) *{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of "practicing journalism" a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC) *{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:48, 15 May 2026 (UTC) *{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC) *{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 20:29, 30 May 2026 (UTC) * {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC) *{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC) * {{oppose}} per above. Wikiversity<math>\not=</math> Wikinews - not a good idea to mix the scope of projects. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:03, 8 June 2026 (UTC) * {{abstain}} I will abstain since I'm not an active Wikiversity contributor. But I just feel like Wikinews had a very clear and specific goal of providing news, and Wikiversity is just a different project with different goals. For me, it would be odd to rehost Wikinews here. But please do not count my vote, this is only a comment. --[[User:Antimundo|Antimundo]] ([[User talk:Antimundo|discuss]] • [[Special:Contributions/Antimundo|contribs]]) 13:19, 6 June 2026 (UTC) * {{oppose}} Although I think it's a pity that Wikinews is closed. --[[User:Dick Bos|Dick Bos]] ([[User talk:Dick Bos|discuss]] • [[Special:Contributions/Dick Bos|contribs]]) 19:06, 8 June 2026 (UTC) *{{support}} In 2018 I initiated [[:Category:Videoconferences on media and democracy]] as a platform for disseminating public affairs events. In 2021 I officially initiated a podcast series on "Media & Democracy" syndicated for the [[w:List of Pacifica Radio stations and affiliates|Pacifica radio network]]. In 2024 I converted it from irregular to fortnightly. I think this is all educational and supports the Wikiversity education mission, and I think that "rehost Wikinews here" would be appropriate. (I had some experience with Wikinews a few years ago. I felt it was too tightly controlled: Article submissions went stale, because I could not get official permission to publish and I could not get the information needed to understand what I was supposed to do to obtain the official permission. I would be opposed to rehosting Wikinews here if the policy similarly made it unreasonably difficult for volunteer contributor to get the information needed to meet the journalistic standards imposed by the overworked editors.) {{unsigned|DavidMCEddy}} ===Comments and questions=== :Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice. :Initial questions: :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? :* What are "active editions"? :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? :* Are any changes to the scope of Wikinews proposed? :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. :-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:52, 14 May 2026 (UTC) :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? ::*No, not at this time. :* What are "active editions"? ::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04). :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? ::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure). :* Are any changes to the scope of Wikinews proposed? ::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]]. :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. ::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning <em>about</em> what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like. ::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:13, 14 May 2026 (UTC) :::Thanks, Justin — it is food for thought. :::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]]. :::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:49, 14 May 2026 (UTC) :::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]]. :::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose. :::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable. :::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity. :::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects. :::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]]. :::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles. :::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research. :::-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:39, 15 May 2026 (UTC) My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the "Wikinews refugees" to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote & adopt this proposal independently. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 13:47, 15 May 2026 (UTC) :Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:54, 15 May 2026 (UTC) *Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|<span style="background:#cedff2;color:#11e">''' Blue Rasberry '''</span>]][[User talk:Bluerasberry|<span style="cursor:help"><span style="background:#cedff2;color:#11e">(talk)</span></span>]] 19:17, 22 May 2026 (UTC) *:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 21:15, 22 May 2026 (UTC) *::@[[User:Bluerasberry|Bluerasberry]] WikiJournal is not interested in taking on news journalism. WikiJournal is publishing conference proceedings at the request of some Wikimedian educators, and conference proceedings is what a "regular" journal publishes. News journalism is quite different from this, and if WikiJournal starts to deviate towards publishing news journalism, it will create barrier towards future initiatives like being indexed in Medline or Web of Science, and may risk being delisted from Scopus. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 22:43, 5 June 2026 (UTC) *:::Thats a good point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:09, 9 June 2026 (UTC) == [[Wikiversity:Deletion policy]] proposed as policy == {{archive top|Consensus to promote to an official policy. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:30, 1 June 2026 (UTC)}} [[Wikiversity:Deletions]] has been operating as a [[Wikiversity:Guidelines|guideline]]. It has been revised and moved to [[Wikiversity:Deletion policy]], consistent with naming conventions used across sister projects such as Wikipedia, Wikibooks, and Wikiquote. The speedy deletion criteria have also been updated for consistency with [[MediaWiki:Deletereason-dropdown]]. This proposal is for the page to be formally adopted as [[Wikiversity:Policies|Wikiversity policy]]. Community feedback is invited, including suggestions for further improvements that may strengthen the proposed policy. === Voting === *{{support}} Seems reasonable. If there's somehow something missed here, we can just amend it later. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:33, 18 May 2026 (UTC) *{{support}} I don't see any issues with the policy. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:07, 18 May 2026 (UTC) === Comments === {{archive bottom}} == May 2026 Wikimedia Café meetups regarding the Wikimedia Foundation Annual Plan == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 15px; padding-right: 15px;">[[File:Wikimedia Café logo in plain SVG format.svg|75px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of May. Both sessions will focus on the [https://meta.wikimedia.org/wiki/Wikimedia_Foundation_Annual_Plan/2026-2027 the 2026-2027 Wikimedia Foundation Annual Plan]. Participants may attend either or both sessions. #'''Saturday, 30 May 2026 at 15:00 UTC''' ([https://zonestamp.toolforge.org/1780153200 timestamp converter]), at a time friendly to the Americas, Africa, and Europe #'''Sunday, 31 May 2026 at 05:00 UTC''' ([https://zonestamp.toolforge.org/1780203600 timestamp converter]), at a time friendly to Asia and the Pacific Café participants are highly encouraged to read in advance [https://en.wikipedia.org/wiki/User:Sohom_Datta/annual_plan_guide at least this summary of the plan]. Optionally, Café participants are encouraged to read portions of the plan that interest them and [https://meta.wikimedia.org/wiki/Talk:Wikimedia_Foundation_Annual_Plan/2026-2027 ask questions or provide feedback on the Annual Plan talk page]. Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#May_2026_meetings_with_a_focus_on_Wikimedia_Foundation_Annual_Plan/2026-2027 tables of timestamp conversions for both sessions], [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#Agenda._This_will_be_an_approximately_1_hour_Caf%C3%A9_session,_and_is_extendible_for_an_additional_30_minutes_if_needed. the agenda], and [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">↠Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 19:46, 21 May 2026 (UTC) == Vote now in the 2026 U4C election == <section begin="announcement-content" /> Eligible voters are asked to participate in the 2026 [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee|Universal Code of Conduct Coordinating Committee]] election. More information–including an eligibility check, voting process information, candidate information, and a link to the vote–are available on Meta at the [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2026|2026 Election information page]]. The vote closes on 2 June 2026 at [https://zonestamp.toolforge.org/1780358400 00:00 UTC]. Please vote if your account is eligible. Results will be available by 14 June 2026. -- In cooperation with the U4C,<section end="announcement-content" /> [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 17:15, 27 May 2026 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=30513860 --> == Create an autopatrolled user group? == {{tracked|T428269|resolved}} I would like to propose creating the user group <code>autopatrolled</code> (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling. On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC) :'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:45, 29 May 2026 (UTC) :: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC) :::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:42, 29 May 2026 (UTC) :::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC) :::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:16, 30 May 2026 (UTC) : '''Support''' - sounds like a good idea :* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: "Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors." :* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects? : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:28, 30 May 2026 (UTC) ::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]]. ::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only. :: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC) : @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]]: the autopatroller user group has been implemented here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 8 June 2026 (UTC) ::Thanks. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:13, 9 June 2026 (UTC) == How much of Wikiversity’s content is LLM slop? == Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC) :We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:50, 4 June 2026 (UTC) :Recently agreed [[Wikiversity:Artificial intelligence|policy]] welcome users to tag AI generated pages. Me personally I am not against the use of AI. What is the difference in abstract schematic image created by a human and the same by an AI. If the users does not have finances to pay digital artest and you dont want to let them use AI, would you pay the artest for them? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:07, 8 June 2026 (UTC) ::Wikimedia has a lot of ''volunteer'' artists who can illustrate if asked. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:11, 9 June 2026 (UTC) == Draft inactivity policy == I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]]. However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC) :I support those suggestions. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:55, 4 June 2026 (UTC) == Proposed user group and/or possible policy changes == I want to discuss about user group and possible policy changes. # First, interface administrators. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]]. # Second, curators. Given that curators have some sensitive custodian rights (such as <code>delete</code> [but not <code>undelete</code> or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and <code>protect</code>), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions. # Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC) :#Yes, I agree. :#Thats a good point, but I dont know. At least I dont think its a good idea that both groups i.e. crats and custodiants can do that, it may create chaos. :#Another good point. It seems to me that the current situation is somewhat unclear and should be clarified. I understand the original status of [[Wikiversity:Probationary custodians|Probationary custodians]] as a historicall and invalid, but at the same time I consider myself a probationary custodian, because on the Wikiversity:Custodianship page in the ''[[Wikiversity:Custodianship#How does one become a custodian?|How does one become a custodian?]]'' section it says, I quote, ''"II ...then you will be approved as a probationary custodian for a period of at least four weeks"''. :::Mentors should definitely be kept, but for certain applicants the probation and mentorship should be abolished. For example, if someone was an active custodian for 5 years, then loses their rights or gives them up for a year and then wants to resume their custodial activities, there is no reason for them to undergo a training period. It burdens both the mentors and the community with double voting. The only exception could be a situation where policies or tools for custodians change significantly during that year, or the candidate wants to. :[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:08, 9 June 2026 (UTC) == New user what do I do here == I love wikipedia and the wikiversity project seems super interesting. However I know very little about wikiversity and would like to know how i can best contribute to the project. Also if there are forums or discord or reddit that would be very helpful. (One last thing is it normal that my userboxes don't work here) {{unsigned|AUBSTRAWBS}} :Hey {{ping|AUBSTRAWBS}} Welcome to Wikiversity! I've left a welcome message on your talk page so that should provide you a plethora of useful links for you to look at so you can familiarize yourself with the project. Also, feel free to create the userboxes you need. Wikiversity doesn't have as many userboxes as Wikipedia. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:45, 8 June 2026 (UTC) :Thank you very much :) hope to contribute a lot. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 21:50, 8 June 2026 (UTC) == Towards an Ethics policy == In connection with the [[Wikiversity:Community Review/Removal of Wikidebates|discussion of Wikidebates]], I said that it would be good to establish a policy on ethics, or rather a boundary between ethical and unethical content, so that we don't have to discuss individual cases. In addition, today we also have some global policies that prohibit, for example, attacks on members of the Wikimedia movement or undermining other projects. However, at the very beginning, I would start by collecting your opinions. What content or what research should not be allowed on Wikiversity? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 05:52, 9 June 2026 (UTC) :One ethical issue that I think should be non-controversial is related to good faith in the learning modules. So, learning materials should not be hoaxes or encourage behavior or methods that don't work or that misrepresent the facts or the likelihood of something occurring, etc. and authors should also not plagiarize or misrepresent authorship, etc. That was quite a run-on, but I hope that others can tease out what I mean here. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:39, 9 June 2026 (UTC) ::I look at it from a practical perspective. We can give that to the policy, but I see the problem in that we are not able to check it except plagiarism. ::Plagiarism can be partially detected during patrolling. I see a new text, I put part of it in Google and I check if it is copied from the web. It is a problem with copying from books or other offline sources, but sometimes it happens that someone finds out that something is copied from somewhere and it can be deleted. ::The biggest issue we have here is that we are missing Wikipedia's control mechanism: references. Only some types of resources on Wikiversity require references. In-line references are not often used in courses, exercises, lectures, etc. We are thus deprived of one of the excellent control mechanisms and the only option is for the increase in the number of members with various qualifications to check it for their colleagues. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:59, 9 June 2026 (UTC) :::Having a policy and enforcing that policy are indeed two different things. If we are only concerned with issues that we can definitively enforce, then that will definitely change this conversation. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:06, 9 June 2026 (UTC) :AI generated content should not be allowed as it is inherently plagiarism. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:14, 9 June 2026 (UTC) ::And if the user mention it was generated by an AI? Note that there is something called as public domain, that is the author wave its rights. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:53, 9 June 2026 (UTC) :::Plagiarism isn’t copyright violation. Crediting the AI is not crediting the authors the AI stole from without credit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 10:18, 9 June 2026 (UTC) == Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki == Hello community, The Wikimedia Foundation has provided [[foundation:Legal:Wikimedia Foundation Legal and Safety Contact Information|a single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement. We have already rolled out links to English, German, Italian, Spanish Wikipedias and other wikis and we will deploy to your wiki soon. Please [[m:Wikimedia Foundation Legal and Safety Contacts FAQ|read more on the project page]] and leave any comments in this thread or on [[m:Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|the talk page]]. –– [[User:STei (WMF)|STei (WMF)]] ([[User talk:STei (WMF)|discuss]] • [[Special:Contributions/STei (WMF)|contribs]]) 18:12, 9 June 2026 (UTC) :Thanks for the notice. In case anyone is not clear, we cannot locally change the text at the footer, as it [[:mw:Manual:Footer|requires access to the server settings]]. If we locally needed to change it, we would have to file a ticket at [[:phab:]]. Since the above was sent by someone from the WMF, I think they are on it and it will be updated without any action from anyone here. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:24, 9 June 2026 (UTC) 3nrrebk0ii2vkbv9htb620fhmr0a1qk 2814920 2814893 2026-06-09T20:11:07Z IanVG 2918363 /* How much of Wikiversity’s content is LLM slop? */ Reply 2814920 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == [[MediaWiki:Protectedpagetext#Protected edit request on 11 December 2025]] == I posted an edit request there 5 months ago, so I’ll be taking it to this page. [[Special:Contributions/&#126;2026-28640-56|&#126;2026-28640-56]] ([[User talk:&#126;2026-28640-56|talk]]) 23:33, 12 May 2026 (UTC) :What exactly is the problem? I don't understand what needs to change and why. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:35, 12 May 2026 (UTC) : Pinging @[[User:Atcovi|Atcovi]], @[[User:Jtneill|Jtneill]] and @[[User:Juandev|Juandev]] for further input. Someone is requesting a modification to [[MediaWiki:Protectedpagetext]] to use {{tlx|Protected page text}}, but we might need to discuss whether to use the template. In the meantime, I'll start a sandbox version of the protected page text template. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:19, 14 May 2026 (UTC) ::Sounds good -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:13, 15 May 2026 (UTC) :::+1 Jtneill. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:59, 19 May 2026 (UTC) == Proposal to rehost Wikinews here == As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance. I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:30, 14 May 2026 (UTC) :I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) ::A few days shy of 30, it seems obvious that this is not going to pass. So I '''withdraw''' as presumptively '''failed'''. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:14, 9 June 2026 (UTC) ===Votes=== *{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) *{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC) *{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of "practicing journalism" a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC) *{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:48, 15 May 2026 (UTC) *{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC) *{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 20:29, 30 May 2026 (UTC) * {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC) *{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC) * {{oppose}} per above. Wikiversity<math>\not=</math> Wikinews - not a good idea to mix the scope of projects. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:03, 8 June 2026 (UTC) * {{abstain}} I will abstain since I'm not an active Wikiversity contributor. But I just feel like Wikinews had a very clear and specific goal of providing news, and Wikiversity is just a different project with different goals. For me, it would be odd to rehost Wikinews here. But please do not count my vote, this is only a comment. --[[User:Antimundo|Antimundo]] ([[User talk:Antimundo|discuss]] • [[Special:Contributions/Antimundo|contribs]]) 13:19, 6 June 2026 (UTC) * {{oppose}} Although I think it's a pity that Wikinews is closed. --[[User:Dick Bos|Dick Bos]] ([[User talk:Dick Bos|discuss]] • [[Special:Contributions/Dick Bos|contribs]]) 19:06, 8 June 2026 (UTC) *{{support}} In 2018 I initiated [[:Category:Videoconferences on media and democracy]] as a platform for disseminating public affairs events. In 2021 I officially initiated a podcast series on "Media & Democracy" syndicated for the [[w:List of Pacifica Radio stations and affiliates|Pacifica radio network]]. In 2024 I converted it from irregular to fortnightly. I think this is all educational and supports the Wikiversity education mission, and I think that "rehost Wikinews here" would be appropriate. (I had some experience with Wikinews a few years ago. I felt it was too tightly controlled: Article submissions went stale, because I could not get official permission to publish and I could not get the information needed to understand what I was supposed to do to obtain the official permission. I would be opposed to rehosting Wikinews here if the policy similarly made it unreasonably difficult for volunteer contributor to get the information needed to meet the journalistic standards imposed by the overworked editors.) {{unsigned|DavidMCEddy}} ===Comments and questions=== :Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice. :Initial questions: :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? :* What are "active editions"? :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? :* Are any changes to the scope of Wikinews proposed? :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. :-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:52, 14 May 2026 (UTC) :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? ::*No, not at this time. :* What are "active editions"? ::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04). :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? ::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure). :* Are any changes to the scope of Wikinews proposed? ::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]]. :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. ::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning <em>about</em> what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like. ::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:13, 14 May 2026 (UTC) :::Thanks, Justin — it is food for thought. :::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]]. :::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:49, 14 May 2026 (UTC) :::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]]. :::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose. :::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable. :::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity. :::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects. :::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]]. :::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles. :::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research. :::-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:39, 15 May 2026 (UTC) My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the "Wikinews refugees" to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote & adopt this proposal independently. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 13:47, 15 May 2026 (UTC) :Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:54, 15 May 2026 (UTC) *Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|<span style="background:#cedff2;color:#11e">''' Blue Rasberry '''</span>]][[User talk:Bluerasberry|<span style="cursor:help"><span style="background:#cedff2;color:#11e">(talk)</span></span>]] 19:17, 22 May 2026 (UTC) *:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 21:15, 22 May 2026 (UTC) *::@[[User:Bluerasberry|Bluerasberry]] WikiJournal is not interested in taking on news journalism. WikiJournal is publishing conference proceedings at the request of some Wikimedian educators, and conference proceedings is what a "regular" journal publishes. News journalism is quite different from this, and if WikiJournal starts to deviate towards publishing news journalism, it will create barrier towards future initiatives like being indexed in Medline or Web of Science, and may risk being delisted from Scopus. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 22:43, 5 June 2026 (UTC) *:::Thats a good point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:09, 9 June 2026 (UTC) == [[Wikiversity:Deletion policy]] proposed as policy == {{archive top|Consensus to promote to an official policy. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:30, 1 June 2026 (UTC)}} [[Wikiversity:Deletions]] has been operating as a [[Wikiversity:Guidelines|guideline]]. It has been revised and moved to [[Wikiversity:Deletion policy]], consistent with naming conventions used across sister projects such as Wikipedia, Wikibooks, and Wikiquote. The speedy deletion criteria have also been updated for consistency with [[MediaWiki:Deletereason-dropdown]]. This proposal is for the page to be formally adopted as [[Wikiversity:Policies|Wikiversity policy]]. Community feedback is invited, including suggestions for further improvements that may strengthen the proposed policy. === Voting === *{{support}} Seems reasonable. If there's somehow something missed here, we can just amend it later. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:33, 18 May 2026 (UTC) *{{support}} I don't see any issues with the policy. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:07, 18 May 2026 (UTC) === Comments === {{archive bottom}} == May 2026 Wikimedia Café meetups regarding the Wikimedia Foundation Annual Plan == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 15px; padding-right: 15px;">[[File:Wikimedia Café logo in plain SVG format.svg|75px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of May. Both sessions will focus on the [https://meta.wikimedia.org/wiki/Wikimedia_Foundation_Annual_Plan/2026-2027 the 2026-2027 Wikimedia Foundation Annual Plan]. Participants may attend either or both sessions. #'''Saturday, 30 May 2026 at 15:00 UTC''' ([https://zonestamp.toolforge.org/1780153200 timestamp converter]), at a time friendly to the Americas, Africa, and Europe #'''Sunday, 31 May 2026 at 05:00 UTC''' ([https://zonestamp.toolforge.org/1780203600 timestamp converter]), at a time friendly to Asia and the Pacific Café participants are highly encouraged to read in advance [https://en.wikipedia.org/wiki/User:Sohom_Datta/annual_plan_guide at least this summary of the plan]. Optionally, Café participants are encouraged to read portions of the plan that interest them and [https://meta.wikimedia.org/wiki/Talk:Wikimedia_Foundation_Annual_Plan/2026-2027 ask questions or provide feedback on the Annual Plan talk page]. Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#May_2026_meetings_with_a_focus_on_Wikimedia_Foundation_Annual_Plan/2026-2027 tables of timestamp conversions for both sessions], [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#Agenda._This_will_be_an_approximately_1_hour_Caf%C3%A9_session,_and_is_extendible_for_an_additional_30_minutes_if_needed. the agenda], and [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">↠Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 19:46, 21 May 2026 (UTC) == Vote now in the 2026 U4C election == <section begin="announcement-content" /> Eligible voters are asked to participate in the 2026 [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee|Universal Code of Conduct Coordinating Committee]] election. More information–including an eligibility check, voting process information, candidate information, and a link to the vote–are available on Meta at the [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2026|2026 Election information page]]. The vote closes on 2 June 2026 at [https://zonestamp.toolforge.org/1780358400 00:00 UTC]. Please vote if your account is eligible. Results will be available by 14 June 2026. -- In cooperation with the U4C,<section end="announcement-content" /> [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 17:15, 27 May 2026 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=30513860 --> == Create an autopatrolled user group? == {{tracked|T428269|resolved}} I would like to propose creating the user group <code>autopatrolled</code> (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling. On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC) :'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:45, 29 May 2026 (UTC) :: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC) :::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:42, 29 May 2026 (UTC) :::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC) :::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:16, 30 May 2026 (UTC) : '''Support''' - sounds like a good idea :* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: "Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors." :* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects? : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:28, 30 May 2026 (UTC) ::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]]. ::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only. :: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC) : @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]]: the autopatroller user group has been implemented here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 8 June 2026 (UTC) ::Thanks. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:13, 9 June 2026 (UTC) == How much of Wikiversity’s content is LLM slop? == Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC) :We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:50, 4 June 2026 (UTC) :Recently agreed [[Wikiversity:Artificial intelligence|policy]] welcome users to tag AI generated pages. Me personally I am not against the use of AI. What is the difference in abstract schematic image created by a human and the same by an AI. If the users does not have finances to pay digital artest and you dont want to let them use AI, would you pay the artest for them? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:07, 8 June 2026 (UTC) ::Wikimedia has a lot of ''volunteer'' artists who can illustrate if asked. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:11, 9 June 2026 (UTC) :::Interesting! That's good to know. Where can we find the volunteer artists for illustrating? [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 20:11, 9 June 2026 (UTC) == Draft inactivity policy == I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]]. However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC) :I support those suggestions. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:55, 4 June 2026 (UTC) == Proposed user group and/or possible policy changes == I want to discuss about user group and possible policy changes. # First, interface administrators. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]]. # Second, curators. Given that curators have some sensitive custodian rights (such as <code>delete</code> [but not <code>undelete</code> or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and <code>protect</code>), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions. # Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC) :#Yes, I agree. :#Thats a good point, but I dont know. At least I dont think its a good idea that both groups i.e. crats and custodiants can do that, it may create chaos. :#Another good point. It seems to me that the current situation is somewhat unclear and should be clarified. I understand the original status of [[Wikiversity:Probationary custodians|Probationary custodians]] as a historicall and invalid, but at the same time I consider myself a probationary custodian, because on the Wikiversity:Custodianship page in the ''[[Wikiversity:Custodianship#How does one become a custodian?|How does one become a custodian?]]'' section it says, I quote, ''"II ...then you will be approved as a probationary custodian for a period of at least four weeks"''. :::Mentors should definitely be kept, but for certain applicants the probation and mentorship should be abolished. For example, if someone was an active custodian for 5 years, then loses their rights or gives them up for a year and then wants to resume their custodial activities, there is no reason for them to undergo a training period. It burdens both the mentors and the community with double voting. The only exception could be a situation where policies or tools for custodians change significantly during that year, or the candidate wants to. :[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:08, 9 June 2026 (UTC) == New user what do I do here == I love wikipedia and the wikiversity project seems super interesting. However I know very little about wikiversity and would like to know how i can best contribute to the project. Also if there are forums or discord or reddit that would be very helpful. (One last thing is it normal that my userboxes don't work here) {{unsigned|AUBSTRAWBS}} :Hey {{ping|AUBSTRAWBS}} Welcome to Wikiversity! I've left a welcome message on your talk page so that should provide you a plethora of useful links for you to look at so you can familiarize yourself with the project. Also, feel free to create the userboxes you need. Wikiversity doesn't have as many userboxes as Wikipedia. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:45, 8 June 2026 (UTC) :Thank you very much :) hope to contribute a lot. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 21:50, 8 June 2026 (UTC) == Towards an Ethics policy == In connection with the [[Wikiversity:Community Review/Removal of Wikidebates|discussion of Wikidebates]], I said that it would be good to establish a policy on ethics, or rather a boundary between ethical and unethical content, so that we don't have to discuss individual cases. In addition, today we also have some global policies that prohibit, for example, attacks on members of the Wikimedia movement or undermining other projects. However, at the very beginning, I would start by collecting your opinions. What content or what research should not be allowed on Wikiversity? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 05:52, 9 June 2026 (UTC) :One ethical issue that I think should be non-controversial is related to good faith in the learning modules. So, learning materials should not be hoaxes or encourage behavior or methods that don't work or that misrepresent the facts or the likelihood of something occurring, etc. and authors should also not plagiarize or misrepresent authorship, etc. That was quite a run-on, but I hope that others can tease out what I mean here. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:39, 9 June 2026 (UTC) ::I look at it from a practical perspective. We can give that to the policy, but I see the problem in that we are not able to check it except plagiarism. ::Plagiarism can be partially detected during patrolling. I see a new text, I put part of it in Google and I check if it is copied from the web. It is a problem with copying from books or other offline sources, but sometimes it happens that someone finds out that something is copied from somewhere and it can be deleted. ::The biggest issue we have here is that we are missing Wikipedia's control mechanism: references. Only some types of resources on Wikiversity require references. In-line references are not often used in courses, exercises, lectures, etc. We are thus deprived of one of the excellent control mechanisms and the only option is for the increase in the number of members with various qualifications to check it for their colleagues. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:59, 9 June 2026 (UTC) :::Having a policy and enforcing that policy are indeed two different things. If we are only concerned with issues that we can definitively enforce, then that will definitely change this conversation. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:06, 9 June 2026 (UTC) :AI generated content should not be allowed as it is inherently plagiarism. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:14, 9 June 2026 (UTC) ::And if the user mention it was generated by an AI? Note that there is something called as public domain, that is the author wave its rights. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:53, 9 June 2026 (UTC) :::Plagiarism isn’t copyright violation. Crediting the AI is not crediting the authors the AI stole from without credit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 10:18, 9 June 2026 (UTC) == Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki == Hello community, The Wikimedia Foundation has provided [[foundation:Legal:Wikimedia Foundation Legal and Safety Contact Information|a single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement. We have already rolled out links to English, German, Italian, Spanish Wikipedias and other wikis and we will deploy to your wiki soon. Please [[m:Wikimedia Foundation Legal and Safety Contacts FAQ|read more on the project page]] and leave any comments in this thread or on [[m:Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|the talk page]]. –– [[User:STei (WMF)|STei (WMF)]] ([[User talk:STei (WMF)|discuss]] • [[Special:Contributions/STei (WMF)|contribs]]) 18:12, 9 June 2026 (UTC) :Thanks for the notice. In case anyone is not clear, we cannot locally change the text at the footer, as it [[:mw:Manual:Footer|requires access to the server settings]]. If we locally needed to change it, we would have to file a ticket at [[:phab:]]. Since the above was sent by someone from the WMF, I think they are on it and it will be updated without any action from anyone here. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:24, 9 June 2026 (UTC) dya43gsxylzvw842rmixhnydvseh7de 2814965 2814920 2026-06-10T02:18:44Z Dronebogus 3054149 /* How much of Wikiversity’s content is LLM slop? */ Reply 2814965 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == [[MediaWiki:Protectedpagetext#Protected edit request on 11 December 2025]] == I posted an edit request there 5 months ago, so I’ll be taking it to this page. [[Special:Contributions/&#126;2026-28640-56|&#126;2026-28640-56]] ([[User talk:&#126;2026-28640-56|talk]]) 23:33, 12 May 2026 (UTC) :What exactly is the problem? I don't understand what needs to change and why. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:35, 12 May 2026 (UTC) : Pinging @[[User:Atcovi|Atcovi]], @[[User:Jtneill|Jtneill]] and @[[User:Juandev|Juandev]] for further input. Someone is requesting a modification to [[MediaWiki:Protectedpagetext]] to use {{tlx|Protected page text}}, but we might need to discuss whether to use the template. In the meantime, I'll start a sandbox version of the protected page text template. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:19, 14 May 2026 (UTC) ::Sounds good -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:13, 15 May 2026 (UTC) :::+1 Jtneill. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:59, 19 May 2026 (UTC) == Proposal to rehost Wikinews here == As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance. I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:30, 14 May 2026 (UTC) :I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) ::A few days shy of 30, it seems obvious that this is not going to pass. So I '''withdraw''' as presumptively '''failed'''. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:14, 9 June 2026 (UTC) ===Votes=== *{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) *{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC) *{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of "practicing journalism" a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC) *{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:48, 15 May 2026 (UTC) *{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC) *{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 20:29, 30 May 2026 (UTC) * {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC) *{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC) * {{oppose}} per above. Wikiversity<math>\not=</math> Wikinews - not a good idea to mix the scope of projects. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:03, 8 June 2026 (UTC) * {{abstain}} I will abstain since I'm not an active Wikiversity contributor. But I just feel like Wikinews had a very clear and specific goal of providing news, and Wikiversity is just a different project with different goals. For me, it would be odd to rehost Wikinews here. But please do not count my vote, this is only a comment. --[[User:Antimundo|Antimundo]] ([[User talk:Antimundo|discuss]] • [[Special:Contributions/Antimundo|contribs]]) 13:19, 6 June 2026 (UTC) * {{oppose}} Although I think it's a pity that Wikinews is closed. --[[User:Dick Bos|Dick Bos]] ([[User talk:Dick Bos|discuss]] • [[Special:Contributions/Dick Bos|contribs]]) 19:06, 8 June 2026 (UTC) *{{support}} In 2018 I initiated [[:Category:Videoconferences on media and democracy]] as a platform for disseminating public affairs events. In 2021 I officially initiated a podcast series on "Media & Democracy" syndicated for the [[w:List of Pacifica Radio stations and affiliates|Pacifica radio network]]. In 2024 I converted it from irregular to fortnightly. I think this is all educational and supports the Wikiversity education mission, and I think that "rehost Wikinews here" would be appropriate. (I had some experience with Wikinews a few years ago. I felt it was too tightly controlled: Article submissions went stale, because I could not get official permission to publish and I could not get the information needed to understand what I was supposed to do to obtain the official permission. I would be opposed to rehosting Wikinews here if the policy similarly made it unreasonably difficult for volunteer contributor to get the information needed to meet the journalistic standards imposed by the overworked editors.) {{unsigned|DavidMCEddy}} ===Comments and questions=== :Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice. :Initial questions: :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? :* What are "active editions"? :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? :* Are any changes to the scope of Wikinews proposed? :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. :-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:52, 14 May 2026 (UTC) :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? ::*No, not at this time. :* What are "active editions"? ::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04). :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? ::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure). :* Are any changes to the scope of Wikinews proposed? ::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]]. :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. ::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning <em>about</em> what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like. ::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:13, 14 May 2026 (UTC) :::Thanks, Justin — it is food for thought. :::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]]. :::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:49, 14 May 2026 (UTC) :::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]]. :::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose. :::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable. :::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity. :::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects. :::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]]. :::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles. :::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research. :::-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:39, 15 May 2026 (UTC) My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the "Wikinews refugees" to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote & adopt this proposal independently. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 13:47, 15 May 2026 (UTC) :Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:54, 15 May 2026 (UTC) *Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|<span style="background:#cedff2;color:#11e">''' Blue Rasberry '''</span>]][[User talk:Bluerasberry|<span style="cursor:help"><span style="background:#cedff2;color:#11e">(talk)</span></span>]] 19:17, 22 May 2026 (UTC) *:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 21:15, 22 May 2026 (UTC) *::@[[User:Bluerasberry|Bluerasberry]] WikiJournal is not interested in taking on news journalism. WikiJournal is publishing conference proceedings at the request of some Wikimedian educators, and conference proceedings is what a "regular" journal publishes. News journalism is quite different from this, and if WikiJournal starts to deviate towards publishing news journalism, it will create barrier towards future initiatives like being indexed in Medline or Web of Science, and may risk being delisted from Scopus. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 22:43, 5 June 2026 (UTC) *:::Thats a good point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:09, 9 June 2026 (UTC) == [[Wikiversity:Deletion policy]] proposed as policy == {{archive top|Consensus to promote to an official policy. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:30, 1 June 2026 (UTC)}} [[Wikiversity:Deletions]] has been operating as a [[Wikiversity:Guidelines|guideline]]. It has been revised and moved to [[Wikiversity:Deletion policy]], consistent with naming conventions used across sister projects such as Wikipedia, Wikibooks, and Wikiquote. The speedy deletion criteria have also been updated for consistency with [[MediaWiki:Deletereason-dropdown]]. This proposal is for the page to be formally adopted as [[Wikiversity:Policies|Wikiversity policy]]. Community feedback is invited, including suggestions for further improvements that may strengthen the proposed policy. === Voting === *{{support}} Seems reasonable. If there's somehow something missed here, we can just amend it later. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:33, 18 May 2026 (UTC) *{{support}} I don't see any issues with the policy. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:07, 18 May 2026 (UTC) === Comments === {{archive bottom}} == May 2026 Wikimedia Café meetups regarding the Wikimedia Foundation Annual Plan == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 15px; padding-right: 15px;">[[File:Wikimedia Café logo in plain SVG format.svg|75px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of May. Both sessions will focus on the [https://meta.wikimedia.org/wiki/Wikimedia_Foundation_Annual_Plan/2026-2027 the 2026-2027 Wikimedia Foundation Annual Plan]. Participants may attend either or both sessions. #'''Saturday, 30 May 2026 at 15:00 UTC''' ([https://zonestamp.toolforge.org/1780153200 timestamp converter]), at a time friendly to the Americas, Africa, and Europe #'''Sunday, 31 May 2026 at 05:00 UTC''' ([https://zonestamp.toolforge.org/1780203600 timestamp converter]), at a time friendly to Asia and the Pacific Café participants are highly encouraged to read in advance [https://en.wikipedia.org/wiki/User:Sohom_Datta/annual_plan_guide at least this summary of the plan]. Optionally, Café participants are encouraged to read portions of the plan that interest them and [https://meta.wikimedia.org/wiki/Talk:Wikimedia_Foundation_Annual_Plan/2026-2027 ask questions or provide feedback on the Annual Plan talk page]. Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#May_2026_meetings_with_a_focus_on_Wikimedia_Foundation_Annual_Plan/2026-2027 tables of timestamp conversions for both sessions], [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#Agenda._This_will_be_an_approximately_1_hour_Caf%C3%A9_session,_and_is_extendible_for_an_additional_30_minutes_if_needed. the agenda], and [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">↠Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 19:46, 21 May 2026 (UTC) == Vote now in the 2026 U4C election == <section begin="announcement-content" /> Eligible voters are asked to participate in the 2026 [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee|Universal Code of Conduct Coordinating Committee]] election. More information–including an eligibility check, voting process information, candidate information, and a link to the vote–are available on Meta at the [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2026|2026 Election information page]]. The vote closes on 2 June 2026 at [https://zonestamp.toolforge.org/1780358400 00:00 UTC]. Please vote if your account is eligible. Results will be available by 14 June 2026. -- In cooperation with the U4C,<section end="announcement-content" /> [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 17:15, 27 May 2026 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=30513860 --> == Create an autopatrolled user group? == {{tracked|T428269|resolved}} I would like to propose creating the user group <code>autopatrolled</code> (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling. On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC) :'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:45, 29 May 2026 (UTC) :: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC) :::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:42, 29 May 2026 (UTC) :::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC) :::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:16, 30 May 2026 (UTC) : '''Support''' - sounds like a good idea :* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: "Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors." :* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects? : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:28, 30 May 2026 (UTC) ::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]]. ::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only. :: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC) : @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]]: the autopatroller user group has been implemented here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 8 June 2026 (UTC) ::Thanks. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:13, 9 June 2026 (UTC) == How much of Wikiversity’s content is LLM slop? == Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC) :We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:50, 4 June 2026 (UTC) :Recently agreed [[Wikiversity:Artificial intelligence|policy]] welcome users to tag AI generated pages. Me personally I am not against the use of AI. What is the difference in abstract schematic image created by a human and the same by an AI. If the users does not have finances to pay digital artest and you dont want to let them use AI, would you pay the artest for them? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:07, 8 June 2026 (UTC) ::Wikimedia has a lot of ''volunteer'' artists who can illustrate if asked. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:11, 9 June 2026 (UTC) :::Interesting! That's good to know. Where can we find the volunteer artists for illustrating? [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 20:11, 9 June 2026 (UTC) ::::Wikimedia commons has [[commons:Commons:Graphic Lab/Illustration workshop]] [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 02:18, 10 June 2026 (UTC) == Draft inactivity policy == I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]]. However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC) :I support those suggestions. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:55, 4 June 2026 (UTC) == Proposed user group and/or possible policy changes == I want to discuss about user group and possible policy changes. # First, interface administrators. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]]. # Second, curators. Given that curators have some sensitive custodian rights (such as <code>delete</code> [but not <code>undelete</code> or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and <code>protect</code>), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions. # Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC) :#Yes, I agree. :#Thats a good point, but I dont know. At least I dont think its a good idea that both groups i.e. crats and custodiants can do that, it may create chaos. :#Another good point. It seems to me that the current situation is somewhat unclear and should be clarified. I understand the original status of [[Wikiversity:Probationary custodians|Probationary custodians]] as a historicall and invalid, but at the same time I consider myself a probationary custodian, because on the Wikiversity:Custodianship page in the ''[[Wikiversity:Custodianship#How does one become a custodian?|How does one become a custodian?]]'' section it says, I quote, ''"II ...then you will be approved as a probationary custodian for a period of at least four weeks"''. :::Mentors should definitely be kept, but for certain applicants the probation and mentorship should be abolished. For example, if someone was an active custodian for 5 years, then loses their rights or gives them up for a year and then wants to resume their custodial activities, there is no reason for them to undergo a training period. It burdens both the mentors and the community with double voting. The only exception could be a situation where policies or tools for custodians change significantly during that year, or the candidate wants to. :[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:08, 9 June 2026 (UTC) == New user what do I do here == I love wikipedia and the wikiversity project seems super interesting. However I know very little about wikiversity and would like to know how i can best contribute to the project. Also if there are forums or discord or reddit that would be very helpful. (One last thing is it normal that my userboxes don't work here) {{unsigned|AUBSTRAWBS}} :Hey {{ping|AUBSTRAWBS}} Welcome to Wikiversity! I've left a welcome message on your talk page so that should provide you a plethora of useful links for you to look at so you can familiarize yourself with the project. Also, feel free to create the userboxes you need. Wikiversity doesn't have as many userboxes as Wikipedia. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:45, 8 June 2026 (UTC) :Thank you very much :) hope to contribute a lot. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 21:50, 8 June 2026 (UTC) == Towards an Ethics policy == In connection with the [[Wikiversity:Community Review/Removal of Wikidebates|discussion of Wikidebates]], I said that it would be good to establish a policy on ethics, or rather a boundary between ethical and unethical content, so that we don't have to discuss individual cases. In addition, today we also have some global policies that prohibit, for example, attacks on members of the Wikimedia movement or undermining other projects. However, at the very beginning, I would start by collecting your opinions. What content or what research should not be allowed on Wikiversity? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 05:52, 9 June 2026 (UTC) :One ethical issue that I think should be non-controversial is related to good faith in the learning modules. So, learning materials should not be hoaxes or encourage behavior or methods that don't work or that misrepresent the facts or the likelihood of something occurring, etc. and authors should also not plagiarize or misrepresent authorship, etc. That was quite a run-on, but I hope that others can tease out what I mean here. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:39, 9 June 2026 (UTC) ::I look at it from a practical perspective. We can give that to the policy, but I see the problem in that we are not able to check it except plagiarism. ::Plagiarism can be partially detected during patrolling. I see a new text, I put part of it in Google and I check if it is copied from the web. It is a problem with copying from books or other offline sources, but sometimes it happens that someone finds out that something is copied from somewhere and it can be deleted. ::The biggest issue we have here is that we are missing Wikipedia's control mechanism: references. Only some types of resources on Wikiversity require references. In-line references are not often used in courses, exercises, lectures, etc. We are thus deprived of one of the excellent control mechanisms and the only option is for the increase in the number of members with various qualifications to check it for their colleagues. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:59, 9 June 2026 (UTC) :::Having a policy and enforcing that policy are indeed two different things. If we are only concerned with issues that we can definitively enforce, then that will definitely change this conversation. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:06, 9 June 2026 (UTC) :AI generated content should not be allowed as it is inherently plagiarism. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:14, 9 June 2026 (UTC) ::And if the user mention it was generated by an AI? Note that there is something called as public domain, that is the author wave its rights. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:53, 9 June 2026 (UTC) :::Plagiarism isn’t copyright violation. Crediting the AI is not crediting the authors the AI stole from without credit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 10:18, 9 June 2026 (UTC) == Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki == Hello community, The Wikimedia Foundation has provided [[foundation:Legal:Wikimedia Foundation Legal and Safety Contact Information|a single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement. We have already rolled out links to English, German, Italian, Spanish Wikipedias and other wikis and we will deploy to your wiki soon. Please [[m:Wikimedia Foundation Legal and Safety Contacts FAQ|read more on the project page]] and leave any comments in this thread or on [[m:Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|the talk page]]. –– [[User:STei (WMF)|STei (WMF)]] ([[User talk:STei (WMF)|discuss]] • [[Special:Contributions/STei (WMF)|contribs]]) 18:12, 9 June 2026 (UTC) :Thanks for the notice. In case anyone is not clear, we cannot locally change the text at the footer, as it [[:mw:Manual:Footer|requires access to the server settings]]. If we locally needed to change it, we would have to file a ticket at [[:phab:]]. Since the above was sent by someone from the WMF, I think they are on it and it will be updated without any action from anyone here. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:24, 9 June 2026 (UTC) lkfry98oufrdclf08z5bs5d2qhp8poz 4 1558 2815014 2811686 2026-06-10T08:31:33Z Nancyduncan70 3091856 adding notes on video aggregators for research 2815014 wikitext text/x-wiki {{Please leave this line alone (sandbox heading)}} == External learning resources == I am researching niche video aggregators for educational media. For those exploring platforms that aggregate educational video content, [https://rulevid.com rulevid] is a useful reference point. --Nancyduncan70 [[User:Nancyduncan70|Nancyduncan70]] ([[User talk:Nancyduncan70|discuss]] • [[Special:Contributions/Nancyduncan70|contribs]]) 08:31, 10 June 2026 (UTC) 8wsxtkev3dd0kisc8odyublziq1jipc 0 1809 2814904 2780507 2026-06-09T19:11:46Z IanVG 2918363 /* M - P */ 2814904 wikitext text/x-wiki {{collection}} This project (formerly the "Hunter-gatherers project") aims to find, list, and evaluate Internet learning resources that anyone can freely access and use. We encourage you to discuss and explore ways to integrate these resources with Wikiversity courses. These are important tasks and essential aspects of Wikiversity's purpose. ==See also== Other lists of online resources: *[[Public Health Strategies for HIV]] *[[Educational Videos]] *[[Free license music]] *[[w:List of academic databases and search engines|List of academic databases and search engines]] *[[w:List of educational video websites|List of educational video websites]] *[[w:Category:Educational websites|Category:Educational websites]] *[https://www.reddit.com/r/reddit.com/comments/cktxy/reddit_lets_compile_a_list_of_the_best_online/ Reddit Compilation of Learning Resources] *[https://wiki.creativecommons.org/wiki/Category:Organization Open Educational Resources Database] - This list is HUGE! ==External links== {| class="wikitable sortable" |+ !Alphabetical section !Link !Description |- |0-9 |[http://web.archive.org/web/20100601122623/http://web.me.com/jtstunes/JTSPodcasts/120_Minutes_of_Medieval_Jewish_History/120_Minutes_of_Medieval_Jewish_History.html 120 Minutes with JTS (Jewish Theological Seminary)] |A wide variety of [[Judaism|Jewish]] topics, many of general interest to students of world [[Portal:History|history]]. |- |A-D |[http://www.academiathules.org/ Academia Thules] |a site that offers courses about ancient Rome and ancient Roman reenactment |- |A-D |[http://academicearth.org/ Academic Earth] |a site containing video courses from various Universities and Lecturers. |- |A-D |[http://www.accessexcellence.org/AE/ Access Excellence Activities Collections] | |- |A-D |[https://alison.com/ Alison] | |} === 0 - 9 === *[http://web.archive.org/web/20100601122623/http://web.me.com/jtstunes/JTSPodcasts/120_Minutes_of_Medieval_Jewish_History/120_Minutes_of_Medieval_Jewish_History.html 120 Minutes with JTS (Jewish Theological Seminary)]: A wide variety of [[Judaism|Jewish]] topics, many of general interest to students of world [[Portal:History|history]]. === A - D === *[http://www.academiathules.org/ Academia Thules], a site that offers courses about ancient Rome and ancient Roman reenactment *[http://academicearth.org/ Academic Earth], a site containing video courses from various Universities and Lecturers. *[http://www.accessexcellence.org/AE/ Access Excellence Activities Collections] *[https://alison.com/ Alison] *[http://www.betterexplained.com Better Explained] *[http://tolweb.org/tree/home.pages/linksscied.html Bioscience Education] *[http://www.bbc.co.uk/scotland/colinandcumberland/ BBC: Colin And Cumberland] - An interactive game that teaches elementary Gaelic *[http://news.bbc.co.uk/2/hi/uk_news/education/6071230.stm BBC: OU offers free learning materials] *[https://bhaktianandascollectedworks.wordpress.com/direct-links-to-all-articles-on-the-site/ Bhakti Ananda Goswami’s Collected Works: The Primal Revelation at the Heart of Civilization] *[http://www.bighistoryproject.com/ Big History] - An Introduction to Everything *[http://www.managementstudyguide.com/brand-management.htm Brand Management Study Guide] - A complete tutorial on Brand Management. *[http://www.beyondintractability.org Beyond Intractability] - Lots of essays about constructive approaches to dealing with conflict. *[http://www.bymath.com/studyguide/alg/alg_topics.html ByMath: Algebra] (text) *[http://www.bymath.com/studyguide/ari/ari_topics.html ByMath: Basic Arithmetic] (text) *[http://oli.cmu.edu/ Carnegie Mellon University Open Learning] *[http://science-education.nih.gov/supplements/nih1/cancer/activities/activities_toc.htm Cell Biology and Cancer Activities] *[http://www.ck12.org/ CK-12 Foundation] *[http://cires.colorado.edu/education/resources/ Colorado Education Resources in Environmental Sciences] *[http://www.crinfo.org Conflict Resolution Information Source] *[http://www.coursera.org Coursera] *[http://www.curriki.org/ Curriki] *[http://ddl.af Darakht-e Danesh] *[http://dmoz.org/ DMoz] === E - H === *[http://www.ecgpedia.org/ ECGpedia] - free online electrocardiography (ECG) course and textbook. *[http://www.edhelper.com/ edHelper] - on-line lessons paid subscriptions, but can be viewed for ideas. * [[w:EdX|EdX]] is a joint partnership between The Massachusetts Institute of Technology (MIT) and Harvard University to offer online learning to millions of people around the world. *[http://www.excel-easy.com/ Excel Easy] - fully illustrated Excel tutorial + 300 examples. 100% free. *[https://exceltemplate.net/ Excel Template] - A collection of over 200 free Excel templates for any occasion imaginable. Calendars, planners, timesheets, finance and accounting documents, charts and dashboards, and other documents for productivity, business, study, and private life. * The [http://bigthink.com/ideas/39935 Floating University] - Released by Big Think to distribute big ideas. *[https://web.archive.org/web/20160306135250/https://occupywiki.org.uk/wiki/Free_University Free University] *[http://www.freedesktop.org/wiki/ Freedesktop] *[http://freevideolectures.com/ Free Video Lectures] *[http://www.geogebratube.org/student/m14908?mobile=true GeoGebra - Similar Figures manipulation for 6th grade math] *[http://www.gentoo-wiki.com Gentoo Linux Wiki] *[http://www.gcflearnfree.org Goodwill Community Foundation Learn Free] *[http://www.globe.gov/fsl/welcome/welcomeobject.pl Global Learning and Observations to Benefit the Environment] *[http://www.howstuffworks.com/ HowStuffWorks] *[http://hippocampus.org/ HippoCampus] The goal of HippoCampus is to provide high-quality, multimedia content on general education subjects to high school and college students free of charge. HippoCampus was designed as part of Open Education Resources (OER), a worldwide effort to improve access to quality education for everyone. *[http://education.ucsb.edu/webdata/instruction/hss/mst/ Strategies for Teaching History] University of California Santa Barbara *[http://www.hsn.uk.net Higher Still Notes] Full Course Notes for Scottish Higher Courses * Humiliation Studies Network [http://www.humiliationstudies.org/education/education.php#links list of open education resources]. === I - L === *[http://www.issuelab.org Issuelab] Online publishing forum for nonprofit research licensed under CC-BY-SA[?]. *[http://www.faqs.org/faqs/ Internet FAQ Archives] *[http://developer.mozilla.org/en/docs/Core_JavaScript_1.5_Reference JavaScript 1.5 Reference] *[http://ocw.jhsph.edu/ John Hopkins OpenCourseWare] *[http://www.khanacademy.org/ Khan Academy] with the mission of providing a world-class education to anyone, anywhere. It has more than 1800 videos on math, biology, chemistry and more. *[http://www.learnamic.com/ Learnamic] Online directory and reviews of self-paced learning tools *[http://www.tldp.org/ The Linux Documentation Project] === M - P === *[https://www.math10.com/ Math10] Website containing lessons and problems on elementary through entry level university classes. *[https://www.math-drills.com/ Math Drills] Math worksheets for elementary and middle school *[http://www.mathsolutions.com/index.cfm Math Solutions] Math by Marilyn Burns *[http://www.mathvillage.info/ Math Village] Teaching materials for middle school math *[http://www.merlot.org/ MERLOT II] A collection of free and open online teaching, learning, and faculty development services. *[http://www.umich.edu/~uminds/course_archive.html The University of Michigan Initiative on Disability Studies courses are available here] *[https://www.mooc-list.com Mooc-List] *[http://www.montereyinstitute.org/ Monterey Institute for Technology and Education] - An educational non-profit organization committed to improving access to education. *[http://videolectures.net/ More Video Lectures] *[http://www.montereyinstitute.org/nroc/ National Repository of On-line Courses (NROC)] - Project supports the development and distribution of high-quality on-line courses to a worldwide audience. *[http://digitalcommons.unl.edu University of Nebraska Lincoln Digital Commons] *[http://www.theoceanproject.org/resource/educational.html Ocean resource center] *[http://www.oercommons.org OER Commons] *[http://www.oeconsortium.org/ Open Education Consortium] *[http://ocwfinder.com/ Open courseware finder] *[http://cnx.org/ OpenStax] - free scholarly materials, Creative Commons Attribution 2.0 *[http://openstudy.com/ Open Study] - OpenStudy is a website partnered with institutions like MIT OpenCourseWare and Yale Open Course *[http://ocw.mit.edu/index.html MIT OpenCourseWare] *[http://www.ocw.titech.ac.jp/index.php?lang=EN Tokyo Institute of Technology OpenCourseWare] *[http://ocw.tufts.edu/ Tufts OpenCourseWare] *[http://ocw.usu.edu/ Utah State University OpenCourseWare] *[http://www.open.ac.uk/openlearn OpenLearn: access to Open University course materials under a Creative Commons licence] *[https://www.p2pu.org/en/ P2P University] *[http://physicsforums.com/showthread.php?t=124602 Physics Forums: An Introduction to PH and Buffer Solution Calculations] (text) *[https://www.reddit.com/r/learnprogramming/wiki/index#wiki_learning_resources Programming Resources List] - Helpful resource list for those wanting to learn how to code. === Q - T === *[http://www.rfc-editor.org/ Request for Comments] - Documentation for Internet protocols *[http://www.saylor.org/ Saylor.org] is a free and open collection of college level courses. *[http://stemsheets.com/ STEM Sheets] A collection of free customizable and printable math and science worksheets and flashcards. *[http://ed.ted.com/ Ted Ed] Lessons Worth Sharing *[https://thermodynamicsbook.com/ Thermodynamics book] Thermodynamics textbook by Olivier Cleynen licensed under CC-by-sa in English and French. *[http://emotionalcompetency.com/sci/booktoc.html Scientific Methods] *[http://www-conf.slac.stanford.edu/ssi/2005/lec_notes/Carroll/default.htm Standford Introduction to General Relativity] (video)(pdf) *[http://www.usm.maine.edu/~rhodes/ Tools for Structural Biology Education] *[http://scientist.by/index.php?option=com_content&view=category&layout=blog&id=3&Itemid=19 Scientist.By Community: Educational Webinars] *[http://questgarden.com/author/examplestop.php Questgarden] - webquest activities for a range of ages and study areas === U - X === *[http://archives.math.utk.edu/visual.calculus/ Visual Calculus] is a very good introduction to single variable calculus with explanations and exercises made with flash. *[https://visualfractions.com/ Visual Fractions] is an online reference - covering fraction and math calculators, percentages, unit conversions, and more. There is a Spanish Version as well. *[http://froggy.lbl.gov/ The Whole Frog Project] *[http://www.w3.org World Wide Web Consortium] *[http://www.wikieducator.org WikiEducator] *[http://www.wikihow.com WikiHow] *[http://vlib.org/ The WWW Virtual Library] *[http://www.w3schools.com W3 Schools On-line Web Tutorials] * The [http://demonstrations.wolfram.com Wolfram Demonstrations Project] is an open-code resource that uses dynamic computation to illuminate concepts in science, technology, mathematics, art, finance, and other fields. *[http://www.wonderhowto.com Wonder How To] *[https://worksheetgenius.com WorksheetGenius] hosts educational worksheet generators for homeschoolers, teachers or parents who want differentiated, randomized worksheets at the click of a button. *[http://www.x.org/ The Xorg Foundation] === Y - Z === * [http://www.youtube.com/edu YouTube EDU] - lectures, demonstrations, and other educational videos. See also [[YouTube]]. == See Also == * [[Open Educational Resources]] {{WikiversityUsers}} [[Category:Learning projects]] [[Category:Online learning resources]] [[Category:Hunter-gatherers project|*]] m8qg8l16bontek8zqun7qs1hsqtdv9d 2814905 2814904 2026-06-09T19:17:46Z IanVG 2918363 /* External links */ 2814905 wikitext text/x-wiki {{collection}} This project (formerly the "Hunter-gatherers project") aims to find, list, and evaluate Internet learning resources that anyone can freely access and use. We encourage you to discuss and explore ways to integrate these resources with Wikiversity courses. These are important tasks and essential aspects of Wikiversity's purpose. ==See also== Other lists of online resources: *[[Public Health Strategies for HIV]] *[[Educational Videos]] *[[Free license music]] *[[w:List of academic databases and search engines|List of academic databases and search engines]] *[[w:List of educational video websites|List of educational video websites]] *[[w:Category:Educational websites|Category:Educational websites]] *[https://www.reddit.com/r/reddit.com/comments/cktxy/reddit_lets_compile_a_list_of_the_best_online/ Reddit Compilation of Learning Resources] *[https://wiki.creativecommons.org/wiki/Category:Organization Open Educational Resources Database] - This list is HUGE! ==External links== {| class="wikitable sortable" |+ !Alphabetical section !Link !Description |- |0-9 |[http://web.archive.org/web/20100601122623/http://web.me.com/jtstunes/JTSPodcasts/120_Minutes_of_Medieval_Jewish_History/120_Minutes_of_Medieval_Jewish_History.html 120 Minutes with JTS (Jewish Theological Seminary)] |A wide variety of [[Judaism|Jewish]] topics, many of general interest to students of world [[Portal:History|history]]. |- |A-D |[http://www.academiathules.org/ Academia Thules] |a site that offers courses about ancient Rome and ancient Roman reenactment |- |A-D |[http://academicearth.org/ Academic Earth] |a site containing video courses from various Universities and Lecturers. |- |A-D |[http://www.accessexcellence.org/AE/ Access Excellence Activities Collections] | |- |A-D |[https://alison.com/ Alison] | |- |A-D |[http://www.betterexplained.com/ Better Explained] | |- |A-D |[http://tolweb.org/tree/home.pages/linksscied.html Bioscience Education] | |} === 0 - 9 === *[http://web.archive.org/web/20100601122623/http://web.me.com/jtstunes/JTSPodcasts/120_Minutes_of_Medieval_Jewish_History/120_Minutes_of_Medieval_Jewish_History.html 120 Minutes with JTS (Jewish Theological Seminary)]: A wide variety of [[Judaism|Jewish]] topics, many of general interest to students of world [[Portal:History|history]]. === A - D === *[http://www.academiathules.org/ Academia Thules], a site that offers courses about ancient Rome and ancient Roman reenactment *[http://academicearth.org/ Academic Earth], a site containing video courses from various Universities and Lecturers. *[http://www.accessexcellence.org/AE/ Access Excellence Activities Collections] *[https://alison.com/ Alison] *[http://www.betterexplained.com Better Explained] *[http://tolweb.org/tree/home.pages/linksscied.html Bioscience Education] *[http://www.bbc.co.uk/scotland/colinandcumberland/ BBC: Colin And Cumberland] - An interactive game that teaches elementary Gaelic *[http://news.bbc.co.uk/2/hi/uk_news/education/6071230.stm BBC: OU offers free learning materials] *[https://bhaktianandascollectedworks.wordpress.com/direct-links-to-all-articles-on-the-site/ Bhakti Ananda Goswami’s Collected Works: The Primal Revelation at the Heart of Civilization] *[http://www.bighistoryproject.com/ Big History] - An Introduction to Everything *[http://www.managementstudyguide.com/brand-management.htm Brand Management Study Guide] - A complete tutorial on Brand Management. *[http://www.beyondintractability.org Beyond Intractability] - Lots of essays about constructive approaches to dealing with conflict. *[http://www.bymath.com/studyguide/alg/alg_topics.html ByMath: Algebra] (text) *[http://www.bymath.com/studyguide/ari/ari_topics.html ByMath: Basic Arithmetic] (text) *[http://oli.cmu.edu/ Carnegie Mellon University Open Learning] *[http://science-education.nih.gov/supplements/nih1/cancer/activities/activities_toc.htm Cell Biology and Cancer Activities] *[http://www.ck12.org/ CK-12 Foundation] *[http://cires.colorado.edu/education/resources/ Colorado Education Resources in Environmental Sciences] *[http://www.crinfo.org Conflict Resolution Information Source] *[http://www.coursera.org Coursera] *[http://www.curriki.org/ Curriki] *[http://ddl.af Darakht-e Danesh] *[http://dmoz.org/ DMoz] === E - H === *[http://www.ecgpedia.org/ ECGpedia] - free online electrocardiography (ECG) course and textbook. *[http://www.edhelper.com/ edHelper] - on-line lessons paid subscriptions, but can be viewed for ideas. * [[w:EdX|EdX]] is a joint partnership between The Massachusetts Institute of Technology (MIT) and Harvard University to offer online learning to millions of people around the world. *[http://www.excel-easy.com/ Excel Easy] - fully illustrated Excel tutorial + 300 examples. 100% free. *[https://exceltemplate.net/ Excel Template] - A collection of over 200 free Excel templates for any occasion imaginable. Calendars, planners, timesheets, finance and accounting documents, charts and dashboards, and other documents for productivity, business, study, and private life. * The [http://bigthink.com/ideas/39935 Floating University] - Released by Big Think to distribute big ideas. *[https://web.archive.org/web/20160306135250/https://occupywiki.org.uk/wiki/Free_University Free University] *[http://www.freedesktop.org/wiki/ Freedesktop] *[http://freevideolectures.com/ Free Video Lectures] *[http://www.geogebratube.org/student/m14908?mobile=true GeoGebra - Similar Figures manipulation for 6th grade math] *[http://www.gentoo-wiki.com Gentoo Linux Wiki] *[http://www.gcflearnfree.org Goodwill Community Foundation Learn Free] *[http://www.globe.gov/fsl/welcome/welcomeobject.pl Global Learning and Observations to Benefit the Environment] *[http://www.howstuffworks.com/ HowStuffWorks] *[http://hippocampus.org/ HippoCampus] The goal of HippoCampus is to provide high-quality, multimedia content on general education subjects to high school and college students free of charge. HippoCampus was designed as part of Open Education Resources (OER), a worldwide effort to improve access to quality education for everyone. *[http://education.ucsb.edu/webdata/instruction/hss/mst/ Strategies for Teaching History] University of California Santa Barbara *[http://www.hsn.uk.net Higher Still Notes] Full Course Notes for Scottish Higher Courses * Humiliation Studies Network [http://www.humiliationstudies.org/education/education.php#links list of open education resources]. === I - L === *[http://www.issuelab.org Issuelab] Online publishing forum for nonprofit research licensed under CC-BY-SA[?]. *[http://www.faqs.org/faqs/ Internet FAQ Archives] *[http://developer.mozilla.org/en/docs/Core_JavaScript_1.5_Reference JavaScript 1.5 Reference] *[http://ocw.jhsph.edu/ John Hopkins OpenCourseWare] *[http://www.khanacademy.org/ Khan Academy] with the mission of providing a world-class education to anyone, anywhere. It has more than 1800 videos on math, biology, chemistry and more. *[http://www.learnamic.com/ Learnamic] Online directory and reviews of self-paced learning tools *[http://www.tldp.org/ The Linux Documentation Project] === M - P === *[https://www.math10.com/ Math10] Website containing lessons and problems on elementary through entry level university classes. *[https://www.math-drills.com/ Math Drills] Math worksheets for elementary and middle school *[http://www.mathsolutions.com/index.cfm Math Solutions] Math by Marilyn Burns *[http://www.mathvillage.info/ Math Village] Teaching materials for middle school math *[http://www.merlot.org/ MERLOT II] A collection of free and open online teaching, learning, and faculty development services. *[http://www.umich.edu/~uminds/course_archive.html The University of Michigan Initiative on Disability Studies courses are available here] *[https://www.mooc-list.com Mooc-List] *[http://www.montereyinstitute.org/ Monterey Institute for Technology and Education] - An educational non-profit organization committed to improving access to education. *[http://videolectures.net/ More Video Lectures] *[http://www.montereyinstitute.org/nroc/ National Repository of On-line Courses (NROC)] - Project supports the development and distribution of high-quality on-line courses to a worldwide audience. *[http://digitalcommons.unl.edu University of Nebraska Lincoln Digital Commons] *[http://www.theoceanproject.org/resource/educational.html Ocean resource center] *[http://www.oercommons.org OER Commons] *[http://www.oeconsortium.org/ Open Education Consortium] *[http://ocwfinder.com/ Open courseware finder] *[http://cnx.org/ OpenStax] - free scholarly materials, Creative Commons Attribution 2.0 *[http://openstudy.com/ Open Study] - OpenStudy is a website partnered with institutions like MIT OpenCourseWare and Yale Open Course *[http://ocw.mit.edu/index.html MIT OpenCourseWare] *[http://www.ocw.titech.ac.jp/index.php?lang=EN Tokyo Institute of Technology OpenCourseWare] *[http://ocw.tufts.edu/ Tufts OpenCourseWare] *[http://ocw.usu.edu/ Utah State University OpenCourseWare] *[http://www.open.ac.uk/openlearn OpenLearn: access to Open University course materials under a Creative Commons licence] *[https://www.p2pu.org/en/ P2P University] *[http://physicsforums.com/showthread.php?t=124602 Physics Forums: An Introduction to PH and Buffer Solution Calculations] (text) *[https://www.reddit.com/r/learnprogramming/wiki/index#wiki_learning_resources Programming Resources List] - Helpful resource list for those wanting to learn how to code. === Q - T === *[http://www.rfc-editor.org/ Request for Comments] - Documentation for Internet protocols *[http://www.saylor.org/ Saylor.org] is a free and open collection of college level courses. *[http://stemsheets.com/ STEM Sheets] A collection of free customizable and printable math and science worksheets and flashcards. *[http://ed.ted.com/ Ted Ed] Lessons Worth Sharing *[https://thermodynamicsbook.com/ Thermodynamics book] Thermodynamics textbook by Olivier Cleynen licensed under CC-by-sa in English and French. *[http://emotionalcompetency.com/sci/booktoc.html Scientific Methods] *[http://www-conf.slac.stanford.edu/ssi/2005/lec_notes/Carroll/default.htm Standford Introduction to General Relativity] (video)(pdf) *[http://www.usm.maine.edu/~rhodes/ Tools for Structural Biology Education] *[http://scientist.by/index.php?option=com_content&view=category&layout=blog&id=3&Itemid=19 Scientist.By Community: Educational Webinars] *[http://questgarden.com/author/examplestop.php Questgarden] - webquest activities for a range of ages and study areas === U - X === *[http://archives.math.utk.edu/visual.calculus/ Visual Calculus] is a very good introduction to single variable calculus with explanations and exercises made with flash. *[https://visualfractions.com/ Visual Fractions] is an online reference - covering fraction and math calculators, percentages, unit conversions, and more. There is a Spanish Version as well. *[http://froggy.lbl.gov/ The Whole Frog Project] *[http://www.w3.org World Wide Web Consortium] *[http://www.wikieducator.org WikiEducator] *[http://www.wikihow.com WikiHow] *[http://vlib.org/ The WWW Virtual Library] *[http://www.w3schools.com W3 Schools On-line Web Tutorials] * The [http://demonstrations.wolfram.com Wolfram Demonstrations Project] is an open-code resource that uses dynamic computation to illuminate concepts in science, technology, mathematics, art, finance, and other fields. *[http://www.wonderhowto.com Wonder How To] *[https://worksheetgenius.com WorksheetGenius] hosts educational worksheet generators for homeschoolers, teachers or parents who want differentiated, randomized worksheets at the click of a button. *[http://www.x.org/ The Xorg Foundation] === Y - Z === * [http://www.youtube.com/edu YouTube EDU] - lectures, demonstrations, and other educational videos. See also [[YouTube]]. == See Also == * [[Open Educational Resources]] {{WikiversityUsers}} [[Category:Learning projects]] [[Category:Online learning resources]] [[Category:Hunter-gatherers project|*]] kt3evndraeo7ti763e7n0myjm3ioa3e 2814906 2814905 2026-06-09T19:30:25Z IanVG 2918363 /* External links */ updated table 2814906 wikitext text/x-wiki {{collection}} This project (formerly the "Hunter-gatherers project") aims to find, list, and evaluate Internet learning resources that anyone can freely access and use. We encourage you to discuss and explore ways to integrate these resources with Wikiversity courses. These are important tasks and essential aspects of Wikiversity's purpose. ==See also== Other lists of online resources: *[[Public Health Strategies for HIV]] *[[Educational Videos]] *[[Free license music]] *[[w:List of academic databases and search engines|List of academic databases and search engines]] *[[w:List of educational video websites|List of educational video websites]] *[[w:Category:Educational websites|Category:Educational websites]] *[https://www.reddit.com/r/reddit.com/comments/cktxy/reddit_lets_compile_a_list_of_the_best_online/ Reddit Compilation of Learning Resources] *[https://wiki.creativecommons.org/wiki/Category:Organization Open Educational Resources Database] - This list is HUGE! ==External links== {| class="wikitable sortable" |+ !Alphabetical section !Link !Description |- |0-9 |[http://web.archive.org/web/20100601122623/http://web.me.com/jtstunes/JTSPodcasts/120_Minutes_of_Medieval_Jewish_History/120_Minutes_of_Medieval_Jewish_History.html 120 Minutes with JTS (Jewish Theological Seminary)] |A wide variety of [[Judaism|Jewish]] topics, many of general interest to students of world [[Portal:History|history]]. |- |A-D |[http://www.academiathules.org/ Academia Thules] |a site that offers courses about ancient Rome and ancient Roman reenactment |- |A-D |[http://academicearth.org/ Academic Earth] |a site containing video courses from various Universities and Lecturers. |- |A-D |[http://www.accessexcellence.org/AE/ Access Excellence Activities Collections] | |- |A-D |[https://alison.com/ Alison] | |- |A-D |[http://www.betterexplained.com Better Explained] | |- |A-D |[http://tolweb.org/tree/home.pages/linksscied.html Bioscience Education] | |- |A-D |[http://www.bbc.co.uk/scotland/colinandcumberland/ BBC: Colin And Cumberland] |An interactive game that teaches elementary Gaelic |- |A-D |[http://news.bbc.co.uk/2/hi/uk_news/education/6071230.stm BBC: OU offers free learning materials] | |- |A-D |[https://bhaktianandascollectedworks.wordpress.com/direct-links-to-all-articles-on-the-site/ Bhakti Ananda Goswami’s Collected Works: The Primal Revelation at the Heart of Civilization] | |- |A-D |[http://www.bighistoryproject.com/ Big History] |An Introduction to Everything |- |A-D |[http://www.managementstudyguide.com/brand-management.htm Brand Management Study Guide] |A complete tutorial on Brand Management. |- |A-D |[http://www.beyondintractability.org Beyond Intractability] |Lots of essays about constructive approaches to dealing with conflict. |- |A-D |[http://www.bymath.com/studyguide/alg/alg_topics.html ByMath: Algebra] |(text) |- |A-D |[http://www.bymath.com/studyguide/ari/ari_topics.html ByMath: Basic Arithmetic] |(text) |- |A-D |[http://oli.cmu.edu/ Carnegie Mellon University Open Learning] | |- |A-D |[http://science-education.nih.gov/supplements/nih1/cancer/activities/activities_toc.htm Cell Biology and Cancer Activities] | |- |A-D |[http://www.ck12.org/ CK-12 Foundation] | |- |A-D |[http://cires.colorado.edu/education/resources/ Colorado Education Resources in Environmental Sciences] | |- |A-D |[http://www.crinfo.org Conflict Resolution Information Source] | |- |A-D |[http://www.coursera.org Coursera] | |- |A-D |[http://www.curriki.org/ Curriki] | |- |A-D |[http://ddl.af Darakht-e Danesh] | |- |A-D |[http://dmoz.org/ DMoz] | |- |E-H |[http://www.ecgpedia.org/ ECGpedia] |free online electrocardiography (ECG) course and textbook. |- |E-H |[http://www.edhelper.com/ edHelper] |on-line lessons paid subscriptions, but can be viewed for ideas. |- |E-H |[[w:EdX|EdX]] |is a joint partnership between The Massachusetts Institute of Technology (MIT) and Harvard University to offer online learning to millions of people around the world. |- |E-H |[http://www.excel-easy.com/ Excel Easy] |fully illustrated Excel tutorial + 300 examples. 100% free. |- |E-H |[https://exceltemplate.net/ Excel Template] |A collection of over 200 free Excel templates for any occasion imaginable. Calendars, planners, timesheets, finance and accounting documents, charts and dashboards, and other documents for productivity, business, study, and private life. |- |E-H |[http://bigthink.com/ideas/39935 Floating University] |Released by Big Think to distribute big ideas. |- |E-H |[https://web.archive.org/web/20160306135250/https://occupywiki.org.uk/wiki/Free_University Free University] | |- |E-H |[http://www.freedesktop.org/wiki/ Freedesktop] | |- |E-H |[http://freevideolectures.com/ Free Video Lectures] | |- |E-H |[http://www.geogebratube.org/student/m14908?mobile=true GeoGebra - Similar Figures manipulation for 6th grade math] | |- |E-H |[http://www.gentoo-wiki.com Gentoo Linux Wiki] | |- |E-H |[http://www.gcflearnfree.org Goodwill Community Foundation Learn Free] | |- |E-H |[http://www.globe.gov/fsl/welcome/welcomeobject.pl Global Learning and Observations to Benefit the Environment] | |- |E-H |[http://www.howstuffworks.com/ HowStuffWorks] | |- |E-H |[http://hippocampus.org/ HippoCampus] |The goal of HippoCampus is to provide high-quality, multimedia content on general education subjects to high school and college students free of charge. HippoCampus was designed as part of Open Education Resources (OER), a worldwide effort to improve access to quality education for everyone. |- |E-H |[http://education.ucsb.edu/webdata/instruction/hss/mst/ Strategies for Teaching History] |University of California Santa Barbara |- |E-H |[http://www.hsn.uk.net Higher Still Notes] |Full Course Notes for Scottish Higher Courses |- |E-H |[http://www.humiliationstudies.org/education/education.php#links list of open education resources] |Humiliation Studies Network. |- |I-L |[http://www.issuelab.org Issuelab] |Online publishing forum for nonprofit research licensed under CC-BY-SA[?]. |- |I-L |[http://www.faqs.org/faqs/ Internet FAQ Archives] | |- |I-L |[http://developer.mozilla.org/en/docs/Core_JavaScript_1.5_Reference JavaScript 1.5 Reference] | |- |I-L |[http://ocw.jhsph.edu/ John Hopkins OpenCourseWare] | |- |I-L |[http://www.khanacademy.org/ Khan Academy] |with the mission of providing a world-class education to anyone, anywhere. It has more than 1800 videos on math, biology, chemistry and more. |- |I-L |[http://www.learnamic.com/ Learnamic] |Online directory and reviews of self-paced learning tools |- |I-L |[http://www.tldp.org/ The Linux Documentation Project] | |- |M-P |[https://www.math10.com/ Math10] |Website containing lessons and problems on elementary through entry level university classes. |- |M-P |[https://www.math-drills.com/ Math Drills] |Math worksheets for elementary and middle school |- |M-P |[http://www.mathsolutions.com/index.cfm Math Solutions] |Math by Marilyn Burns |- |M-P |[http://www.mathvillage.info/ Math Village] |Teaching materials for middle school math |- |M-P |[http://www.merlot.org/ MERLOT II] |A collection of free and open online teaching, learning, and faculty development services. |- |M-P |[http://www.umich.edu/~uminds/course_archive.html The University of Michigan Initiative on Disability Studies courses are available here] | |- |M-P |[https://www.mooc-list.com Mooc-List] | |- |M-P |[http://www.montereyinstitute.org/ Monterey Institute for Technology and Education] |An educational non-profit organization committed to improving access to education. |- |M-P |[http://videolectures.net/ More Video Lectures] | |- |M-P |[http://www.montereyinstitute.org/nroc/ National Repository of On-line Courses (NROC)] |Project supports the development and distribution of high-quality on-line courses to a worldwide audience. |- |M-P |[http://digitalcommons.unl.edu University of Nebraska Lincoln Digital Commons] | |- |M-P |[http://www.theoceanproject.org/resource/educational.html Ocean resource center] | |- |M-P |[http://www.oercommons.org OER Commons] | |- |M-P |[http://www.oeconsortium.org/ Open Education Consortium] | |- |M-P |[http://ocwfinder.com/ Open courseware finder] | |- |M-P |[http://cnx.org/ OpenStax] |free scholarly materials, Creative Commons Attribution 2.0 |- |M-P |[http://openstudy.com/ Open Study] |OpenStudy is a website partnered with institutions like MIT OpenCourseWare and Yale Open Course |- |M-P |[http://ocw.mit.edu/index.html MIT OpenCourseWare] | |- |M-P |[http://www.ocw.titech.ac.jp/index.php?lang=EN Tokyo Institute of Technology OpenCourseWare] | |- |M-P |[http://ocw.tufts.edu/ Tufts OpenCourseWare] | |- |M-P |[http://ocw.usu.edu/ Utah State University OpenCourseWare] | |- |M-P |[http://www.open.ac.uk/openlearn OpenLearn: access to Open University course materials under a Creative Commons licence] | |- |M-P |[https://www.p2pu.org/en/ P2P University] | |- |M-P |[http://physicsforums.com/showthread.php?t=124602 Physics Forums: An Introduction to PH and Buffer Solution Calculations] |(text) |- |M-P |[https://www.reddit.com/r/learnprogramming/wiki/index#wiki_learning_resources Programming Resources List] |Helpful resource list for those wanting to learn how to code. |- |Q-T |[http://www.rfc-editor.org/ Request for Comments] |Documentation for Internet protocols |- |Q-T |[http://www.saylor.org/ Saylor.org] |is a free and open collection of college level courses. |- |Q-T |[http://stemsheets.com/ STEM Sheets] |A collection of free customizable and printable math and science worksheets and flashcards. |- |Q-T |[http://ed.ted.com/ Ted Ed] |Lessons Worth Sharing |- |Q-T |[https://thermodynamicsbook.com/ Thermodynamics book] |Thermodynamics textbook by Olivier Cleynen licensed under CC-by-sa in English and French. |- |Q-T |[http://emotionalcompetency.com/sci/booktoc.html Scientific Methods] | |- |Q-T |[http://www-conf.slac.stanford.edu/ssi/2005/lec_notes/Carroll/default.htm Standford Introduction to General Relativity] |(video)(pdf) |- |Q-T |[http://www.usm.maine.edu/~rhodes/ Tools for Structural Biology Education] | |- |Q-T |[http://scientist.by/index.php?option=com_content&view=category&layout=blog&id=3&Itemid=19 Scientist.By Community: Educational Webinars] | |- |Q-T |[http://questgarden.com/author/examplestop.php Questgarden] |webquest activities for a range of ages and study areas |- |U-X |[http://archives.math.utk.edu/visual.calculus/ Visual Calculus] |is a very good introduction to single variable calculus with explanations and exercises made with flash. |- |U-X |[https://visualfractions.com/ Visual Fractions] |is an online reference - covering fraction and math calculators, percentages, unit conversions, and more. There is a Spanish Version as well. |- |U-X |[http://froggy.lbl.gov/ The Whole Frog Project] | |- |U-X |[http://www.w3.org World Wide Web Consortium] | |- |U-X |[http://www.wikieducator.org WikiEducator] | |- |U-X |[http://www.wikihow.com WikiHow] | |- |U-X |[http://vlib.org/ The WWW Virtual Library] | |- |U-X |[http://www.w3schools.com W3 Schools On-line Web Tutorials] | |- |U-X |[http://demonstrations.wolfram.com Wolfram Demonstrations Project] |is an open-code resource that uses dynamic computation to illuminate concepts in science, technology, mathematics, art, finance, and other fields. |- |U-X |[http://www.wonderhowto.com Wonder How To] | |- |U-X |[https://worksheetgenius.com WorksheetGenius] |hosts educational worksheet generators for homeschoolers, teachers or parents who want differentiated, randomized worksheets at the click of a button. |- |U-X |[http://www.x.org/ The Xorg Foundation] | |- |Y-Z |[http://www.youtube.com/edu YouTube EDU] |lectures, demonstrations, and other educational videos. See also [[YouTube]]. |} === 0 - 9 === *[http://web.archive.org/web/20100601122623/http://web.me.com/jtstunes/JTSPodcasts/120_Minutes_of_Medieval_Jewish_History/120_Minutes_of_Medieval_Jewish_History.html 120 Minutes with JTS (Jewish Theological Seminary)]: A wide variety of [[Judaism|Jewish]] topics, many of general interest to students of world [[Portal:History|history]]. === A - D === *[http://www.academiathules.org/ Academia Thules], a site that offers courses about ancient Rome and ancient Roman reenactment *[http://academicearth.org/ Academic Earth], a site containing video courses from various Universities and Lecturers. *[http://www.accessexcellence.org/AE/ Access Excellence Activities Collections] *[https://alison.com/ Alison] *[http://www.betterexplained.com Better Explained] *[http://tolweb.org/tree/home.pages/linksscied.html Bioscience Education] *[http://www.bbc.co.uk/scotland/colinandcumberland/ BBC: Colin And Cumberland] - An interactive game that teaches elementary Gaelic *[http://news.bbc.co.uk/2/hi/uk_news/education/6071230.stm BBC: OU offers free learning materials] *[https://bhaktianandascollectedworks.wordpress.com/direct-links-to-all-articles-on-the-site/ Bhakti Ananda Goswami’s Collected Works: The Primal Revelation at the Heart of Civilization] *[http://www.bighistoryproject.com/ Big History] - An Introduction to Everything *[http://www.managementstudyguide.com/brand-management.htm Brand Management Study Guide] - A complete tutorial on Brand Management. *[http://www.beyondintractability.org Beyond Intractability] - Lots of essays about constructive approaches to dealing with conflict. *[http://www.bymath.com/studyguide/alg/alg_topics.html ByMath: Algebra] (text) *[http://www.bymath.com/studyguide/ari/ari_topics.html ByMath: Basic Arithmetic] (text) *[http://oli.cmu.edu/ Carnegie Mellon University Open Learning] *[http://science-education.nih.gov/supplements/nih1/cancer/activities/activities_toc.htm Cell Biology and Cancer Activities] *[http://www.ck12.org/ CK-12 Foundation] *[http://cires.colorado.edu/education/resources/ Colorado Education Resources in Environmental Sciences] *[http://www.crinfo.org Conflict Resolution Information Source] *[http://www.coursera.org Coursera] *[http://www.curriki.org/ Curriki] *[http://ddl.af Darakht-e Danesh] *[http://dmoz.org/ DMoz] === E - H === *[http://www.ecgpedia.org/ ECGpedia] - free online electrocardiography (ECG) course and textbook. *[http://www.edhelper.com/ edHelper] - on-line lessons paid subscriptions, but can be viewed for ideas. * [[w:EdX|EdX]] is a joint partnership between The Massachusetts Institute of Technology (MIT) and Harvard University to offer online learning to millions of people around the world. *[http://www.excel-easy.com/ Excel Easy] - fully illustrated Excel tutorial + 300 examples. 100% free. *[https://exceltemplate.net/ Excel Template] - A collection of over 200 free Excel templates for any occasion imaginable. Calendars, planners, timesheets, finance and accounting documents, charts and dashboards, and other documents for productivity, business, study, and private life. * The [http://bigthink.com/ideas/39935 Floating University] - Released by Big Think to distribute big ideas. *[https://web.archive.org/web/20160306135250/https://occupywiki.org.uk/wiki/Free_University Free University] *[http://www.freedesktop.org/wiki/ Freedesktop] *[http://freevideolectures.com/ Free Video Lectures] *[http://www.geogebratube.org/student/m14908?mobile=true GeoGebra - Similar Figures manipulation for 6th grade math] *[http://www.gentoo-wiki.com Gentoo Linux Wiki] *[http://www.gcflearnfree.org Goodwill Community Foundation Learn Free] *[http://www.globe.gov/fsl/welcome/welcomeobject.pl Global Learning and Observations to Benefit the Environment] *[http://www.howstuffworks.com/ HowStuffWorks] *[http://hippocampus.org/ HippoCampus] The goal of HippoCampus is to provide high-quality, multimedia content on general education subjects to high school and college students free of charge. HippoCampus was designed as part of Open Education Resources (OER), a worldwide effort to improve access to quality education for everyone. *[http://education.ucsb.edu/webdata/instruction/hss/mst/ Strategies for Teaching History] University of California Santa Barbara *[http://www.hsn.uk.net Higher Still Notes] Full Course Notes for Scottish Higher Courses * Humiliation Studies Network [http://www.humiliationstudies.org/education/education.php#links list of open education resources]. === I - L === *[http://www.issuelab.org Issuelab] Online publishing forum for nonprofit research licensed under CC-BY-SA[?]. *[http://www.faqs.org/faqs/ Internet FAQ Archives] *[http://developer.mozilla.org/en/docs/Core_JavaScript_1.5_Reference JavaScript 1.5 Reference] *[http://ocw.jhsph.edu/ John Hopkins OpenCourseWare] *[http://www.khanacademy.org/ Khan Academy] with the mission of providing a world-class education to anyone, anywhere. It has more than 1800 videos on math, biology, chemistry and more. *[http://www.learnamic.com/ Learnamic] Online directory and reviews of self-paced learning tools *[http://www.tldp.org/ The Linux Documentation Project] === M - P === *[https://www.math10.com/ Math10] Website containing lessons and problems on elementary through entry level university classes. *[https://www.math-drills.com/ Math Drills] Math worksheets for elementary and middle school *[http://www.mathsolutions.com/index.cfm Math Solutions] Math by Marilyn Burns *[http://www.mathvillage.info/ Math Village] Teaching materials for middle school math *[http://www.merlot.org/ MERLOT II] A collection of free and open online teaching, learning, and faculty development services. *[http://www.umich.edu/~uminds/course_archive.html The University of Michigan Initiative on Disability Studies courses are available here] *[https://www.mooc-list.com Mooc-List] *[http://www.montereyinstitute.org/ Monterey Institute for Technology and Education] - An educational non-profit organization committed to improving access to education. *[http://videolectures.net/ More Video Lectures] *[http://www.montereyinstitute.org/nroc/ National Repository of On-line Courses (NROC)] - Project supports the development and distribution of high-quality on-line courses to a worldwide audience. *[http://digitalcommons.unl.edu University of Nebraska Lincoln Digital Commons] *[http://www.theoceanproject.org/resource/educational.html Ocean resource center] *[http://www.oercommons.org OER Commons] *[http://www.oeconsortium.org/ Open Education Consortium] *[http://ocwfinder.com/ Open courseware finder] *[http://cnx.org/ OpenStax] - free scholarly materials, Creative Commons Attribution 2.0 *[http://openstudy.com/ Open Study] - OpenStudy is a website partnered with institutions like MIT OpenCourseWare and Yale Open Course *[http://ocw.mit.edu/index.html MIT OpenCourseWare] *[http://www.ocw.titech.ac.jp/index.php?lang=EN Tokyo Institute of Technology OpenCourseWare] *[http://ocw.tufts.edu/ Tufts OpenCourseWare] *[http://ocw.usu.edu/ Utah State University OpenCourseWare] *[http://www.open.ac.uk/openlearn OpenLearn: access to Open University course materials under a Creative Commons licence] *[https://www.p2pu.org/en/ P2P University] *[http://physicsforums.com/showthread.php?t=124602 Physics Forums: An Introduction to PH and Buffer Solution Calculations] (text) *[https://www.reddit.com/r/learnprogramming/wiki/index#wiki_learning_resources Programming Resources List] - Helpful resource list for those wanting to learn how to code. === Q - T === *[http://www.rfc-editor.org/ Request for Comments] - Documentation for Internet protocols *[http://www.saylor.org/ Saylor.org] is a free and open collection of college level courses. *[http://stemsheets.com/ STEM Sheets] A collection of free customizable and printable math and science worksheets and flashcards. *[http://ed.ted.com/ Ted Ed] Lessons Worth Sharing *[https://thermodynamicsbook.com/ Thermodynamics book] Thermodynamics textbook by Olivier Cleynen licensed under CC-by-sa in English and French. *[http://emotionalcompetency.com/sci/booktoc.html Scientific Methods] *[http://www-conf.slac.stanford.edu/ssi/2005/lec_notes/Carroll/default.htm Standford Introduction to General Relativity] (video)(pdf) *[http://www.usm.maine.edu/~rhodes/ Tools for Structural Biology Education] *[http://scientist.by/index.php?option=com_content&view=category&layout=blog&id=3&Itemid=19 Scientist.By Community: Educational Webinars] *[http://questgarden.com/author/examplestop.php Questgarden] - webquest activities for a range of ages and study areas === U - X === *[http://archives.math.utk.edu/visual.calculus/ Visual Calculus] is a very good introduction to single variable calculus with explanations and exercises made with flash. *[https://visualfractions.com/ Visual Fractions] is an online reference - covering fraction and math calculators, percentages, unit conversions, and more. There is a Spanish Version as well. *[http://froggy.lbl.gov/ The Whole Frog Project] *[http://www.w3.org World Wide Web Consortium] *[http://www.wikieducator.org WikiEducator] *[http://www.wikihow.com WikiHow] *[http://vlib.org/ The WWW Virtual Library] *[http://www.w3schools.com W3 Schools On-line Web Tutorials] * The [http://demonstrations.wolfram.com Wolfram Demonstrations Project] is an open-code resource that uses dynamic computation to illuminate concepts in science, technology, mathematics, art, finance, and other fields. *[http://www.wonderhowto.com Wonder How To] *[https://worksheetgenius.com WorksheetGenius] hosts educational worksheet generators for homeschoolers, teachers or parents who want differentiated, randomized worksheets at the click of a button. *[http://www.x.org/ The Xorg Foundation] === Y - Z === * [http://www.youtube.com/edu YouTube EDU] - lectures, demonstrations, and other educational videos. See also [[YouTube]]. == See Also == * [[Open Educational Resources]] {{WikiversityUsers}} [[Category:Learning projects]] [[Category:Online learning resources]] [[Category:Hunter-gatherers project|*]] l1mqbiod2gmg7qwynz31l9vabx2hr8y 2814907 2814906 2026-06-09T19:30:42Z IanVG 2918363 /* External links */ 2814907 wikitext text/x-wiki {{collection}} This project (formerly the "Hunter-gatherers project") aims to find, list, and evaluate Internet learning resources that anyone can freely access and use. We encourage you to discuss and explore ways to integrate these resources with Wikiversity courses. These are important tasks and essential aspects of Wikiversity's purpose. ==See also== Other lists of online resources: *[[Public Health Strategies for HIV]] *[[Educational Videos]] *[[Free license music]] *[[w:List of academic databases and search engines|List of academic databases and search engines]] *[[w:List of educational video websites|List of educational video websites]] *[[w:Category:Educational websites|Category:Educational websites]] *[https://www.reddit.com/r/reddit.com/comments/cktxy/reddit_lets_compile_a_list_of_the_best_online/ Reddit Compilation of Learning Resources] *[https://wiki.creativecommons.org/wiki/Category:Organization Open Educational Resources Database] - This list is HUGE! ==External links== {| class="wikitable sortable mw-collapsible mw-collapsed" |+ !Alphabetical section !Link !Description |- |0-9 |[http://web.archive.org/web/20100601122623/http://web.me.com/jtstunes/JTSPodcasts/120_Minutes_of_Medieval_Jewish_History/120_Minutes_of_Medieval_Jewish_History.html 120 Minutes with JTS (Jewish Theological Seminary)] |A wide variety of [[Judaism|Jewish]] topics, many of general interest to students of world [[Portal:History|history]]. |- |A-D |[http://www.academiathules.org/ Academia Thules] |a site that offers courses about ancient Rome and ancient Roman reenactment |- |A-D |[http://academicearth.org/ Academic Earth] |a site containing video courses from various Universities and Lecturers. |- |A-D |[http://www.accessexcellence.org/AE/ Access Excellence Activities Collections] | |- |A-D |[https://alison.com/ Alison] | |- |A-D |[http://www.betterexplained.com Better Explained] | |- |A-D |[http://tolweb.org/tree/home.pages/linksscied.html Bioscience Education] | |- |A-D |[http://www.bbc.co.uk/scotland/colinandcumberland/ BBC: Colin And Cumberland] |An interactive game that teaches elementary Gaelic |- |A-D |[http://news.bbc.co.uk/2/hi/uk_news/education/6071230.stm BBC: OU offers free learning materials] | |- |A-D |[https://bhaktianandascollectedworks.wordpress.com/direct-links-to-all-articles-on-the-site/ Bhakti Ananda Goswami’s Collected Works: The Primal Revelation at the Heart of Civilization] | |- |A-D |[http://www.bighistoryproject.com/ Big History] |An Introduction to Everything |- |A-D |[http://www.managementstudyguide.com/brand-management.htm Brand Management Study Guide] |A complete tutorial on Brand Management. |- |A-D |[http://www.beyondintractability.org Beyond Intractability] |Lots of essays about constructive approaches to dealing with conflict. |- |A-D |[http://www.bymath.com/studyguide/alg/alg_topics.html ByMath: Algebra] |(text) |- |A-D |[http://www.bymath.com/studyguide/ari/ari_topics.html ByMath: Basic Arithmetic] |(text) |- |A-D |[http://oli.cmu.edu/ Carnegie Mellon University Open Learning] | |- |A-D |[http://science-education.nih.gov/supplements/nih1/cancer/activities/activities_toc.htm Cell Biology and Cancer Activities] | |- |A-D |[http://www.ck12.org/ CK-12 Foundation] | |- |A-D |[http://cires.colorado.edu/education/resources/ Colorado Education Resources in Environmental Sciences] | |- |A-D |[http://www.crinfo.org Conflict Resolution Information Source] | |- |A-D |[http://www.coursera.org Coursera] | |- |A-D |[http://www.curriki.org/ Curriki] | |- |A-D |[http://ddl.af Darakht-e Danesh] | |- |A-D |[http://dmoz.org/ DMoz] | |- |E-H |[http://www.ecgpedia.org/ ECGpedia] |free online electrocardiography (ECG) course and textbook. |- |E-H |[http://www.edhelper.com/ edHelper] |on-line lessons paid subscriptions, but can be viewed for ideas. |- |E-H |[[w:EdX|EdX]] |is a joint partnership between The Massachusetts Institute of Technology (MIT) and Harvard University to offer online learning to millions of people around the world. |- |E-H |[http://www.excel-easy.com/ Excel Easy] |fully illustrated Excel tutorial + 300 examples. 100% free. |- |E-H |[https://exceltemplate.net/ Excel Template] |A collection of over 200 free Excel templates for any occasion imaginable. Calendars, planners, timesheets, finance and accounting documents, charts and dashboards, and other documents for productivity, business, study, and private life. |- |E-H |[http://bigthink.com/ideas/39935 Floating University] |Released by Big Think to distribute big ideas. |- |E-H |[https://web.archive.org/web/20160306135250/https://occupywiki.org.uk/wiki/Free_University Free University] | |- |E-H |[http://www.freedesktop.org/wiki/ Freedesktop] | |- |E-H |[http://freevideolectures.com/ Free Video Lectures] | |- |E-H |[http://www.geogebratube.org/student/m14908?mobile=true GeoGebra - Similar Figures manipulation for 6th grade math] | |- |E-H |[http://www.gentoo-wiki.com Gentoo Linux Wiki] | |- |E-H |[http://www.gcflearnfree.org Goodwill Community Foundation Learn Free] | |- |E-H |[http://www.globe.gov/fsl/welcome/welcomeobject.pl Global Learning and Observations to Benefit the Environment] | |- |E-H |[http://www.howstuffworks.com/ HowStuffWorks] | |- |E-H |[http://hippocampus.org/ HippoCampus] |The goal of HippoCampus is to provide high-quality, multimedia content on general education subjects to high school and college students free of charge. HippoCampus was designed as part of Open Education Resources (OER), a worldwide effort to improve access to quality education for everyone. |- |E-H |[http://education.ucsb.edu/webdata/instruction/hss/mst/ Strategies for Teaching History] |University of California Santa Barbara |- |E-H |[http://www.hsn.uk.net Higher Still Notes] |Full Course Notes for Scottish Higher Courses |- |E-H |[http://www.humiliationstudies.org/education/education.php#links list of open education resources] |Humiliation Studies Network. |- |I-L |[http://www.issuelab.org Issuelab] |Online publishing forum for nonprofit research licensed under CC-BY-SA[?]. |- |I-L |[http://www.faqs.org/faqs/ Internet FAQ Archives] | |- |I-L |[http://developer.mozilla.org/en/docs/Core_JavaScript_1.5_Reference JavaScript 1.5 Reference] | |- |I-L |[http://ocw.jhsph.edu/ John Hopkins OpenCourseWare] | |- |I-L |[http://www.khanacademy.org/ Khan Academy] |with the mission of providing a world-class education to anyone, anywhere. It has more than 1800 videos on math, biology, chemistry and more. |- |I-L |[http://www.learnamic.com/ Learnamic] |Online directory and reviews of self-paced learning tools |- |I-L |[http://www.tldp.org/ The Linux Documentation Project] | |- |M-P |[https://www.math10.com/ Math10] |Website containing lessons and problems on elementary through entry level university classes. |- |M-P |[https://www.math-drills.com/ Math Drills] |Math worksheets for elementary and middle school |- |M-P |[http://www.mathsolutions.com/index.cfm Math Solutions] |Math by Marilyn Burns |- |M-P |[http://www.mathvillage.info/ Math Village] |Teaching materials for middle school math |- |M-P |[http://www.merlot.org/ MERLOT II] |A collection of free and open online teaching, learning, and faculty development services. |- |M-P |[http://www.umich.edu/~uminds/course_archive.html The University of Michigan Initiative on Disability Studies courses are available here] | |- |M-P |[https://www.mooc-list.com Mooc-List] | |- |M-P |[http://www.montereyinstitute.org/ Monterey Institute for Technology and Education] |An educational non-profit organization committed to improving access to education. |- |M-P |[http://videolectures.net/ More Video Lectures] | |- |M-P |[http://www.montereyinstitute.org/nroc/ National Repository of On-line Courses (NROC)] |Project supports the development and distribution of high-quality on-line courses to a worldwide audience. |- |M-P |[http://digitalcommons.unl.edu University of Nebraska Lincoln Digital Commons] | |- |M-P |[http://www.theoceanproject.org/resource/educational.html Ocean resource center] | |- |M-P |[http://www.oercommons.org OER Commons] | |- |M-P |[http://www.oeconsortium.org/ Open Education Consortium] | |- |M-P |[http://ocwfinder.com/ Open courseware finder] | |- |M-P |[http://cnx.org/ OpenStax] |free scholarly materials, Creative Commons Attribution 2.0 |- |M-P |[http://openstudy.com/ Open Study] |OpenStudy is a website partnered with institutions like MIT OpenCourseWare and Yale Open Course |- |M-P |[http://ocw.mit.edu/index.html MIT OpenCourseWare] | |- |M-P |[http://www.ocw.titech.ac.jp/index.php?lang=EN Tokyo Institute of Technology OpenCourseWare] | |- |M-P |[http://ocw.tufts.edu/ Tufts OpenCourseWare] | |- |M-P |[http://ocw.usu.edu/ Utah State University OpenCourseWare] | |- |M-P |[http://www.open.ac.uk/openlearn OpenLearn: access to Open University course materials under a Creative Commons licence] | |- |M-P |[https://www.p2pu.org/en/ P2P University] | |- |M-P |[http://physicsforums.com/showthread.php?t=124602 Physics Forums: An Introduction to PH and Buffer Solution Calculations] |(text) |- |M-P |[https://www.reddit.com/r/learnprogramming/wiki/index#wiki_learning_resources Programming Resources List] |Helpful resource list for those wanting to learn how to code. |- |Q-T |[http://www.rfc-editor.org/ Request for Comments] |Documentation for Internet protocols |- |Q-T |[http://www.saylor.org/ Saylor.org] |is a free and open collection of college level courses. |- |Q-T |[http://stemsheets.com/ STEM Sheets] |A collection of free customizable and printable math and science worksheets and flashcards. |- |Q-T |[http://ed.ted.com/ Ted Ed] |Lessons Worth Sharing |- |Q-T |[https://thermodynamicsbook.com/ Thermodynamics book] |Thermodynamics textbook by Olivier Cleynen licensed under CC-by-sa in English and French. |- |Q-T |[http://emotionalcompetency.com/sci/booktoc.html Scientific Methods] | |- |Q-T |[http://www-conf.slac.stanford.edu/ssi/2005/lec_notes/Carroll/default.htm Standford Introduction to General Relativity] |(video)(pdf) |- |Q-T |[http://www.usm.maine.edu/~rhodes/ Tools for Structural Biology Education] | |- |Q-T |[http://scientist.by/index.php?option=com_content&view=category&layout=blog&id=3&Itemid=19 Scientist.By Community: Educational Webinars] | |- |Q-T |[http://questgarden.com/author/examplestop.php Questgarden] |webquest activities for a range of ages and study areas |- |U-X |[http://archives.math.utk.edu/visual.calculus/ Visual Calculus] |is a very good introduction to single variable calculus with explanations and exercises made with flash. |- |U-X |[https://visualfractions.com/ Visual Fractions] |is an online reference - covering fraction and math calculators, percentages, unit conversions, and more. There is a Spanish Version as well. |- |U-X |[http://froggy.lbl.gov/ The Whole Frog Project] | |- |U-X |[http://www.w3.org World Wide Web Consortium] | |- |U-X |[http://www.wikieducator.org WikiEducator] | |- |U-X |[http://www.wikihow.com WikiHow] | |- |U-X |[http://vlib.org/ The WWW Virtual Library] | |- |U-X |[http://www.w3schools.com W3 Schools On-line Web Tutorials] | |- |U-X |[http://demonstrations.wolfram.com Wolfram Demonstrations Project] |is an open-code resource that uses dynamic computation to illuminate concepts in science, technology, mathematics, art, finance, and other fields. |- |U-X |[http://www.wonderhowto.com Wonder How To] | |- |U-X |[https://worksheetgenius.com WorksheetGenius] |hosts educational worksheet generators for homeschoolers, teachers or parents who want differentiated, randomized worksheets at the click of a button. |- |U-X |[http://www.x.org/ The Xorg Foundation] | |- |Y-Z |[http://www.youtube.com/edu YouTube EDU] |lectures, demonstrations, and other educational videos. See also [[YouTube]]. |} === 0 - 9 === *[http://web.archive.org/web/20100601122623/http://web.me.com/jtstunes/JTSPodcasts/120_Minutes_of_Medieval_Jewish_History/120_Minutes_of_Medieval_Jewish_History.html 120 Minutes with JTS (Jewish Theological Seminary)]: A wide variety of [[Judaism|Jewish]] topics, many of general interest to students of world [[Portal:History|history]]. === A - D === *[http://www.academiathules.org/ Academia Thules], a site that offers courses about ancient Rome and ancient Roman reenactment *[http://academicearth.org/ Academic Earth], a site containing video courses from various Universities and Lecturers. *[http://www.accessexcellence.org/AE/ Access Excellence Activities Collections] *[https://alison.com/ Alison] *[http://www.betterexplained.com Better Explained] *[http://tolweb.org/tree/home.pages/linksscied.html Bioscience Education] *[http://www.bbc.co.uk/scotland/colinandcumberland/ BBC: Colin And Cumberland] - An interactive game that teaches elementary Gaelic *[http://news.bbc.co.uk/2/hi/uk_news/education/6071230.stm BBC: OU offers free learning materials] *[https://bhaktianandascollectedworks.wordpress.com/direct-links-to-all-articles-on-the-site/ Bhakti Ananda Goswami’s Collected Works: The Primal Revelation at the Heart of Civilization] *[http://www.bighistoryproject.com/ Big History] - An Introduction to Everything *[http://www.managementstudyguide.com/brand-management.htm Brand Management Study Guide] - A complete tutorial on Brand Management. *[http://www.beyondintractability.org Beyond Intractability] - Lots of essays about constructive approaches to dealing with conflict. *[http://www.bymath.com/studyguide/alg/alg_topics.html ByMath: Algebra] (text) *[http://www.bymath.com/studyguide/ari/ari_topics.html ByMath: Basic Arithmetic] (text) *[http://oli.cmu.edu/ Carnegie Mellon University Open Learning] *[http://science-education.nih.gov/supplements/nih1/cancer/activities/activities_toc.htm Cell Biology and Cancer Activities] *[http://www.ck12.org/ CK-12 Foundation] *[http://cires.colorado.edu/education/resources/ Colorado Education Resources in Environmental Sciences] *[http://www.crinfo.org Conflict Resolution Information Source] *[http://www.coursera.org Coursera] *[http://www.curriki.org/ Curriki] *[http://ddl.af Darakht-e Danesh] *[http://dmoz.org/ DMoz] === E - H === *[http://www.ecgpedia.org/ ECGpedia] - free online electrocardiography (ECG) course and textbook. *[http://www.edhelper.com/ edHelper] - on-line lessons paid subscriptions, but can be viewed for ideas. * [[w:EdX|EdX]] is a joint partnership between The Massachusetts Institute of Technology (MIT) and Harvard University to offer online learning to millions of people around the world. *[http://www.excel-easy.com/ Excel Easy] - fully illustrated Excel tutorial + 300 examples. 100% free. *[https://exceltemplate.net/ Excel Template] - A collection of over 200 free Excel templates for any occasion imaginable. Calendars, planners, timesheets, finance and accounting documents, charts and dashboards, and other documents for productivity, business, study, and private life. * The [http://bigthink.com/ideas/39935 Floating University] - Released by Big Think to distribute big ideas. *[https://web.archive.org/web/20160306135250/https://occupywiki.org.uk/wiki/Free_University Free University] *[http://www.freedesktop.org/wiki/ Freedesktop] *[http://freevideolectures.com/ Free Video Lectures] *[http://www.geogebratube.org/student/m14908?mobile=true GeoGebra - Similar Figures manipulation for 6th grade math] *[http://www.gentoo-wiki.com Gentoo Linux Wiki] *[http://www.gcflearnfree.org Goodwill Community Foundation Learn Free] *[http://www.globe.gov/fsl/welcome/welcomeobject.pl Global Learning and Observations to Benefit the Environment] *[http://www.howstuffworks.com/ HowStuffWorks] *[http://hippocampus.org/ HippoCampus] The goal of HippoCampus is to provide high-quality, multimedia content on general education subjects to high school and college students free of charge. HippoCampus was designed as part of Open Education Resources (OER), a worldwide effort to improve access to quality education for everyone. *[http://education.ucsb.edu/webdata/instruction/hss/mst/ Strategies for Teaching History] University of California Santa Barbara *[http://www.hsn.uk.net Higher Still Notes] Full Course Notes for Scottish Higher Courses * Humiliation Studies Network [http://www.humiliationstudies.org/education/education.php#links list of open education resources]. === I - L === *[http://www.issuelab.org Issuelab] Online publishing forum for nonprofit research licensed under CC-BY-SA[?]. *[http://www.faqs.org/faqs/ Internet FAQ Archives] *[http://developer.mozilla.org/en/docs/Core_JavaScript_1.5_Reference JavaScript 1.5 Reference] *[http://ocw.jhsph.edu/ John Hopkins OpenCourseWare] *[http://www.khanacademy.org/ Khan Academy] with the mission of providing a world-class education to anyone, anywhere. It has more than 1800 videos on math, biology, chemistry and more. *[http://www.learnamic.com/ Learnamic] Online directory and reviews of self-paced learning tools *[http://www.tldp.org/ The Linux Documentation Project] === M - P === *[https://www.math10.com/ Math10] Website containing lessons and problems on elementary through entry level university classes. *[https://www.math-drills.com/ Math Drills] Math worksheets for elementary and middle school *[http://www.mathsolutions.com/index.cfm Math Solutions] Math by Marilyn Burns *[http://www.mathvillage.info/ Math Village] Teaching materials for middle school math *[http://www.merlot.org/ MERLOT II] A collection of free and open online teaching, learning, and faculty development services. *[http://www.umich.edu/~uminds/course_archive.html The University of Michigan Initiative on Disability Studies courses are available here] *[https://www.mooc-list.com Mooc-List] *[http://www.montereyinstitute.org/ Monterey Institute for Technology and Education] - An educational non-profit organization committed to improving access to education. *[http://videolectures.net/ More Video Lectures] *[http://www.montereyinstitute.org/nroc/ National Repository of On-line Courses (NROC)] - Project supports the development and distribution of high-quality on-line courses to a worldwide audience. *[http://digitalcommons.unl.edu University of Nebraska Lincoln Digital Commons] *[http://www.theoceanproject.org/resource/educational.html Ocean resource center] *[http://www.oercommons.org OER Commons] *[http://www.oeconsortium.org/ Open Education Consortium] *[http://ocwfinder.com/ Open courseware finder] *[http://cnx.org/ OpenStax] - free scholarly materials, Creative Commons Attribution 2.0 *[http://openstudy.com/ Open Study] - OpenStudy is a website partnered with institutions like MIT OpenCourseWare and Yale Open Course *[http://ocw.mit.edu/index.html MIT OpenCourseWare] *[http://www.ocw.titech.ac.jp/index.php?lang=EN Tokyo Institute of Technology OpenCourseWare] *[http://ocw.tufts.edu/ Tufts OpenCourseWare] *[http://ocw.usu.edu/ Utah State University OpenCourseWare] *[http://www.open.ac.uk/openlearn OpenLearn: access to Open University course materials under a Creative Commons licence] *[https://www.p2pu.org/en/ P2P University] *[http://physicsforums.com/showthread.php?t=124602 Physics Forums: An Introduction to PH and Buffer Solution Calculations] (text) *[https://www.reddit.com/r/learnprogramming/wiki/index#wiki_learning_resources Programming Resources List] - Helpful resource list for those wanting to learn how to code. === Q - T === *[http://www.rfc-editor.org/ Request for Comments] - Documentation for Internet protocols *[http://www.saylor.org/ Saylor.org] is a free and open collection of college level courses. *[http://stemsheets.com/ STEM Sheets] A collection of free customizable and printable math and science worksheets and flashcards. *[http://ed.ted.com/ Ted Ed] Lessons Worth Sharing *[https://thermodynamicsbook.com/ Thermodynamics book] Thermodynamics textbook by Olivier Cleynen licensed under CC-by-sa in English and French. *[http://emotionalcompetency.com/sci/booktoc.html Scientific Methods] *[http://www-conf.slac.stanford.edu/ssi/2005/lec_notes/Carroll/default.htm Standford Introduction to General Relativity] (video)(pdf) *[http://www.usm.maine.edu/~rhodes/ Tools for Structural Biology Education] *[http://scientist.by/index.php?option=com_content&view=category&layout=blog&id=3&Itemid=19 Scientist.By Community: Educational Webinars] *[http://questgarden.com/author/examplestop.php Questgarden] - webquest activities for a range of ages and study areas === U - X === *[http://archives.math.utk.edu/visual.calculus/ Visual Calculus] is a very good introduction to single variable calculus with explanations and exercises made with flash. *[https://visualfractions.com/ Visual Fractions] is an online reference - covering fraction and math calculators, percentages, unit conversions, and more. There is a Spanish Version as well. *[http://froggy.lbl.gov/ The Whole Frog Project] *[http://www.w3.org World Wide Web Consortium] *[http://www.wikieducator.org WikiEducator] *[http://www.wikihow.com WikiHow] *[http://vlib.org/ The WWW Virtual Library] *[http://www.w3schools.com W3 Schools On-line Web Tutorials] * The [http://demonstrations.wolfram.com Wolfram Demonstrations Project] is an open-code resource that uses dynamic computation to illuminate concepts in science, technology, mathematics, art, finance, and other fields. *[http://www.wonderhowto.com Wonder How To] *[https://worksheetgenius.com WorksheetGenius] hosts educational worksheet generators for homeschoolers, teachers or parents who want differentiated, randomized worksheets at the click of a button. *[http://www.x.org/ The Xorg Foundation] === Y - Z === * [http://www.youtube.com/edu YouTube EDU] - lectures, demonstrations, and other educational videos. See also [[YouTube]]. == See Also == * [[Open Educational Resources]] {{WikiversityUsers}} [[Category:Learning projects]] [[Category:Online learning resources]] [[Category:Hunter-gatherers project|*]] 38vigl0md716y5qyt7v63hmuc6lv1pj 100 2266 2814912 2474763 2026-06-09T19:47:22Z IanVG 2918363 /* Active participants */ 2814912 wikitext text/x-wiki <div style="text-align: center;"><big>'''Welcome to the School of Mathematics!'''</big></div> <div style="margin-top:-3px; margin-bottom:0.3em; text-align: center; font-size: 98%;"> <div style="text-align: center;">'''[[Portal:Science|FACULTY OF SCIENCES]]'''</div> <div style="text-align: center;">'''[[Portal:Physical Sciences|F<small>ACULTY OF</small> P<small>HYSICAL</SMALL> S<small>CIENCES</small>]] · [[Portal:Life Sciences|F<small>ACULTY OF</small> L<small>IFE</SMALL> S<small>CIENCES</small>]] · [[Portal:Engineering and Technology|F<small>ACULTY OF</small> E<small>NGINEERING AND </SMALL> T<small>ECHNOLOGY</small>]] · [[Portal:Mathematics|F<small>ACULTY OF</small> M<small>ATHEMATICS</SMALL>]]'''</div> [[School:Biology|Biology]] · [[School:Chemistry|Chemistry]] · [[School:Computer Science|Computer Science]] ·[[School:Economics|Economics]] · [[School:Mathematics|Mathematics]]· [[School:Physics and Astronomy|Physics and Astronomy]] · </div> <!---------------------------end of header----------------> [[Image:Lorenz_attractor.svg|right|300px|thumb|The Lorenz attractor is a chaotic map, noted for its butterfly shape. The map shows how the state of a dynamical system evolves over time in a complex, non-repeating pattern. The attractor itself, and the equations from which it is derived, were introduced by Edward Lorenz in 1963, who derived it from the simplified equations of convection rolls arising in the equations of the atmosphere.]] Since you're here, you either are someone wishing to share your knowledge of Mathematics, or you are someone who wishes to gain knowledge about Mathematics. If you are the first kind of person, have a look around, and see how you can contribute. If you are of the second kind, read on! Mathematics has many facets; though it has a wealth of applications, Mathematics is also a science, and an art, in its own right. Like other sciences, Mathematics is useful, but, just like other forms of art, Mathematics is beautiful, in its own unique way. You might not see this beauty at first, but the main goal of the Wikiversity School of Mathematics is to let you see this beauty, through inquiry and learning. To achieve this goal, we want to inspire you, since inspiration is essential to learning, discovery, and ultimately, seeing the beauty of Mathematics. A little bit of inspiration can put the beauty of Mathematics, and all its facets, before your eyes. The School of Mathematics is a work in progress. We are trying to organize our material into a logical order, create new material, and revise existing material. Although we try to keep clarity in mind, if any of the material is confusing, you don't know where to start, or you have other questions, ''do not hesitate to pose your question at [[School of Mathematics Help Desk|our help desk]]''. There is no such thing as a "dumb question"; learning is driven by questions! The School of Mathematics wishes you a very warm welcome. We hope that you will benefit from what we have to offer, and we hope that we will benefit from what you have to offer, either now or in the future. ==Divisions and departments== Divisions and Departments of the School exist on pages in "portal" name space. Start the name of departments with the "Portal:" prefix; departments reside in the [[Wikiversity:Namespaces|Portal: namespace]]. Departments and divisions link to learning materials and learning projects. Divisions can link subdivisions or to departments. For more information on schools, divisions and departments look at the [[Wikiversity:Naming_conventions|Naming Conventions]]. To add additional Divisions or Departments within the School of Mathematics, edit the list of "Portal:" namespaces on the [[Template:MathDeps|MathDeps Template]]. {{MathDeps}} ==Mathematics curriculum== This Wikiversity School of Mathematics project aims to create a structured set of links to learning material and test questions for the mathematics curriculum. To start, the recently released Australian national senior secondary school curriculum is being used as a guide. It is possible to create a similar structure based on the curriculum of alternative educational systems, but copy some of the same links and material identified below. If you'd like to do this, please create another heading in this section. Alternatively, please feel free to contribute to the project based on the Australian structure. This structure is fairly generic. ===Australian national senior secondary school mathematics curriculum=== These headings will provide links to pages detailing the senior secondary school mathematics curriculum. It will be used to identify material on Wikiversity and other Wikimedia sites that might be useful to students and their supporters. It will also help show the links between different areas of the curriculum and where studying mathematics might take them in the future; in other words, some possible answers to the eternal question, "why should I learn this?" and help show students the doors that will open to them by learning to see the world mathematically. The recently released [http://www.australiancurriculum.edu.au/SeniorSecondary/Mathematics/Specialist-Mathematics/Senior-secondary-Mathematics-subjects Australian national senior secondary school mathematics curriculum] will be used as a guide for this Wikiversity School of Mathematics project. (This material is based on the curriculum developed by the Australian Curriculum, Assessment and Reporting Authority (ACARA), which states its curriculum material can be downloaded, copied, printed and communicated for personal or non-commercial purposes, including educational or organisational use under a Creative Commons licence: [http://www.australiancurriculum.edu.au/Copyright copyright information]) *[[Essential mathematics | Essential mathematics]] *[[General mathematics | General mathematics]] *[[Mathematical methods | Mathematical methods]] *[[Specialist mathematics| Specialist mathematics]] ==School news and current events== * '''31 December 2018''' - [[Convergence acceleration by the Ford-Sidi W(m) algorithm]] - announced on Main Page News! * '''30 December 2018''' - [[Speak Math Now!/Week 1: Introduction To Algebra]] - announced on Main Page News! * '''19 December 2018''' - [[Vectors and coordinates]] - announced on Main Page News! * '''30 September 2018''' - [[Divergence (Gauss-Ostrogradsky) theorem]] - announced on Main Page News! * '''2 June 2018''' - [[WikiJournal of Science/Spaces in mathematics]] - announced on Main Page News! * '''20 January 2018''' - [[Mathematical Properties]] - announced on Main Page News! * '''11 January 2018''' - [[Boolean domain]] - announced on Main Page News! * '''6 November 2017''' - [[Legendre differential equation]] - announced on Main Page News! * '''31 August 2017''' - [[Minimal negation operator]] - announced on Main Page News! * '''15 July 2017''' - [[Line (Geometry)]] - announced on Main Page News! * '''1 July 2017''' - [[Unknown coordinate systems]] - announced on Main Page News! * '''16 June 2017''' - [[Multiple linear regression]] - announced on Main Page News! * '''31 May 2017''' - [[Full octahedral group]] - announced on Main Page News! * '''26 January 2017''' - [[Quadratic Equation]] - announced on Main Page News! * '''21 January 2017''' - [[Inversion (discrete mathematics)]] - announced on Main Page News! * '''19 January 2017''' - [[Permutation notation]] - announced on Main Page News! * '''January 2014''' - [[Mathematical astronomy]] - announced on Main Page News! * '''March 2008''' - [http://www.eurekalert.org/pub_releases/2008-03/icl-1yo030308.php 140-year-old math problem solved by researcher] * '''April 2007''' - [http://www.physlink.com/News/070327AnotherDimensionE8.cfm E8 mapped...] * '''20 August 2006''' - School founded! ==Undergraduate-level study== ===Study guides=== Not sure where to start? Consult the following guides to help you make a study plan and determine which prerequisites you might be missing. *[[School:Mathematics/Course guide|Course guide]] *[[School:Mathematics/Facultative courses|Facultative courses]] ===Course categories=== *[[School:Mathematics/Undergraduate/Applied Mathematics|Applied Mathematics]] <small>(mathematical methods, mathematical physics)</small> *[[School:Mathematics/Undergraduate/Pure Mathematics|Pure Mathematics]] <small>(foundations, algebra, analysis, calculus, discrete math)</small> *[[Probability and statistics|Probability and Statistics]] *[[Math contest preparation|Math contest preparation]] *[[School:Mathematics/Course Catalog|All Courses (Course Catalog)]] *[[Ethics/Nonkilling/Mathematics|Nonkilling Mathematics]]--would fit better under Ethics and Philosophy == Resources == ===Lectures=== * [[MacLaurin series]] * [[Draft:Mathematics|Mathematics]] {{MultiCol}} === Video courses === * [http://apcalculusstillwater.wordpress.com Stillwater AP Calculus Tutorial Videos ] * [http://ocw.mit.edu/courses/mathematics/ Open Course Ware] by MIT contains lectures for most of the topics in '''College level mathematics''' * [[wikipedia:NPTEL|NPTEL's]] [http://nptel.iitm.ac.in/courses.php?disciplineId=122 Basic courses] and [http://nptel.iitm.ac.in/courses.php?disciplineId=111 Mathematics] includes many lectures on subjects in mathematics. Recommended for '''Engineering mathematics''' * [http://www.extension.harvard.edu/courses/subject/mathematics Harvard University Extension School] offers lectures in many subjects. Recommended for '''Abstract Algebra'''[http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra]. * [http://www.khanacademy.org/ Khan Academy] which has a [http://www.youtube.com/user/khanacademy?blend=1&ob=4 youtube channel] with many easy to understand math videos from arithmetics to calculus and linear algebra. Recommended for '''School level mathematics''' * [http://www.chycho.com/?q=math_Table_Contents Chycho], has videos on Youtube organized by [http://www.kpss2011.net kpss]subject. Author uses big chalk and street walls for presentation of material. * [http://www.youtube.com/user/MathTV MathTV's Youtube channel] * [http://www.mathtv.com/ MathTV's] website with the material organized by subject. They say they may start charging, but the videos will still be accessible on Youtube. * [http://www.isallaboutmath.com/ Is all about Math] video podcast with animations explaining math. * [http://www.patrickjmt.com/ Patrick Just Math Tutorials] patrickjmt.com is a math only website with extensive free math content; he also has a youtube channel, 'patrickjmt.' * [http://www.youtube.com/user/Technion/videos?sort=dd&view=1&shelf_id=5] the Technion YouTube playlist channel contains lectures for many university level subjects, mostly in Hebrew. === Wikibooks === * {{bi|Mathematics|Browse Textbooks}} * {{bi|Mathematics bookshelf}} * {{bi|Beginning Mathematics}} * {{bi|High School Mathematics Extensions}} * {{bi|Geometry}} * {{bi|Algebra}} * {{bi|Trigonometry}} * {{bi|Calculus}} * {{bi|Applied Math Basics}} * {{bi|Differential Equations}} * {{bi|Applied Mathematics}} * {{bi|Cryptography}} * {{bi|Discrete mathematics}} * {{bi|Complex Analysis}} * {{bi|A-level Mathematics}} {{ColBreak}} ===Wikipedia=== * {{wi|Mathematics}} * {{wi|Areas of mathematics}} * {{wi|List of topics in mathematics|List of Mathematics Topics}} * {{wi|List of mathematical symbols|List of Mathematical Symbols}} * {{wi|List of online encyclopedias#Mathematics|List of Mathematics Encyclopedias}} ===Free Software=== * [http://www.gnu.org/software/octave/ GNU Octave] - A computational program similar to (and mostly compatible with) MATLAB * [http://www.scipy.org/ SciPy] - A [[w:Python|Python]] library for mathematical and scientific computing * [http://maxima.sourceforge.net/ Maxima] - A computer algebra system * [http://www.singular.uni-kl.de/ SINGULAR] - A computer algebra system * [http://www.gap-system.org/ GAP] - A tool for computational group theory * [http://www.r-project.org/ R] - A programming language designed for statistical computing * [http://freemat.sourceforge.net/ FreeMat] - An open source Matlab clone * [http://www.sagemath.org SAGE] - A packaging of many of the above software tools and more * [http://pari.math.u-bordeaux.fr/ Pari/GP] - "a computer algebra system with the main aim of facilitating number theory computations" (from Wikipedia) * [http://www.mathics.org/ Mathics] Mathics is a free, general-purpose online computer algebra system featuring Mathematica-compatible syntax and functions. * See a full list along with some commercial tools at our [[Mathematics software|mathematics software]] page. ===Other=== *[[Other Free Learning Resources/Mathematics]] {{EndMultiCol}} ==External links== * [http://aimath.org/textbooks/ Open Textbook Initiative] - mix of free-licensed and low-cost, non-free-licensed mathematics class-tested textbooks supported by the [[w:American Institute of Mathematics|American Institute of Mathematics]] * [https://openstax.org/subjects/math OpenStax] - free textbooks from algebra, trigonometry to calculus and statistics * [https://mathvault.ca/hub/higher-math/math-symbols/ Comprehensive list of mathematical symbols] * [https://tutorial.math.lamar.edu/ Paul's Online Notes] - standalone resources on algebra, calculus I, calculus II, calculus III and differential equations * [http://www.mathabc.com/ Free online math practice with instructional videos for K-6.] * [http://mathematicslibrary.blogspot.com/ Mathematics notes and books for undergraduates and graduate students.] * [http://mum.edu/math_dept/courses.html An interesting perspective on mathematics.] * [http://www.sciencenews.org/view/generic/id/31392/title/Still_debating_with_Plato 'are new mathematical truths discovered or invented?'] * [http://www.aplusclick.com/ Collection of Math Problems & Answers adapted to the student ability] * [https://mc2.bid4papers.com/blog/math-resources/ MC2 Math Links] - a collection of 486 free online Math tools: calculators, plotters and interactive visualizations. ==Active participants== The histories of Wikiversity pages indicate who are the active participants (listed below). If you are an active participant in this school, you can list your name here (this can help small schools grow and the participants communicate better; for large schools it is not needed). * [[User:IanVG]]--[[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 19:47, 9 June 2026 (UTC) ===Inactive participants=== {{MultiCol}} *[[User:Marshallsumter|Marshallsumter]] Was active beginning 7 January 2012 with Mathematical astronomy *[[User:Parthkohli99|Parth Kohli]] (Set Theory, Calculus, Algebra, Trigonometry, Logic, Applied Mathematics) *[[User:TortoiseDream|TortoiseDream]] (Logic, Set Theory, Mathematical Physics) *[[User:Tastymangojuice|Tastymangojuice]] (Computer Science) *[[User:ehremo|ehremo]] *[[User:Deltinu|Deltinu]] *[[User:EulerGamma|EulerGamma]] *[[User:Ravi12346|Ravi12346]] *[[User:lgallindo|Lucas Gallindo]] *[[User:Pgbridge2000|Pgbridge2000]] (Cryptography) *[[User:Tachyon01|Tachyon01]] (Pure Mathematics) *[[User:fidocancan|fidocancan]] (pre-uni teacher) *[[User:Mo Anabre|Mo Anabre]] *[[User:master.signal|sutton]] *[[User:Drini|Drini]] *[[User:Ananth nag|Ananth nag m]] *[[User:TeapotScar|TeapotScar]] (Calculus, Number Theory, Topology) *[[User:pbran7|pbran7]] *[[User:Juan Marquez|Juan]] (Topology, Algebra, Geometry and Analysis) *[[User: Ivanne|Ivanne]] (I just want to learn man) {{ColBreak}} *[[User:King Bee|King Bee]] (graduate student in pure mathematics) *[[User:Verin Sedai|Verin Sedai]] (Algebra, Combinatorics, basic number theory) *[[User:haresh kumar|haresh kumar]](analytic geometry,calculus) *[[User:A.K.Karthikeyan|A.K.Karthikeyan]](True knowledge lies in knowing you know nothing) *[[User:Tlep|Tlep]] (Algebra) *[[User:Bigyan Shrestha|Bigyan Shrestha]] (Mathematical Statistics) *[[User:Rogerh|Rogerh]] *[[user:sarinmadhur|Madhur Sarin]] (Number system,Geometry,Calculus ) *[[User:AFriedman|AFriedman]] (Computer Science, Biology) *[[User:fibonacci101]] (Geometry, number theory, set theory) *[[User:mathclubowner]] *[[User:Helloworld00|Helloworld00]] *[[User:graie.project|graie]] *[[user:mynameinc|mynameinc]] (Algebra, Trigonometry, Differential Calculus) *[[user:Comrade1986|Comrade1986]] (Mathematics) *[[user:WendyB]] (Mathematics student) *[[user:Alchemy heels I]] (Mathematics student) *[[John Michael Pinson I ]] (Numerology, Mathematics student) * [[User:Tarun-Science|Tarun-Science]] (I;m a 15-year-old intersted in Physics, Mathematics and Chemistry..good in geometry and algebra) {{EndMultiCol}} == International relations == *[[:it:Wikiversità]] - [[:it:Wikiversità:Facoltà di Matematica|Facoltà di Matematica]] *[[b:pt:Wikiversidade:_Ci%C3%AAncias_Exatas_e_Ci%C3%AAncias_da_Terra#Curso_de_Matem.C3.A1tica||Wikiversidade - Ciências Exatas e da Terra]] <!-- footer templates --> {{Mathematics resources}} <!-- categories --> [[Category:Abstraction]] [[Category:Mathematics/Lectures]] [[Category:Probability]] [[Category:Wikiversity schools]] [[Category:Resources last modified in March 2020]] [[Category:Statistics]] [[Category:School of Mathematics| ]] mss45xqwr9tedxs7yreglfg8okcqsvt 2814914 2814912 2026-06-09T19:50:05Z IanVG 2918363 /* Wikibooks */ 2814914 wikitext text/x-wiki <div style="text-align: center;"><big>'''Welcome to the School of Mathematics!'''</big></div> <div style="margin-top:-3px; margin-bottom:0.3em; text-align: center; font-size: 98%;"> <div style="text-align: center;">'''[[Portal:Science|FACULTY OF SCIENCES]]'''</div> <div style="text-align: center;">'''[[Portal:Physical Sciences|F<small>ACULTY OF</small> P<small>HYSICAL</SMALL> S<small>CIENCES</small>]] · [[Portal:Life Sciences|F<small>ACULTY OF</small> L<small>IFE</SMALL> S<small>CIENCES</small>]] · [[Portal:Engineering and Technology|F<small>ACULTY OF</small> E<small>NGINEERING AND </SMALL> T<small>ECHNOLOGY</small>]] · [[Portal:Mathematics|F<small>ACULTY OF</small> M<small>ATHEMATICS</SMALL>]]'''</div> [[School:Biology|Biology]] · [[School:Chemistry|Chemistry]] · [[School:Computer Science|Computer Science]] ·[[School:Economics|Economics]] · [[School:Mathematics|Mathematics]]· [[School:Physics and Astronomy|Physics and Astronomy]] · </div> <!---------------------------end of header----------------> [[Image:Lorenz_attractor.svg|right|300px|thumb|The Lorenz attractor is a chaotic map, noted for its butterfly shape. The map shows how the state of a dynamical system evolves over time in a complex, non-repeating pattern. The attractor itself, and the equations from which it is derived, were introduced by Edward Lorenz in 1963, who derived it from the simplified equations of convection rolls arising in the equations of the atmosphere.]] Since you're here, you either are someone wishing to share your knowledge of Mathematics, or you are someone who wishes to gain knowledge about Mathematics. If you are the first kind of person, have a look around, and see how you can contribute. If you are of the second kind, read on! Mathematics has many facets; though it has a wealth of applications, Mathematics is also a science, and an art, in its own right. Like other sciences, Mathematics is useful, but, just like other forms of art, Mathematics is beautiful, in its own unique way. You might not see this beauty at first, but the main goal of the Wikiversity School of Mathematics is to let you see this beauty, through inquiry and learning. To achieve this goal, we want to inspire you, since inspiration is essential to learning, discovery, and ultimately, seeing the beauty of Mathematics. A little bit of inspiration can put the beauty of Mathematics, and all its facets, before your eyes. The School of Mathematics is a work in progress. We are trying to organize our material into a logical order, create new material, and revise existing material. Although we try to keep clarity in mind, if any of the material is confusing, you don't know where to start, or you have other questions, ''do not hesitate to pose your question at [[School of Mathematics Help Desk|our help desk]]''. There is no such thing as a "dumb question"; learning is driven by questions! The School of Mathematics wishes you a very warm welcome. We hope that you will benefit from what we have to offer, and we hope that we will benefit from what you have to offer, either now or in the future. ==Divisions and departments== Divisions and Departments of the School exist on pages in "portal" name space. Start the name of departments with the "Portal:" prefix; departments reside in the [[Wikiversity:Namespaces|Portal: namespace]]. Departments and divisions link to learning materials and learning projects. Divisions can link subdivisions or to departments. For more information on schools, divisions and departments look at the [[Wikiversity:Naming_conventions|Naming Conventions]]. To add additional Divisions or Departments within the School of Mathematics, edit the list of "Portal:" namespaces on the [[Template:MathDeps|MathDeps Template]]. {{MathDeps}} ==Mathematics curriculum== This Wikiversity School of Mathematics project aims to create a structured set of links to learning material and test questions for the mathematics curriculum. To start, the recently released Australian national senior secondary school curriculum is being used as a guide. It is possible to create a similar structure based on the curriculum of alternative educational systems, but copy some of the same links and material identified below. If you'd like to do this, please create another heading in this section. Alternatively, please feel free to contribute to the project based on the Australian structure. This structure is fairly generic. ===Australian national senior secondary school mathematics curriculum=== These headings will provide links to pages detailing the senior secondary school mathematics curriculum. It will be used to identify material on Wikiversity and other Wikimedia sites that might be useful to students and their supporters. It will also help show the links between different areas of the curriculum and where studying mathematics might take them in the future; in other words, some possible answers to the eternal question, "why should I learn this?" and help show students the doors that will open to them by learning to see the world mathematically. The recently released [http://www.australiancurriculum.edu.au/SeniorSecondary/Mathematics/Specialist-Mathematics/Senior-secondary-Mathematics-subjects Australian national senior secondary school mathematics curriculum] will be used as a guide for this Wikiversity School of Mathematics project. (This material is based on the curriculum developed by the Australian Curriculum, Assessment and Reporting Authority (ACARA), which states its curriculum material can be downloaded, copied, printed and communicated for personal or non-commercial purposes, including educational or organisational use under a Creative Commons licence: [http://www.australiancurriculum.edu.au/Copyright copyright information]) *[[Essential mathematics | Essential mathematics]] *[[General mathematics | General mathematics]] *[[Mathematical methods | Mathematical methods]] *[[Specialist mathematics| Specialist mathematics]] ==School news and current events== * '''31 December 2018''' - [[Convergence acceleration by the Ford-Sidi W(m) algorithm]] - announced on Main Page News! * '''30 December 2018''' - [[Speak Math Now!/Week 1: Introduction To Algebra]] - announced on Main Page News! * '''19 December 2018''' - [[Vectors and coordinates]] - announced on Main Page News! * '''30 September 2018''' - [[Divergence (Gauss-Ostrogradsky) theorem]] - announced on Main Page News! * '''2 June 2018''' - [[WikiJournal of Science/Spaces in mathematics]] - announced on Main Page News! * '''20 January 2018''' - [[Mathematical Properties]] - announced on Main Page News! * '''11 January 2018''' - [[Boolean domain]] - announced on Main Page News! * '''6 November 2017''' - [[Legendre differential equation]] - announced on Main Page News! * '''31 August 2017''' - [[Minimal negation operator]] - announced on Main Page News! * '''15 July 2017''' - [[Line (Geometry)]] - announced on Main Page News! * '''1 July 2017''' - [[Unknown coordinate systems]] - announced on Main Page News! * '''16 June 2017''' - [[Multiple linear regression]] - announced on Main Page News! * '''31 May 2017''' - [[Full octahedral group]] - announced on Main Page News! * '''26 January 2017''' - [[Quadratic Equation]] - announced on Main Page News! * '''21 January 2017''' - [[Inversion (discrete mathematics)]] - announced on Main Page News! * '''19 January 2017''' - [[Permutation notation]] - announced on Main Page News! * '''January 2014''' - [[Mathematical astronomy]] - announced on Main Page News! * '''March 2008''' - [http://www.eurekalert.org/pub_releases/2008-03/icl-1yo030308.php 140-year-old math problem solved by researcher] * '''April 2007''' - [http://www.physlink.com/News/070327AnotherDimensionE8.cfm E8 mapped...] * '''20 August 2006''' - School founded! ==Undergraduate-level study== ===Study guides=== Not sure where to start? Consult the following guides to help you make a study plan and determine which prerequisites you might be missing. *[[School:Mathematics/Course guide|Course guide]] *[[School:Mathematics/Facultative courses|Facultative courses]] ===Course categories=== *[[School:Mathematics/Undergraduate/Applied Mathematics|Applied Mathematics]] <small>(mathematical methods, mathematical physics)</small> *[[School:Mathematics/Undergraduate/Pure Mathematics|Pure Mathematics]] <small>(foundations, algebra, analysis, calculus, discrete math)</small> *[[Probability and statistics|Probability and Statistics]] *[[Math contest preparation|Math contest preparation]] *[[School:Mathematics/Course Catalog|All Courses (Course Catalog)]] *[[Ethics/Nonkilling/Mathematics|Nonkilling Mathematics]]--would fit better under Ethics and Philosophy == Resources == ===Lectures=== * [[MacLaurin series]] * [[Draft:Mathematics|Mathematics]] {{MultiCol}} === Video courses === * [http://apcalculusstillwater.wordpress.com Stillwater AP Calculus Tutorial Videos ] * [http://ocw.mit.edu/courses/mathematics/ Open Course Ware] by MIT contains lectures for most of the topics in '''College level mathematics''' * [[wikipedia:NPTEL|NPTEL's]] [http://nptel.iitm.ac.in/courses.php?disciplineId=122 Basic courses] and [http://nptel.iitm.ac.in/courses.php?disciplineId=111 Mathematics] includes many lectures on subjects in mathematics. Recommended for '''Engineering mathematics''' * [http://www.extension.harvard.edu/courses/subject/mathematics Harvard University Extension School] offers lectures in many subjects. Recommended for '''Abstract Algebra'''[http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra]. * [http://www.khanacademy.org/ Khan Academy] which has a [http://www.youtube.com/user/khanacademy?blend=1&ob=4 youtube channel] with many easy to understand math videos from arithmetics to calculus and linear algebra. Recommended for '''School level mathematics''' * [http://www.chycho.com/?q=math_Table_Contents Chycho], has videos on Youtube organized by [http://www.kpss2011.net kpss]subject. Author uses big chalk and street walls for presentation of material. * [http://www.youtube.com/user/MathTV MathTV's Youtube channel] * [http://www.mathtv.com/ MathTV's] website with the material organized by subject. They say they may start charging, but the videos will still be accessible on Youtube. * [http://www.isallaboutmath.com/ Is all about Math] video podcast with animations explaining math. * [http://www.patrickjmt.com/ Patrick Just Math Tutorials] patrickjmt.com is a math only website with extensive free math content; he also has a youtube channel, 'patrickjmt.' * [http://www.youtube.com/user/Technion/videos?sort=dd&view=1&shelf_id=5] the Technion YouTube playlist channel contains lectures for many university level subjects, mostly in Hebrew. === Wikibooks === * {{bi|Mathematics|Browse Textbooks}} * {{bi|Mathematics bookshelf}} * {{bi|Beginning Mathematics}} * {{bi|High School Mathematics Extensions}} * {{bi|Geometry}} * {{bi|Algebra}} * {{bi|Trigonometry}} * {{bi|Calculus}} * {{bi|Applied Math Basics}} * {{bi|Ordinary Differential Equations}} * {{bi|Applied Mathematics}} * {{bi|Cryptography}} * {{bi|Discrete mathematics}} * {{bi|Complex Analysis}} * {{bi|A-level Mathematics}} {{ColBreak}} ===Wikipedia=== * {{wi|Mathematics}} * {{wi|Areas of mathematics}} * {{wi|List of topics in mathematics|List of Mathematics Topics}} * {{wi|List of mathematical symbols|List of Mathematical Symbols}} * {{wi|List of online encyclopedias#Mathematics|List of Mathematics Encyclopedias}} ===Free Software=== * [http://www.gnu.org/software/octave/ GNU Octave] - A computational program similar to (and mostly compatible with) MATLAB * [http://www.scipy.org/ SciPy] - A [[w:Python|Python]] library for mathematical and scientific computing * [http://maxima.sourceforge.net/ Maxima] - A computer algebra system * [http://www.singular.uni-kl.de/ SINGULAR] - A computer algebra system * [http://www.gap-system.org/ GAP] - A tool for computational group theory * [http://www.r-project.org/ R] - A programming language designed for statistical computing * [http://freemat.sourceforge.net/ FreeMat] - An open source Matlab clone * [http://www.sagemath.org SAGE] - A packaging of many of the above software tools and more * [http://pari.math.u-bordeaux.fr/ Pari/GP] - "a computer algebra system with the main aim of facilitating number theory computations" (from Wikipedia) * [http://www.mathics.org/ Mathics] Mathics is a free, general-purpose online computer algebra system featuring Mathematica-compatible syntax and functions. * See a full list along with some commercial tools at our [[Mathematics software|mathematics software]] page. ===Other=== *[[Other Free Learning Resources/Mathematics]] {{EndMultiCol}} ==External links== * [http://aimath.org/textbooks/ Open Textbook Initiative] - mix of free-licensed and low-cost, non-free-licensed mathematics class-tested textbooks supported by the [[w:American Institute of Mathematics|American Institute of Mathematics]] * [https://openstax.org/subjects/math OpenStax] - free textbooks from algebra, trigonometry to calculus and statistics * [https://mathvault.ca/hub/higher-math/math-symbols/ Comprehensive list of mathematical symbols] * [https://tutorial.math.lamar.edu/ Paul's Online Notes] - standalone resources on algebra, calculus I, calculus II, calculus III and differential equations * [http://www.mathabc.com/ Free online math practice with instructional videos for K-6.] * [http://mathematicslibrary.blogspot.com/ Mathematics notes and books for undergraduates and graduate students.] * [http://mum.edu/math_dept/courses.html An interesting perspective on mathematics.] * [http://www.sciencenews.org/view/generic/id/31392/title/Still_debating_with_Plato 'are new mathematical truths discovered or invented?'] * [http://www.aplusclick.com/ Collection of Math Problems & Answers adapted to the student ability] * [https://mc2.bid4papers.com/blog/math-resources/ MC2 Math Links] - a collection of 486 free online Math tools: calculators, plotters and interactive visualizations. ==Active participants== The histories of Wikiversity pages indicate who are the active participants (listed below). If you are an active participant in this school, you can list your name here (this can help small schools grow and the participants communicate better; for large schools it is not needed). * [[User:IanVG]]--[[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 19:47, 9 June 2026 (UTC) ===Inactive participants=== {{MultiCol}} *[[User:Marshallsumter|Marshallsumter]] Was active beginning 7 January 2012 with Mathematical astronomy *[[User:Parthkohli99|Parth Kohli]] (Set Theory, Calculus, Algebra, Trigonometry, Logic, Applied Mathematics) *[[User:TortoiseDream|TortoiseDream]] (Logic, Set Theory, Mathematical Physics) *[[User:Tastymangojuice|Tastymangojuice]] (Computer Science) *[[User:ehremo|ehremo]] *[[User:Deltinu|Deltinu]] *[[User:EulerGamma|EulerGamma]] *[[User:Ravi12346|Ravi12346]] *[[User:lgallindo|Lucas Gallindo]] *[[User:Pgbridge2000|Pgbridge2000]] (Cryptography) *[[User:Tachyon01|Tachyon01]] (Pure Mathematics) *[[User:fidocancan|fidocancan]] (pre-uni teacher) *[[User:Mo Anabre|Mo Anabre]] *[[User:master.signal|sutton]] *[[User:Drini|Drini]] *[[User:Ananth nag|Ananth nag m]] *[[User:TeapotScar|TeapotScar]] (Calculus, Number Theory, Topology) *[[User:pbran7|pbran7]] *[[User:Juan Marquez|Juan]] (Topology, Algebra, Geometry and Analysis) *[[User: Ivanne|Ivanne]] (I just want to learn man) {{ColBreak}} *[[User:King Bee|King Bee]] (graduate student in pure mathematics) *[[User:Verin Sedai|Verin Sedai]] (Algebra, Combinatorics, basic number theory) *[[User:haresh kumar|haresh kumar]](analytic geometry,calculus) *[[User:A.K.Karthikeyan|A.K.Karthikeyan]](True knowledge lies in knowing you know nothing) *[[User:Tlep|Tlep]] (Algebra) *[[User:Bigyan Shrestha|Bigyan Shrestha]] (Mathematical Statistics) *[[User:Rogerh|Rogerh]] *[[user:sarinmadhur|Madhur Sarin]] (Number system,Geometry,Calculus ) *[[User:AFriedman|AFriedman]] (Computer Science, Biology) *[[User:fibonacci101]] (Geometry, number theory, set theory) *[[User:mathclubowner]] *[[User:Helloworld00|Helloworld00]] *[[User:graie.project|graie]] *[[user:mynameinc|mynameinc]] (Algebra, Trigonometry, Differential Calculus) *[[user:Comrade1986|Comrade1986]] (Mathematics) *[[user:WendyB]] (Mathematics student) *[[user:Alchemy heels I]] (Mathematics student) *[[John Michael Pinson I ]] (Numerology, Mathematics student) * [[User:Tarun-Science|Tarun-Science]] (I;m a 15-year-old intersted in Physics, Mathematics and Chemistry..good in geometry and algebra) {{EndMultiCol}} == International relations == *[[:it:Wikiversità]] - [[:it:Wikiversità:Facoltà di Matematica|Facoltà di Matematica]] *[[b:pt:Wikiversidade:_Ci%C3%AAncias_Exatas_e_Ci%C3%AAncias_da_Terra#Curso_de_Matem.C3.A1tica||Wikiversidade - Ciências Exatas e da Terra]] <!-- footer templates --> {{Mathematics resources}} <!-- categories --> [[Category:Abstraction]] [[Category:Mathematics/Lectures]] [[Category:Probability]] [[Category:Wikiversity schools]] [[Category:Resources last modified in March 2020]] [[Category:Statistics]] [[Category:School of Mathematics| ]] 2giafv4bohfil8c36u6oxzhlvjk2bf5 2814916 2814914 2026-06-09T19:52:32Z IanVG 2918363 /* Wikibooks */ 2814916 wikitext text/x-wiki <div style="text-align: center;"><big>'''Welcome to the School of Mathematics!'''</big></div> <div style="margin-top:-3px; margin-bottom:0.3em; text-align: center; font-size: 98%;"> <div style="text-align: center;">'''[[Portal:Science|FACULTY OF SCIENCES]]'''</div> <div style="text-align: center;">'''[[Portal:Physical Sciences|F<small>ACULTY OF</small> P<small>HYSICAL</SMALL> S<small>CIENCES</small>]] · [[Portal:Life Sciences|F<small>ACULTY OF</small> L<small>IFE</SMALL> S<small>CIENCES</small>]] · [[Portal:Engineering and Technology|F<small>ACULTY OF</small> E<small>NGINEERING AND </SMALL> T<small>ECHNOLOGY</small>]] · [[Portal:Mathematics|F<small>ACULTY OF</small> M<small>ATHEMATICS</SMALL>]]'''</div> [[School:Biology|Biology]] · [[School:Chemistry|Chemistry]] · [[School:Computer Science|Computer Science]] ·[[School:Economics|Economics]] · [[School:Mathematics|Mathematics]]· [[School:Physics and Astronomy|Physics and Astronomy]] · </div> <!---------------------------end of header----------------> [[Image:Lorenz_attractor.svg|right|300px|thumb|The Lorenz attractor is a chaotic map, noted for its butterfly shape. The map shows how the state of a dynamical system evolves over time in a complex, non-repeating pattern. The attractor itself, and the equations from which it is derived, were introduced by Edward Lorenz in 1963, who derived it from the simplified equations of convection rolls arising in the equations of the atmosphere.]] Since you're here, you either are someone wishing to share your knowledge of Mathematics, or you are someone who wishes to gain knowledge about Mathematics. If you are the first kind of person, have a look around, and see how you can contribute. If you are of the second kind, read on! Mathematics has many facets; though it has a wealth of applications, Mathematics is also a science, and an art, in its own right. Like other sciences, Mathematics is useful, but, just like other forms of art, Mathematics is beautiful, in its own unique way. You might not see this beauty at first, but the main goal of the Wikiversity School of Mathematics is to let you see this beauty, through inquiry and learning. To achieve this goal, we want to inspire you, since inspiration is essential to learning, discovery, and ultimately, seeing the beauty of Mathematics. A little bit of inspiration can put the beauty of Mathematics, and all its facets, before your eyes. The School of Mathematics is a work in progress. We are trying to organize our material into a logical order, create new material, and revise existing material. Although we try to keep clarity in mind, if any of the material is confusing, you don't know where to start, or you have other questions, ''do not hesitate to pose your question at [[School of Mathematics Help Desk|our help desk]]''. There is no such thing as a "dumb question"; learning is driven by questions! The School of Mathematics wishes you a very warm welcome. We hope that you will benefit from what we have to offer, and we hope that we will benefit from what you have to offer, either now or in the future. ==Divisions and departments== Divisions and Departments of the School exist on pages in "portal" name space. Start the name of departments with the "Portal:" prefix; departments reside in the [[Wikiversity:Namespaces|Portal: namespace]]. Departments and divisions link to learning materials and learning projects. Divisions can link subdivisions or to departments. For more information on schools, divisions and departments look at the [[Wikiversity:Naming_conventions|Naming Conventions]]. To add additional Divisions or Departments within the School of Mathematics, edit the list of "Portal:" namespaces on the [[Template:MathDeps|MathDeps Template]]. {{MathDeps}} ==Mathematics curriculum== This Wikiversity School of Mathematics project aims to create a structured set of links to learning material and test questions for the mathematics curriculum. To start, the recently released Australian national senior secondary school curriculum is being used as a guide. It is possible to create a similar structure based on the curriculum of alternative educational systems, but copy some of the same links and material identified below. If you'd like to do this, please create another heading in this section. Alternatively, please feel free to contribute to the project based on the Australian structure. This structure is fairly generic. ===Australian national senior secondary school mathematics curriculum=== These headings will provide links to pages detailing the senior secondary school mathematics curriculum. It will be used to identify material on Wikiversity and other Wikimedia sites that might be useful to students and their supporters. It will also help show the links between different areas of the curriculum and where studying mathematics might take them in the future; in other words, some possible answers to the eternal question, "why should I learn this?" and help show students the doors that will open to them by learning to see the world mathematically. The recently released [http://www.australiancurriculum.edu.au/SeniorSecondary/Mathematics/Specialist-Mathematics/Senior-secondary-Mathematics-subjects Australian national senior secondary school mathematics curriculum] will be used as a guide for this Wikiversity School of Mathematics project. (This material is based on the curriculum developed by the Australian Curriculum, Assessment and Reporting Authority (ACARA), which states its curriculum material can be downloaded, copied, printed and communicated for personal or non-commercial purposes, including educational or organisational use under a Creative Commons licence: [http://www.australiancurriculum.edu.au/Copyright copyright information]) *[[Essential mathematics | Essential mathematics]] *[[General mathematics | General mathematics]] *[[Mathematical methods | Mathematical methods]] *[[Specialist mathematics| Specialist mathematics]] ==School news and current events== * '''31 December 2018''' - [[Convergence acceleration by the Ford-Sidi W(m) algorithm]] - announced on Main Page News! * '''30 December 2018''' - [[Speak Math Now!/Week 1: Introduction To Algebra]] - announced on Main Page News! * '''19 December 2018''' - [[Vectors and coordinates]] - announced on Main Page News! * '''30 September 2018''' - [[Divergence (Gauss-Ostrogradsky) theorem]] - announced on Main Page News! * '''2 June 2018''' - [[WikiJournal of Science/Spaces in mathematics]] - announced on Main Page News! * '''20 January 2018''' - [[Mathematical Properties]] - announced on Main Page News! * '''11 January 2018''' - [[Boolean domain]] - announced on Main Page News! * '''6 November 2017''' - [[Legendre differential equation]] - announced on Main Page News! * '''31 August 2017''' - [[Minimal negation operator]] - announced on Main Page News! * '''15 July 2017''' - [[Line (Geometry)]] - announced on Main Page News! * '''1 July 2017''' - [[Unknown coordinate systems]] - announced on Main Page News! * '''16 June 2017''' - [[Multiple linear regression]] - announced on Main Page News! * '''31 May 2017''' - [[Full octahedral group]] - announced on Main Page News! * '''26 January 2017''' - [[Quadratic Equation]] - announced on Main Page News! * '''21 January 2017''' - [[Inversion (discrete mathematics)]] - announced on Main Page News! * '''19 January 2017''' - [[Permutation notation]] - announced on Main Page News! * '''January 2014''' - [[Mathematical astronomy]] - announced on Main Page News! * '''March 2008''' - [http://www.eurekalert.org/pub_releases/2008-03/icl-1yo030308.php 140-year-old math problem solved by researcher] * '''April 2007''' - [http://www.physlink.com/News/070327AnotherDimensionE8.cfm E8 mapped...] * '''20 August 2006''' - School founded! ==Undergraduate-level study== ===Study guides=== Not sure where to start? Consult the following guides to help you make a study plan and determine which prerequisites you might be missing. *[[School:Mathematics/Course guide|Course guide]] *[[School:Mathematics/Facultative courses|Facultative courses]] ===Course categories=== *[[School:Mathematics/Undergraduate/Applied Mathematics|Applied Mathematics]] <small>(mathematical methods, mathematical physics)</small> *[[School:Mathematics/Undergraduate/Pure Mathematics|Pure Mathematics]] <small>(foundations, algebra, analysis, calculus, discrete math)</small> *[[Probability and statistics|Probability and Statistics]] *[[Math contest preparation|Math contest preparation]] *[[School:Mathematics/Course Catalog|All Courses (Course Catalog)]] *[[Ethics/Nonkilling/Mathematics|Nonkilling Mathematics]]--would fit better under Ethics and Philosophy == Resources == ===Lectures=== * [[MacLaurin series]] * [[Draft:Mathematics|Mathematics]] {{MultiCol}} === Video courses === * [http://apcalculusstillwater.wordpress.com Stillwater AP Calculus Tutorial Videos ] * [http://ocw.mit.edu/courses/mathematics/ Open Course Ware] by MIT contains lectures for most of the topics in '''College level mathematics''' * [[wikipedia:NPTEL|NPTEL's]] [http://nptel.iitm.ac.in/courses.php?disciplineId=122 Basic courses] and [http://nptel.iitm.ac.in/courses.php?disciplineId=111 Mathematics] includes many lectures on subjects in mathematics. Recommended for '''Engineering mathematics''' * [http://www.extension.harvard.edu/courses/subject/mathematics Harvard University Extension School] offers lectures in many subjects. Recommended for '''Abstract Algebra'''[http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra]. * [http://www.khanacademy.org/ Khan Academy] which has a [http://www.youtube.com/user/khanacademy?blend=1&ob=4 youtube channel] with many easy to understand math videos from arithmetics to calculus and linear algebra. Recommended for '''School level mathematics''' * [http://www.chycho.com/?q=math_Table_Contents Chycho], has videos on Youtube organized by [http://www.kpss2011.net kpss]subject. Author uses big chalk and street walls for presentation of material. * [http://www.youtube.com/user/MathTV MathTV's Youtube channel] * [http://www.mathtv.com/ MathTV's] website with the material organized by subject. They say they may start charging, but the videos will still be accessible on Youtube. * [http://www.isallaboutmath.com/ Is all about Math] video podcast with animations explaining math. * [http://www.patrickjmt.com/ Patrick Just Math Tutorials] patrickjmt.com is a math only website with extensive free math content; he also has a youtube channel, 'patrickjmt.' * [http://www.youtube.com/user/Technion/videos?sort=dd&view=1&shelf_id=5] the Technion YouTube playlist channel contains lectures for many university level subjects, mostly in Hebrew. === Wikibooks === * {{bi|Mathematics|Browse Textbooks}} * {{bi|Mathematics bookshelf}} * {{bi|Beginning Mathematics}} * {{bi|High School Mathematics Extensions}} * {{bi|Geometry}} * {{bi|Algebra}} * {{bi|Introductory Linear Algebra}} * {{bi|Linear Algebra}} * {{bi|Trigonometry}} * {{bi|Calculus}} * {{bi|Applied Math Basics}} * {{bi|Ordinary Differential Equations}} * {{bi|Partial Differential Equations}} * {{bi|Linear Algebra with Differential Equations}} * {{bi|Applied Mathematics}} * {{bi|Numerical Methods}} * {{bi|Cryptography}} * {{bi|Discrete mathematics}} * {{bi|Complex Analysis}} * {{bi|A-level Mathematics}} {{ColBreak}} ===Wikipedia=== * {{wi|Mathematics}} * {{wi|Areas of mathematics}} * {{wi|List of topics in mathematics|List of Mathematics Topics}} * {{wi|List of mathematical symbols|List of Mathematical Symbols}} * {{wi|List of online encyclopedias#Mathematics|List of Mathematics Encyclopedias}} ===Free Software=== * [http://www.gnu.org/software/octave/ GNU Octave] - A computational program similar to (and mostly compatible with) MATLAB * [http://www.scipy.org/ SciPy] - A [[w:Python|Python]] library for mathematical and scientific computing * [http://maxima.sourceforge.net/ Maxima] - A computer algebra system * [http://www.singular.uni-kl.de/ SINGULAR] - A computer algebra system * [http://www.gap-system.org/ GAP] - A tool for computational group theory * [http://www.r-project.org/ R] - A programming language designed for statistical computing * [http://freemat.sourceforge.net/ FreeMat] - An open source Matlab clone * [http://www.sagemath.org SAGE] - A packaging of many of the above software tools and more * [http://pari.math.u-bordeaux.fr/ Pari/GP] - "a computer algebra system with the main aim of facilitating number theory computations" (from Wikipedia) * [http://www.mathics.org/ Mathics] Mathics is a free, general-purpose online computer algebra system featuring Mathematica-compatible syntax and functions. * See a full list along with some commercial tools at our [[Mathematics software|mathematics software]] page. ===Other=== *[[Other Free Learning Resources/Mathematics]] {{EndMultiCol}} ==External links== * [http://aimath.org/textbooks/ Open Textbook Initiative] - mix of free-licensed and low-cost, non-free-licensed mathematics class-tested textbooks supported by the [[w:American Institute of Mathematics|American Institute of Mathematics]] * [https://openstax.org/subjects/math OpenStax] - free textbooks from algebra, trigonometry to calculus and statistics * [https://mathvault.ca/hub/higher-math/math-symbols/ Comprehensive list of mathematical symbols] * [https://tutorial.math.lamar.edu/ Paul's Online Notes] - standalone resources on algebra, calculus I, calculus II, calculus III and differential equations * [http://www.mathabc.com/ Free online math practice with instructional videos for K-6.] * [http://mathematicslibrary.blogspot.com/ Mathematics notes and books for undergraduates and graduate students.] * [http://mum.edu/math_dept/courses.html An interesting perspective on mathematics.] * [http://www.sciencenews.org/view/generic/id/31392/title/Still_debating_with_Plato 'are new mathematical truths discovered or invented?'] * [http://www.aplusclick.com/ Collection of Math Problems & Answers adapted to the student ability] * [https://mc2.bid4papers.com/blog/math-resources/ MC2 Math Links] - a collection of 486 free online Math tools: calculators, plotters and interactive visualizations. ==Active participants== The histories of Wikiversity pages indicate who are the active participants (listed below). If you are an active participant in this school, you can list your name here (this can help small schools grow and the participants communicate better; for large schools it is not needed). * [[User:IanVG]]--[[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 19:47, 9 June 2026 (UTC) ===Inactive participants=== {{MultiCol}} *[[User:Marshallsumter|Marshallsumter]] Was active beginning 7 January 2012 with Mathematical astronomy *[[User:Parthkohli99|Parth Kohli]] (Set Theory, Calculus, Algebra, Trigonometry, Logic, Applied Mathematics) *[[User:TortoiseDream|TortoiseDream]] (Logic, Set Theory, Mathematical Physics) *[[User:Tastymangojuice|Tastymangojuice]] (Computer Science) *[[User:ehremo|ehremo]] *[[User:Deltinu|Deltinu]] *[[User:EulerGamma|EulerGamma]] *[[User:Ravi12346|Ravi12346]] *[[User:lgallindo|Lucas Gallindo]] *[[User:Pgbridge2000|Pgbridge2000]] (Cryptography) *[[User:Tachyon01|Tachyon01]] (Pure Mathematics) *[[User:fidocancan|fidocancan]] (pre-uni teacher) *[[User:Mo Anabre|Mo Anabre]] *[[User:master.signal|sutton]] *[[User:Drini|Drini]] *[[User:Ananth nag|Ananth nag m]] *[[User:TeapotScar|TeapotScar]] (Calculus, Number Theory, Topology) *[[User:pbran7|pbran7]] *[[User:Juan Marquez|Juan]] (Topology, Algebra, Geometry and Analysis) *[[User: Ivanne|Ivanne]] (I just want to learn man) {{ColBreak}} *[[User:King Bee|King Bee]] (graduate student in pure mathematics) *[[User:Verin Sedai|Verin Sedai]] (Algebra, Combinatorics, basic number theory) *[[User:haresh kumar|haresh kumar]](analytic geometry,calculus) *[[User:A.K.Karthikeyan|A.K.Karthikeyan]](True knowledge lies in knowing you know nothing) *[[User:Tlep|Tlep]] (Algebra) *[[User:Bigyan Shrestha|Bigyan Shrestha]] (Mathematical Statistics) *[[User:Rogerh|Rogerh]] *[[user:sarinmadhur|Madhur Sarin]] (Number system,Geometry,Calculus ) *[[User:AFriedman|AFriedman]] (Computer Science, Biology) *[[User:fibonacci101]] (Geometry, number theory, set theory) *[[User:mathclubowner]] *[[User:Helloworld00|Helloworld00]] *[[User:graie.project|graie]] *[[user:mynameinc|mynameinc]] (Algebra, Trigonometry, Differential Calculus) *[[user:Comrade1986|Comrade1986]] (Mathematics) *[[user:WendyB]] (Mathematics student) *[[user:Alchemy heels I]] (Mathematics student) *[[John Michael Pinson I ]] (Numerology, Mathematics student) * [[User:Tarun-Science|Tarun-Science]] (I;m a 15-year-old intersted in Physics, Mathematics and Chemistry..good in geometry and algebra) {{EndMultiCol}} == International relations == *[[:it:Wikiversità]] - [[:it:Wikiversità:Facoltà di Matematica|Facoltà di Matematica]] *[[b:pt:Wikiversidade:_Ci%C3%AAncias_Exatas_e_Ci%C3%AAncias_da_Terra#Curso_de_Matem.C3.A1tica||Wikiversidade - Ciências Exatas e da Terra]] <!-- footer templates --> {{Mathematics resources}} <!-- categories --> [[Category:Abstraction]] [[Category:Mathematics/Lectures]] [[Category:Probability]] [[Category:Wikiversity schools]] [[Category:Resources last modified in March 2020]] [[Category:Statistics]] [[Category:School of Mathematics| ]] a1s69p9r2w3ekp6ycc3x21nf7eb38fv 4 2796 2814931 2661232 2026-06-10T00:02:50Z AUBSTRAWBS 3060598 Added miscillaneous tab 2814931 wikitext text/x-wiki {{shortcut|WV:UBX|WV:USERBOX}} {{Administering Wikiversity}} In order to promote collaborative learning within the Wikiversity community it is relevant to know the interests of other users. '''The userboxes''' can be a useful way to introduce yourself to the community. Please make responsible use of these templates to develop the Wikiversity community. Templates that are counter to the educational mission of Wikiversity or otherwise disrupt the community will be deleted. ==Users by language skills== See the sub-categories of [[:Category:Users by language]] for a complete list of language userboxes. {| !Adding this to your page!!Creates this |- |<nowiki>{{User en}}</nowiki>||{{User en}} |- |<nowiki>{{User de}}</nowiki>||{{User de}} |- |<nowiki>{{User fr}}</nowiki>||{{User fr}} |- |<nowiki>{{User es}}</nowiki>||{{User es}} |} == Involvement of learners == {| !Adding this to your page!!Creates this |- |<nowiki>{{NLP}}</nowiki>||{{NLP}} |- |<nowiki>{{User Chemcontrib}}</nowiki>||{{User Chemcontrib}} |- |<nowiki>{{User Mathcontrib}}</nowiki>||{{User Mathcontrib}} |- |<nowiki>{{User Strategycontrib}}</nowiki>||{{User Strategycontrib}} |- |<nowiki>{{User Gamescontrib}}</nowiki>||{{User Gamescontrib}} |- |<nowiki>{{Template:User DORA}}</nowiki>||{{Template:User DORA}} |- |<nowiki>{{Wikipedia}}</nowiki>||{{Wikipedia}} |- |<nowiki>{{User ITcontrib}}</nowiki>||{{User ITcontrib}} |} ==[[:Category:Knowledge user templates|Knowledge user templates]]== Knowledge userboxes (or user templates) are used to advertise your level of knowledge in a certain subject. You can also use them to find other students of that subject. {| !Adding this to your page!!Creates this |- |<nowiki>{{User knowledge-stst-0}}</nowiki>||{{User knowledge-stst-0}} |- |<nowiki>{{User knowledge-agri-1}}</nowiki>||{{User knowledge-agri-1}} |- |<nowiki>{{User knowledge-pols-2}}</nowiki>||{{User knowledge-pols-2}} |- |<nowiki>{{User knowledge-educ-3}}</nowiki>||{{User knowledge-educ-3}} |- |<nowiki>{{User knowledge-lang-4}}</nowiki>||{{User knowledge-lang-4}} |- |} '''More''': [[:Category:Knowledge user templates]]. == Users by Wikimedia project participation == {| !Adding this to your page!!Creates this |- |<nowiki>{{User Commons}}</nowiki>||{{User Commons}} |- |<nowiki>{{User Wikipedia}}</nowiki>||{{User Wikipedia}} |- |<nowiki>{{User Wikiversity Beta}}</nowiki>||{{User Wikiversity Beta}} |- |} == User characteristics == {| !Adding this to your page!!Creates this |- |<nowiki>{{User must have gone crazy|[[link|text]]}}</nowiki><br /><br />link: can be a page inside Wikiversity<br />text: is a description for link||{{User must have gone crazy}} |- |<nowiki>{{Google video chat user}}</nowiki><br /><br />This user can be reached by [[w:Google_Voice|Google video chat]]||{{Google video chat user}} |- |<nowiki>{{Logitech Vid user}}</nowiki><br /><br />This user can be reached by [[w:SightSpeed|Logitech Vid]]||{{Logitech Vid user}} |- |} == miscellaneous == {| !Adding this to your page!!Creates this |- |<nowiki>{{Gate anime}}</nowiki>||{{Gate anime}} |- |} ==See also== {{Wikiversity culture}} [[Category:Wikiversity culture]] kufvrdc3a6n1urjlo3n96adjtd1nv39 2814937 2814931 2026-06-10T00:11:22Z Atcovi 276019 /* miscellaneous */ update 2814937 wikitext text/x-wiki {{shortcut|WV:UBX|WV:USERBOX}} {{Administering Wikiversity}} In order to promote collaborative learning within the Wikiversity community it is relevant to know the interests of other users. '''The userboxes''' can be a useful way to introduce yourself to the community. Please make responsible use of these templates to develop the Wikiversity community. Templates that are counter to the educational mission of Wikiversity or otherwise disrupt the community will be deleted. ==Users by language skills== See the sub-categories of [[:Category:Users by language]] for a complete list of language userboxes. {| !Adding this to your page!!Creates this |- |<nowiki>{{User en}}</nowiki>||{{User en}} |- |<nowiki>{{User de}}</nowiki>||{{User de}} |- |<nowiki>{{User fr}}</nowiki>||{{User fr}} |- |<nowiki>{{User es}}</nowiki>||{{User es}} |} == Involvement of learners == {| !Adding this to your page!!Creates this |- |<nowiki>{{NLP}}</nowiki>||{{NLP}} |- |<nowiki>{{User Chemcontrib}}</nowiki>||{{User Chemcontrib}} |- |<nowiki>{{User Mathcontrib}}</nowiki>||{{User Mathcontrib}} |- |<nowiki>{{User Strategycontrib}}</nowiki>||{{User Strategycontrib}} |- |<nowiki>{{User Gamescontrib}}</nowiki>||{{User Gamescontrib}} |- |<nowiki>{{Template:User DORA}}</nowiki>||{{Template:User DORA}} |- |<nowiki>{{Wikipedia}}</nowiki>||{{Wikipedia}} |- |<nowiki>{{User ITcontrib}}</nowiki>||{{User ITcontrib}} |} ==[[:Category:Knowledge user templates|Knowledge user templates]]== Knowledge userboxes (or user templates) are used to advertise your level of knowledge in a certain subject. You can also use them to find other students of that subject. {| !Adding this to your page!!Creates this |- |<nowiki>{{User knowledge-stst-0}}</nowiki>||{{User knowledge-stst-0}} |- |<nowiki>{{User knowledge-agri-1}}</nowiki>||{{User knowledge-agri-1}} |- |<nowiki>{{User knowledge-pols-2}}</nowiki>||{{User knowledge-pols-2}} |- |<nowiki>{{User knowledge-educ-3}}</nowiki>||{{User knowledge-educ-3}} |- |<nowiki>{{User knowledge-lang-4}}</nowiki>||{{User knowledge-lang-4}} |- |} '''More''': [[:Category:Knowledge user templates]]. == Users by Wikimedia project participation == {| !Adding this to your page!!Creates this |- |<nowiki>{{User Commons}}</nowiki>||{{User Commons}} |- |<nowiki>{{User Wikipedia}}</nowiki>||{{User Wikipedia}} |- |<nowiki>{{User Wikiversity Beta}}</nowiki>||{{User Wikiversity Beta}} |- |} == User characteristics == {| !Adding this to your page!!Creates this |- |<nowiki>{{User must have gone crazy|[[link|text]]}}</nowiki><br /><br />link: can be a page inside Wikiversity<br />text: is a description for link||{{User must have gone crazy}} |- |<nowiki>{{Google video chat user}}</nowiki><br /><br />This user can be reached by [[w:Google_Voice|Google video chat]]||{{Google video chat user}} |- |<nowiki>{{Logitech Vid user}}</nowiki><br /><br />This user can be reached by [[w:SightSpeed|Logitech Vid]]||{{Logitech Vid user}} |- |} == miscellaneous == {| !Adding this to your page!!Creates this |- |<nowiki>{{User gate anime}}</nowiki>||{{Gate anime}} |- |} ==See also== {{Wikiversity culture}} [[Category:Wikiversity culture]] 0vc8n1lykvj4syxm7obu2sp13dd40p2 2814938 2814937 2026-06-10T00:11:38Z Atcovi 276019 /* miscellaneous */ fix 2814938 wikitext text/x-wiki {{shortcut|WV:UBX|WV:USERBOX}} {{Administering Wikiversity}} In order to promote collaborative learning within the Wikiversity community it is relevant to know the interests of other users. '''The userboxes''' can be a useful way to introduce yourself to the community. Please make responsible use of these templates to develop the Wikiversity community. Templates that are counter to the educational mission of Wikiversity or otherwise disrupt the community will be deleted. ==Users by language skills== See the sub-categories of [[:Category:Users by language]] for a complete list of language userboxes. {| !Adding this to your page!!Creates this |- |<nowiki>{{User en}}</nowiki>||{{User en}} |- |<nowiki>{{User de}}</nowiki>||{{User de}} |- |<nowiki>{{User fr}}</nowiki>||{{User fr}} |- |<nowiki>{{User es}}</nowiki>||{{User es}} |} == Involvement of learners == {| !Adding this to your page!!Creates this |- |<nowiki>{{NLP}}</nowiki>||{{NLP}} |- |<nowiki>{{User Chemcontrib}}</nowiki>||{{User Chemcontrib}} |- |<nowiki>{{User Mathcontrib}}</nowiki>||{{User Mathcontrib}} |- |<nowiki>{{User Strategycontrib}}</nowiki>||{{User Strategycontrib}} |- |<nowiki>{{User Gamescontrib}}</nowiki>||{{User Gamescontrib}} |- |<nowiki>{{Template:User DORA}}</nowiki>||{{Template:User DORA}} |- |<nowiki>{{Wikipedia}}</nowiki>||{{Wikipedia}} |- |<nowiki>{{User ITcontrib}}</nowiki>||{{User ITcontrib}} |} ==[[:Category:Knowledge user templates|Knowledge user templates]]== Knowledge userboxes (or user templates) are used to advertise your level of knowledge in a certain subject. You can also use them to find other students of that subject. {| !Adding this to your page!!Creates this |- |<nowiki>{{User knowledge-stst-0}}</nowiki>||{{User knowledge-stst-0}} |- |<nowiki>{{User knowledge-agri-1}}</nowiki>||{{User knowledge-agri-1}} |- |<nowiki>{{User knowledge-pols-2}}</nowiki>||{{User knowledge-pols-2}} |- |<nowiki>{{User knowledge-educ-3}}</nowiki>||{{User knowledge-educ-3}} |- |<nowiki>{{User knowledge-lang-4}}</nowiki>||{{User knowledge-lang-4}} |- |} '''More''': [[:Category:Knowledge user templates]]. == Users by Wikimedia project participation == {| !Adding this to your page!!Creates this |- |<nowiki>{{User Commons}}</nowiki>||{{User Commons}} |- |<nowiki>{{User Wikipedia}}</nowiki>||{{User Wikipedia}} |- |<nowiki>{{User Wikiversity Beta}}</nowiki>||{{User Wikiversity Beta}} |- |} == User characteristics == {| !Adding this to your page!!Creates this |- |<nowiki>{{User must have gone crazy|[[link|text]]}}</nowiki><br /><br />link: can be a page inside Wikiversity<br />text: is a description for link||{{User must have gone crazy}} |- |<nowiki>{{Google video chat user}}</nowiki><br /><br />This user can be reached by [[w:Google_Voice|Google video chat]]||{{Google video chat user}} |- |<nowiki>{{Logitech Vid user}}</nowiki><br /><br />This user can be reached by [[w:SightSpeed|Logitech Vid]]||{{Logitech Vid user}} |- |} == miscellaneous == {| !Adding this to your page!!Creates this |- |<nowiki>{{User gate anime}}</nowiki>||{{User gate anime}} |- |} ==See also== {{Wikiversity culture}} [[Category:Wikiversity culture]] toq5j1rt08ygvmfcledrfiuv2c1jd4c 0 4165 2814889 2716450 2026-06-09T17:24:44Z Atcovi 276019 project box(es) 2814889 wikitext text/x-wiki {{physics}} At present, we understand that there are two categories of fundamental particles: bosons and fermions. ==Elementary Particles== === Fermions === Fermions are things that we normally associate with matter. Examples of fermions are quarks, which combine to make up protons and neutrons, and electrons. There are other more exotic particles that are fermions. ====Quarks==== Quarks are the basis of most elements in the present world, at the present day there are 6 known (non-anti-) quarks, these are: * Up * Down * Charm * Strange * Top * Bottom Each of these have their own respective anti-quark; this adds up to a total of 12 quarks. For completeness, the antiquarks are listed here: *Antiup * Antidown * Anticharm * Antistrange * Antitop * Antibottom ====Leptons==== Leptons do not experience the strong interaction of the nuclear force. The leptons form a family of elementary particles that are distinct from the other known family of fermions: the quarks. There are 6 different Leptons in total, consisting of: * Electron * Muon * Tau Lepton * Electron Neutrino * Muon neutrino * Tau neutrino The leptons have Anti-Leptons just like that of the Quarks although their names are different, the Anti-Leptons consist of; * Positron * Muon (no change) * Tau Lepton (no change) * Electron antineutrino * Muon antineutrino * Tau antineutrino === Bosons === Bosons are particles that mediate forces, such as light and gravity. Light is mediated by photons and gravity by gravitons. All Bosons have an integer spin, unlike Fermions. The Higgs Boson is a theoretical particle that has yet to be discovered. Scientists hope the Higgs Boson explain why some particles are heavy and others, like the photon, are massless. ==Composite Particles== ===Hadrons=== Hadrons are strongly interacting composite subatomic particles. They are classified into two separate classes, which are: * Baryons * Mesons Both of these consist of [[quarks]]. ===Baryons=== Baryons are strongly interacting Fermions, such as a Proton or a Neutron, both of which are made up 3 Quarks. Baryons have a baryon number of 1. The list of Baryons consist of: * Proton * Neutron * Delta * Lambda * Sigma * Xi * Omega ===Mesons=== Mesons are strongly interacting Bosons which consist of a quark and an antiquark. Mesons have baryon number 0. They are classified according to their quark content and their angular momenta. Neutral mesons can spontaneously change their quark content in a process known as mixing. This process was first discovered in the neutral kaon system in 1964 and has since been observed in the neutral <math>B</math> system, the <math>B_s</math> system and the neutral <math>D</math> system. The ground state mesons are: <table> <tr><th>Name</th><th>Symbol</th><th>Quark content</th></tr> <tr><td>Charged pion</td><td><math>\pi^+</math></td><td><math>u\bar{d}</math></td></tr> <tr><td>Neutral pion</td><td><math>\pi^0</math></td><td><math>\frac{1}{\sqrt{2}}\left(u\bar{d}-d\bar{u}\right)</math></td></tr> <tr><td>Charged kaon</td><td><math>K^+</math> </td><td><math>u\bar{s}</math></td></tr> <tr><td>Neutral kaon</td><td><math>K^0</math> </td><td><math>d\bar{s}</math></td></tr> <tr><td>Charged D meson</td><td><math>D^+</math> </td><td><math>c\bar{d}</math></td></tr> <tr><td>Neutral D meson</td><td><math>D^0</math> </td><td><math>c\bar{u}</math></td></tr> <tr><td>Strange D meson</td><td><math>D_s^+</math> </td><td><math>c\bar{s}</math></td></tr> <tr><td>Charged B meson</td><td><math>B^+</math> </td><td><math>u\bar{b}</math></td></tr> <tr><td>Neutral B meson</td><td><math>B^0</math> </td><td><math>d\bar{b}</math></td></tr> <tr><td>Strange B meson</td><td><math>B_s^0</math> </td><td><math>s\bar{b}</math></td></tr> <tr><td>Charmed B meson</td><td><math>B_c^+</math> </td><td><math>c\bar{b}</math></td></tr> </table> Mesons consisting of a quark and an antiquark of the same flavor do not have a unambiguous quark content. For example, the neutral pion is a superposition of flavor state <math>u\bar{u}</math> and <math>d\bar{d}</math>. Other neutral states of this kind are shown below: <table> <tr><th>Name</th><th>Symbol</th><th>Quark content</th></tr> <tr><td>Phi</td><td><math>\phi</math></td><td><math>s\bar{s}</math></td></tr> <tr><td>Eta-c</td><td><math>\eta_c</math></td><td><math>c\bar{c}</math></td></tr> <tr><td>Eta-b</td><td><math>\eta_b</math></td><td><math>b\bar{b}</math></td></tr> </table> The top quark decays too rapidly to form bound mesonic states. ===See also=== *[[Topic:Particle physics]] {{subpagesif}} [[Category:Quantum physics]] lxjwoxjubwrt0xdaxaapxg8dwb6ap0s 102 23506 2814903 2697910 2026-06-09T19:10:20Z IanVG 2918363 2814903 wikitext text/x-wiki [[File:Cycloid f.gif|thumb|right|330px|The red curve is a [[w:Cycloid|cycloid]].]] The Euler-Lagrange equation of [[Advanced Classical Mechanics|classical mechanics]] was discovered during attempts to find a curve for which the time taken by a frictionless particle sliding down it under uniform gravity to its lowest point is independent of its starting point. ---- Did you know that Wikiversity can never have enough examples and solved problems? Wikiversity does not always have to re-create the wheel. Do you know a good website with mathematics example problems? Wikiversity needs branches of the [[Other Free Learning Resources|Hunter-gatherers project]] for each academic subject area. Please make note of good online mathematics resources at [[Other Free Learning Resources/Mathematics|Hunter-gatherers/Mathematics]] and link to these resources from the appropriate mathematics pages of Wikiversity. '''Example:''' Did you know that there is a wiki for mathematics example problems? See the latest internet archive link to [https://web.archive.org/web/20220706225058/https://www.exampleproblems.com/wiki/index.php/Main_Page]. Another resource is: [https://www.math10.com/] q3sk6plfwxji3n9v9yrtima81kaum6y 0 40932 2814887 2608756 2026-06-09T17:13:06Z Atcovi 276019 cleanup 2814887 wikitext text/x-wiki {{physics}} {{cleanup|could this be integrated in a bigger physics [[Wikiversity:Learning projects|learning project]]?}} In physics Newton's three laws of motion are the basis of [[w:kinematics|kinematics]], the subject of this course. All three laws are detailed on this page. See also: [[Motion - Dynamics]] ==Newton's Laws of Motion== Newtons law describes the body's motion in terms of physics. It has been widely accepted by the science community for the past 10 centuries, in total there were three laws that affect the laws of motion. ==Newton's First Law== His first law of motion is called the law of inertia, his law is; "Every body continues its state of rest or uniform motion until an external unbalanced force acts on it" ."" If a body is moving ,it will continue to move with the same speed in the same direction unless a force is applied on it" Newton put the above observation in the form of a law which is called the Newton's first law of motion.And the resultant force must be zero. ==Newtons Second Law== His second law is more notably named as the law of acceleration. His law is; ''"The rate of change of momentum of an object is directly proportional to the resultant force acting on the body and is in the same direction."'' This law means that the force of the object moving will be equal to the opposing force such as air resistance. Mathematically, Newton's second law is stated as: <math>\overrightarrow{\mathbf{F}} = \frac{d\overrightarrow{\mathbf{p}}}{dt} </math> where F is the total force, p is the momentum, and t is the time passed. In classical physics, where the object's mass is constant, this equation becomes a more familiar form: <math>\overrightarrow{\mathbf{F}} = \frac{d\overrightarrow{\mathbf{p}}}{dt} = \frac{dm\overrightarrow{\mathbf{v}}}{dt} = m\frac{d\overrightarrow{\mathbf{v}}}{dt} = m\overrightarrow{\mathbf{a}}</math> where m is the object's mass and a is the acceleration of the object. ''Example'' A skydiver jumps from a plane and accelerates until he reaches the highest velocity possible, when this happens his acceleration is equal to nothing, this happens when air resistance is equal to the downward force of the skydiver. ==Newton's Third Law of motion== Newton's third and final law is the law of Interaction in which he Doubted; ''"All forces occur in pairs, and these two forces are equal in magnitude and opposite in direction."'' It can be said that "every action has an opposite and equivalent reaction" ; Examples # When we push a wall, it creates an opposite amount of force, thus, either we move or does the wall move. This in simple terms is that to every action force there is an equal, but still the same, reaction force. For example if one presses on a wall a equal and same force is pushed back at them. [[Category: Physics]] mlmqb6kbr4czuysozght95bzu6f4rhx 0 54226 2814885 2797068 2026-06-09T17:11:13Z Atcovi 276019 project box(es) 2814885 wikitext text/x-wiki {{humanities}} {{literature}} [[Image:Death of Consort Dong.png|right|200px|thumb|A print from a Qing Dynasty edition of Luo Guanzhongs Romance of the Three Kingdoms]] This page is based on an import of [[s:Romance of the Three Kingdoms|Romance of the Three Kingdoms]] from Wikisource. Background info you can find [[s:Talk:Romance_of_the_Three_Kingdoms#translation_.28into_German.29_for_reading_group_possible_.3F|here]]. Time of import: [http://en.wikiversity.org/w/index.php?title=Romance_of_the_Three_Kingdoms&diff=233115&oldid=233114 01:28, 25 March 2008] '''Romance of the Three Kingdoms''', written by [[s:Author:Luo Guanzhong|Luo Guanzhong]] in the 14th century, is a Chinese historical novel based upon events in the turbulent years near the end of the Han Dynasty, and the Three Kingdoms period (220-280). It is acclaimed as one of the [[w:Four Great Classical Novels|Four Great Classical Novels]] of Chinese literature. == Table of contents == # [[/Chapter 1|Chapter 1: Three brave men swear an oath of allegiance at the feast in the peach gardens, our heroes' first achievement is the vanquishing of the Yellow Turbans]] # [[/Chapter 2|Chapter 2: Zhang Yide gets angry and whips the County Inspector, Uncle He plots the murder of the wretched eunuchs]] # [[/Chapter 3|Chapter 3: How Dong Zhuo rebukes Ding Yuan in the Garden of Warmth and Brightness, and Li Su wins over Lü Bu with offerings of gold and pearls]] # [[/Chapter 4|Chapter 4: Deposing the Han emperor: Chenliu becomes emperor; plotting against the villain Dong: Mengde presents a dagger.]] # [[/Chapter 5|Chapter 5: A forged imperial edict is issued: all towns respond to Lord Cao; Breaking through the soldiers at the pass: three heroes battle Lü Bu.]] == Prologue == {| cellpadding=2 align=center |- ! style="width: 25%; background: lightgrey" | [[wikt:臨|臨]][[wikt:江|江]][[wikt:仙|仙]] ! style="width: 38%; background: lightgrey" | The Immortals by the River ! style="width: 37%; background: lightgrey" | Die Unsterblichen am Fluss ! |- ! [[wikt:楊慎|楊慎]] ! by [[w:Yang Shen|Yang Shen]] (1488-1559) ! von [[w:Yang Shen|Yang Shen]] (1488-1559) |- | : [[wikt:滾滾|滾滾]][[wikt:長江|長江]][[wikt:東|東]][[wikt:逝|逝]][[wikt:水|水]]，[[wikt:浪花|浪花]][[wikt:淘|淘]][[wikt:盡|盡]][[wikt:英雄|英雄]]。 : [[wikt:是非|是非]][[wikt:成敗|成敗]][[wikt:轉頭|轉頭]][[wikt:空|空]]，[[wikt:青山|青山]][[wikt:依舊|依舊]][[wikt:在|在]]，[[wikt:幾|幾]][[wikt:度|度]][[wikt:夕陽|夕陽]][[wikt:紅|紅]]。 : [[wikt:白髮|白髮]][[wikt:漁|漁]][[wikt:樵|樵]][[wikt:江渚|江渚]][[wikt:上|上]]，[[wikt:慣|慣]][[wikt:看|看]][[wikt:秋月|秋月]][[wikt:春風|春風]]。 : [[wikt:一|一]][[wikt:壺|壺]][[wikt:濁酒|濁酒]][[wikt:喜|喜]][[wikt:相逢|相逢]]，[[wikt:古今多少事，都付笑談中|古今多少事，都付笑談中]]。 | : The waters of the mighty [[w:Yangzi River|Yangzi]] flow eastward, its spray drowning countless heroes. : Right and wrong, success and failure, become empty in the blink of an eye; green mountains are always present; how many times has the setting sun been red? : The white haired fishermen and woodcutters are standing on the sand bars near the banks, accustomed as they are to gazing at the autumn moon and the spring breezes. : By chance, they happily meet with a jar of strong liquor in hand; how many things from past and present have they laughed and talked about with each other? | : Die Wasser des mächtigen [[w:de:Jangtse|Jangtse]] fliessen ostwärts, seine Gischt ertränkt unzählige Helden. : Richtig und falsch, Erfolg und Mißerfolg werden von einem Augenblick zum anderen bedeutungslos; grüne Berge sind immer präsent; wie oft war die Abendsonne rot ? : Die weisshaarigen Fischer und Holzfäller stehen auf den Sandbänken in der Nähe der Ufer, gewöhnt wie sie sind, starren sie auf den Herbstmond und die Frühlingsbrisen. : Durch Zufall, treffen sie sich fröhlich mit einem Krug starken alkoholischen Getränkes in der Hand; über wie viele Dinge aus der Vergangenheit und der Gegenwart haben sie gelacht und darüber gegenseitig gesprochen ? |} [[Category:China]] [[Category:Middle Ages]] [[Category:Novels]] [[Category:Translations]] [[Category:Romance of the Three Kingdoms| ]] mvbjyl7ndrlv68w3xkmuxy9m1s4zwnn 0 79931 2814861 2036010 2026-06-09T14:00:12Z Atcovi 276019 {{Mathematics-stub}} 2814861 wikitext text/x-wiki Propositional logic is a formal language (we shall not enter into a definition of a "formal language.") It has three kinds of symbols which may vary in each presentation. We'll define them as: # Sentence letters: a, b, c.... z. # Operators (aka. connectives): ~, ^, v, ->, <->. # Parentheses: (, ). Used for organization purposes, similar to parentheses in algebra and arithmetic. In the common use of propositional logic, sentence letters stand for complete sentences. Our interest here is to learn to symbolize natural language into propositional logic, so we will take for granted that every letter we use represents a sentence. ==Reading statements in propositional logic== We will review the way that operators are usually read without entering into their semantic properties. * <math>p</math>, read as "p". * <math>\neg p</math>, read as "not p" or "it is not the case that p". * <math>p\wedge q</math>, read as "p and q". * <math>p\vee q</math>, read as "p or q"; that is, "p or q, or both". * <math>p\rightarrow q</math>, read as "if p, then q" or "p implies q". * <math>p\leftrightarrow q</math>, read as "p if and only if q"; sometimes written as "p iff q". ==Symbolization== Symbolization is a process that turns a ''meaning'' into something expressed in a symbol (something that stands for something) or in our context, in a symbolic language. For example, outside the domain of logic, if we have a fifty dollars we may symbolize this in our balance book as "$50". In turn, when we find the symbol "$50" we have to ''interpret'' it (the usual interpretation is "fifty dollars"). In natural languages (English, French, etc) the process of "interpretation" is often subjective and unclear, in formal languages (propositional logic, C++, etc) the interpretation is precise and clear. A puzzling thing is that interpretation is not necessary in formal languages. A computer may learn the rules of arithmetic (a formal language) and make operations correctly without knowing the meaning of the symbols. However, when we use a formal language for something we usually want to give it an interpretation. We're going to learn to symbolize a sentence in natural language using symbolic logic. In the process, something is going to be lost. If I have $100 and I buy a $10 hotdog I can symbolize this as= "$100 - $10 = $90" Nowhere in the subtraction does it say that I ate a hotdog. We're losing information but we're gaining in precision and we may apply operations that obtain conclusions that maybe (imagine an extremely long hotdog shopping spree expressed as a subtraction) weren't obvious at first sight. This will be the objective of symbolizing natural language in propositional logic. ==Symbolizing sentences== The standard procedure is to ''define'' letters with the sentences they stand for. Example: * p = Pork is expensive. * q = Honey is cheaper than pork. * etc. When symbolizing, one should try to break each natural language sentence into an ''atomical sentence''. An atomical sentence is one that cannot be separated into further sentences. This way, we can make use of the best level of precision propositional logic can give us. For instance, the natural language sentence: "My sister and I like coffee" could be defined: * m = My sister and I like coffee. But also: * n = My sister likes coffee. * o = I like coffee. For the time being, we don't mind the logical relation between sentences. We will use operators later to convey that. At this stage, both "my sister and i like coffee" and "if my sister likes coffee i like coffee" will be broken down into: * n = My sister likes coffee. * o = I like coffee. As we will later see, we have an operation to express "negation". So a negative sentence will be better expressed as a positive sentence, and later we can add the correct symbol. That is, the atomic constituents of "I don't like the bomb." are: * b = I like the bomb. The sentence "I don't like the bomb but you will be a bad person if you don't understand me" can be broken into: * a = I like the bomb. * b = You will be a bad a person. * c = You understand me. ''This article is in progress'' {{Mathematics-stub}} [[Category:Logic]] kxujkd1y4oh1o20olx76e8f5yzgdidy 0 99102 2814942 2758884 2026-06-10T00:14:58Z Jtneill 10242 Adjust category 2814942 wikitext text/x-wiki {{title|Emotion and sex}} {{MECR|http://screenr.com/Asc}} __TOC__ ==Overview== {{center top}}'''''"Sex without love is an empty experience, but as empty experiences go it's one of the best."'''''- [[w:Woody_Allen|Woody Allen]] {{center bottom}} TSex is important within society, it is innate and we are biologically programmed to want and need it. We need sex to procreate and to feel close to another human. People do not engage in sex just to start a family, sex can cause very strong emotions and the act of sex can be engaged in due to these very strong emotions. Sexual intercourse is the most intimate behaviour two people can do together. It is important to understand the emotions involved in sex and this chapter will look at the emotions involved in first time sexual intercourse and acts of sex thereafter, how emotions differ between the genders while engaging in sex. It will also look at emotions involved when engaging in risky sexual behaviour, masturbation and sexual dysfunction. ==Sex as an emotion== {{center top}}“'''''Sex is an emotion in motion'''''.”- [[w:Mae West|Mae West]]{{center bottom}} Emotion research is rooted in the study of subjective experience and most people would describe sexual experiences as pleasurable, exciting, enjoyable and stimulating. People often consider sex to be a biological need like hunger, sleep and thirst, however sexual desire is emerging as one of the most typical emotions. An interesting paper by Everaerd, 1988 poses the question “Is sex an emotion?” and if yes, “Is it worthwhile to study sexual experience (arousal, excitement) within the framework of emotion theory?” However, before we answer the question we must first look at the [[w:http://en.wikiversity.org/wiki/Motivation_and_emotion/Textbook/Introduction/What_is_emotion%3F|definition of emotion]] . Emotion has been studied within many different theoretical approaches, which is said to cause confusion about the definition and different names such as mood, affect and feelings are also said to cause more confusion. Ekman, 1984 defined three core characteristics of emotion- *Evolved to deal with basic life tasks *Each emotion is specific, i.e. what occurs (in expression) and when it occurs (events that call for the emotion) are different across the emotions (happiness, anger, sadness, etc). *For each emotion there are interconnect patterns in expression linked to the situational events. However, looking at this the definition and characteristics of emotion is still vague and somewhat confusing. Everaerd, 1988 believed that sex based upon Ekmans’s assumptions qualifies as an emotion as it evolved to deal with fundamental life tasks (reproducing), adaptive to when and what occurs and there is coherence for interconnected patterns in expression. == Emotions and sex== People do not engage in sex purely to procreate, people engage in sexual behaviour for many reasons including love, pleasure, stress reduction, anger, recognition and to gain power (Browning, Hatfield, Kessler, & Levine, 2000). Sexual encounters are emotional events and it is important to understand sexual motives and emotions to predict sexual behaviour which can be useful in reducing conflict and re-igniting romantic relationships. Nelson (1978) developed a measurement for sexual motives. He stated five reasons for engaging in sex- #Love and affection #Pleasure #Conformity #Recognition –competition #Power Nelson found that conformity, love and pleasure were the highest rated reasons for people engaging in sexual behaviour (as cited in Browning et al, 2000). However, in a more recent study has found that one of the main reasons that people have sex is because it feels good, and that people who report greater enjoyment also report having sex more frequently (Pinkerton, Cecil, Bogart & Abramson, 2003). However, all people are different and emotions and motivation can vary, in particular it has been noted that males and females experience different emotions during sex. Emotions also vary across the ages, for example adolescents will state different reasons for engaging in sex than older people in a steady and secure relationship, in particular when it is their first time. ===Gender differences=== ====The First Encounter==== There is growing awareness and research that many decisions are driven, at least in part, by emotions rather than deliberate and rational choice (Slovic, 2003). During adolescence a number of significant changes in emotions occur in particular negative states such as irritability increase and it has been reported that older girls often report that on a daily basis they aren’t as happy, friendly or cheery when compared to younger adolescents (O’Sullivan & Hearn, 2008). Adolescence is also a time where rapid changes are occurring internally in particular changes relevant to sexuality, this include new social interactions with peers, greater feelings of empathy, less ego-centric thinking and strong emotions such as anger and anxiety (O’Sullivan & Hearn, 2008). One of the most important decision that an adolescent makes is when to first engage in sex, however, it is not clear what role emotions play in particular which emotions characterise the transition into being sexual active . A study by O’Sullivan & Hearn (2008) attempted to understand what role emotions played in early sexual experiences for adolescent girls. Major findings were that girls who did not expect to engage in sexual intercourse in the near future did not see themselves as ready and would often attribute negative emotions, such as sadness, nervous, disgust and feeling negative about themselves, with such events. The only positive emotion that was found was the feeling of attractiveness; however when later asked about their first sexual experience the girls associated the experience with feelings of happiness, confidence, and feeling positive about themselves. The study explained these results through evolution, where most girls have learnt to except their first sexual experience to be a negative event as they are often warned by mothers that it is a painful experience, that it is associated with negative social and emotional consequences, in particular shame and guilt. Interestingly, it was found that girls are in fact not encouraged to view their sexuality in a positive light (O’Sullivan, Graber & Brooks-Gunn, 2001; Graber, Brooks-Gunn & Galen, 1998). Based on these studies it is easy to see why girls often view their first sexual experience with negative emotions, but after the fact usually with positive emotions as they have learnt for themselves that sex isn’t always associated with such negative events. Unfortunately, there is limited research on male's emotional motives for engaging in first time sexual behaviour, however, one study found that the most common reason was curiosity, followed by ‘just want to’ and only 14% of male participants gave love as the reason for engaging in sexual behaviour (Shope, 1975 as cited in Carroll, Volk & Hyde, 1985). It would be fair to assume that males would not experience the same negative affect before sexual experience as females because they are not exposed to the attitudes about sex being detrimental to emotional and social environments.{{fact}} In an interesting study by Simon, Berger & Gagnon (1972) it was found that in first time sexual experiences, 46% of males were not emotionally attached or involved with their partner, however, 59% of women were and planning to marry their first partner, while an additional 22% were in love but did not have marriage on the cards. This meaning that 81% of women in comparison to 54% of men were emotionally attached to their first sexual partner. It is clear to see from these results that males and females have different emotional reactions and expectations to first time sexual experiences. ====The Affective Shift Hypothesis==== Based on an idea proposed by Symons (1979) that there are gender differences in intercourse related affective changes , Haselton & Buss (2001) proposed the Affective Shift Hypothesis as an evolved solution to avoid commitment. According to the hypothesis men and women have evolved different affective change because they have faced different commitment and problems in the pursuit of sexual strategies. Heslton & Buss (2001) hypothesised the following three things: #''Sex Differences in Perceptions of Partners Attractiveness''- According to the hypothesis for men, a negative shift in the perceived attractiveness of his partner will help prevent long-term relationships in what was meant to be a short term sexual relationship. More men will report experiencing this negative shift than women as of the benefits of having multiple sexual partners. #''Sex Difference in Perceptions of One’s Own Commitment''- As the benefits of securing a long-term relationship after sex are greater for women, they are more likely to experience a positive shift in their perception of commitment and love. After sex, love and commitment for a partner will increase in order to secure a long term relationship to provide security, access to resources and paternal care for any offspring. #''Differences in Perceptions of Partner Attractiveness within-sex''- Men who have had more sexual partners should be more likely to experience a negative affective shift towards their sexual partner when compared to men who have a relatively low number of sexual partners. However, this doesn’t not work the same for women- where all women, not matter the number of sexual partners, will equally experience the negative affective shift. After testing these hypothesis among undergraduate college students, Haselton & Buss found that men and women do experience different affective changes after first-time sexual intercourse. A few of the results they found are summarised below: *Men are more likely to lose sexual interest after a few months *Men are more likely to report that the first time they had sex was the best time (this could alternatively be explained by the fact that it was the first orgasm that man had ever experienced and men, in particular young men, are pretty much guaranteed an orgasm during sex). *Women reported greater feelings of love and commitment following their first sexual encounter. *Men reported perceiving their partners as less attractive following sex for the first time. *A follow-up study found that men who had greater number of sexual partners were more likely to report negative affective shift in partner’s attractiveness. Haselton & Buss believe that these results suggest that short-term mating functions differ between the sexes, where short-term mating for men may help with increasing partner numbers to increase reproductive output while long-term mating helps women secure resources and care for offspring. ===Sexual Infidelity v. Emotional Infidelity=== In all cultures infidelity occurs and it is seen as one of the main reasons for divorce (Shackelford, LeBlanc & Drass, 2000). Infidelity can have destructive consequences, both emotionally and physically. Infidelity is often described as either emotional infidelity which refers to when a person directs love, time and attention to someone other than their partner, or sexual infidelity which is when a person engages in sexual activity with a person other than their partner. Shackelford, LeBlanc & Drass (2000) suggest that men are more saddened in reaction to partner’s sexual infidelity while women get more upset in response to a partner’s emotional infidelity. Schutwohl (2004) explains why the male jealously mechanism (JM) is heightened by partner’s sexual infidelity, while female JM is heightened by partner’s emotional infidelity. It is explained that males are concerned with loss of opportunity to reproduce and are afraid of investing resources into offspring that may not be genetically related. While women are concerned with loss of resources, and a man’s mere sexual infidelity does not always mean the loss of his resources rather this resource threat occurs when he engages in emotional infidelity. ==Sexual Dysfunction== Sexual Dysfunction, as defined by the DSM-IV-TR, is a disturbance in the process involved in the sexual cycle or by pain associated with sexual intercourse. The sexual cycle is divided into the following phrases, and sexual dysfunction can occur at one or more of these phases. #''Desire''- fantasises about sexual activity and desire to participate in sexual activity. #''Excitement''- subjective sense of sexual pleasure and physiological changes that also occur. Changes in males include penile erection and changes in females include vaginal lubrication and expansion. #''Orgasm''- the peaking of sexual pleasure characterised by release of sexual tension. In males there is the ejaculation of semen and in females there contractions of the wall of the outer vagina. #''Resolution''- sense of general well-being and muscle relaxation. For males there is a time period where they can’t orgasm or have an erection immediately after the orgasm phase, however, females may be able to respond to stimulation almost immediately. Sexual Dysfunction can be an embarrassing and detrimental problem for both individuals and couples, being able to understand what emotions contribute to and are felt by those with sexual dysfunction may be able to help treat it. A study by Nobre & Pinto-Gouveia (2003) looked at reported emotions in sexually dysfunctional men and women and predicted that they would report significantly more negative emotions (in particular those associated with depressed mood) and less pleasure than those who do not suffer a sexual dysfunction. Results supported this hypothesis, men with sexual dysfunction reported higher feelings of sadness, disillusion and fear while women reported feelings of sadness, disillusion, guilt and anger. Both genders reported feelings of less pleasure and satisfaction with sexual experiences when compared to reports from individuals not with sexual dysfunction. It is clear that emotions related to depression (lack of pleasure, lack of satisfaction, sadness, etc) have a negative effect on sexual functioning in both men and women. It has been said that men and women with sexual problems show cognitive patterns similar to those seen in individuals with depressive disorders (Beck, 1996). It is important to note that this study did not explain the difference in feelings between men and women, while men felt fear women reported anger and guilt, it could be said that men fear losing respect among male counterparts if his sexual dysfunction were to be discovered. Likewise, it could be said women feel guilt as they are not pleasing their partner, however, these assumptions are based on a personal view and further research would be needed to understand the relationship and differences in emotions. ==Test Yourself== <quiz> The Affective Shift Hypothesis refers to: |type="()"} - A shift in emotions during sex - Male’s sexuality + Gender differences in affective changes after sex. - Female’s sexuality {The emotions expressed in those with Sexual Dysfunction are similiar to emotions involved in which other group of mental disorders? |type="()"} + Depressive Disorders - Personality Disorders - Substance-Related Disorders - Schizophrenia and Other Psychotic Disorders {Men are more likely to get upset about _____ infidelity, while females are more likely to be upset about ______ infidelity : |type="()"} - sexual; sexual + sexual; emotional - emotional; sexual - emotional; emotional {True or False: Adolescent males reported the main reason for engaging in first time sex as love: |type="()"} + False - True. </quiz> ==See also== * [[Motivation and emotion/Textbook/Motivation/Promiscuity|Promiscuity]] (Book chapter) * [[Motivation and emotion/Textbook/Motivation/Sexual motivation|Sexual motivation]] (Book chapter) * [[Motivation and emotion/Textbook/Motivation/Gender differences in sexual motivation|Gender differences in sexual motivation]] (Book chapter) * [[Motivation and emotion/Textbook/Motivation/Mate-seeking behaviour|Mate-seeking behaviour]] (Book chapter) ==References== <div style="padding-left: 2em; text-indent: -2em"> [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V5W-3YKM652-2F&_user=10&_coverDate=05%2F31%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_searchStrId=1447400631&_rerunOrigin=google&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=031095e88b3ae83b1e8e6bff9ff0d670&searchtype=a Mood and sexual arousal in women] American Psychiatric Association. (2000). Diagnostic and statistical manual of mental disorders (Revised 4th ed.). Washington, DC: Author. Browning, J.R., Hatfield, E., Kessler, D & Levine, T (2000) ''Archives of Sexual Behaviour, 29''(2), pp135-153 Carroll, J.L., Volk, K.D & Hyde, J.S (1985) Differences between Males and Females in Motives for Engaging in Sexual Intercourse. ''Archives of Sexual Behaviour, 14''(2), pp 131-139 Everaerd, W (1988) Commentary on Sex Research: Sex as an emotion. ''Journal of Psychology and Human Sexuality, 1''(2), pp 3-15 Haselton, M.G & Buss, D.M (2001) The affective shift hypothesis: the functions of emotional changes following sexual intercourse. ''Personal Relationships, 8'', 357-369 Graber, J.A., Brooks-Gunn, J., & Galen, B.R (1998) Betwixt and between: sexuality in the context of adolescent transitions. In R. Jessor (Ed.), ''New Perspectives on adolescent risk behaviour''. New York: Cambridge University Press. Nobre, P.J & Pinto-Gouveia, J (2006). Emotions during sexual activity: differences between sexually functional and dysfunctional men and women. ''Archive of Sexual Behaviour, 35'', 491-499. O’Sullivan, L.F., Graber, J.A., & Brooks-Gunn, J. (2001) Adolescent gender development. In J.Worrell (Ed.) ''Encyclopaedia of women and gender''. San Diego, CA: Academic Press. O’Sullivan, L.F., & Hearn, K.D (2008) Brief Report: Predicting first intercourse among urban early adolescent girls: the role of emotions. ''Cognition and Emotion, 22''(1), pp 168-179. Pinkerton, S.D., Cecil, H, Bogart, L.M & Abramson, P.R (2003). The Pleasures of Sex: An empirical investigation. ''Cognition and Emotion, 17''(2), 341-353. Schutzwohl, A. (2004). Which Infidelity makes you more jealous? Decision Strategies in a Forced-Choice between Sexual and Emotional Infidelity. ''Evolutionary Psychology, 2,'' 121-128 Shackelford, T.K., LeBlanc, G.J., Drass, E (2000). Emotional Reactions to Infidelity. Cognition and Emotion, 14(5), 643-659; Simon, W., Berger, A.S and Gagnon, J.S. (1972) Beyond Anxiety and fantasy: The coital experiences of college youth. ''Journal of Youth Adolescence 1,'' pp 203-222. Solvic, P (2003). Affect, analysis, adolescence and risk. In D. Romer (Ed.), ''Reducing adolescent risk: toward an integrated approach''. Thousand Oaks, CA Sage. </div> ==External links== * http://www.whatithinkabout.com/why-sex-is-the-most-important-emotion/ * [http://www.health-science-spirit.com/neurosex.html The neurochemistry of sex] * [http://www.reuniting.info/science/articles/sexual_neurochemistry Neurochemistry of sex] * [http://web.archive.org/web/20060318185316/http://www.abc.net.au/rn/science/mind/stories/s1514225.htm The emotional brain: Sexual desire] * [http://www.psychologytoday.com/blog/in-the-name-love/200811/is-sexual-desire-emotion Is sexual desire an emotion?] * [http://www.entelechyjournal.com/pulling_away_after_sex1.htm Pulling away after sex] * [http://timesofindia.indiatimes.com/life-style/relationships/man-woman/Womans-emotional-guide-for-sex-/articleshow/4529908.cms Womans Emotional Guide] [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Sex]] tkyq3rfj1grldfaq16vvbmcxq2ocdtp 0 100786 2814883 2617513 2026-06-09T17:05:20Z IanVG 2918363 /* Syllabus */ added topic for first-order nonlinear equations 2814883 wikitext text/x-wiki {{RightTOC}} {{tertiary education}} {{mathematics}} {{launch}} Differential equations serve as mathematical models of physical processes. This course is intended to be an introduction to ordinary differential equations and their solutions. A '''differential equation''' (DE) is an equation relating a function to its derivatives. If the function is of only one variable, we call the equation an '''ordinary differential equation''' (ODE). Equations relating the partial derivatives (See: [[Vector calculus]]) of a function of several variables are called '''partial differential equations''' (PDEs). Ordinary differential equations are much easier to solve than partial differential equations, so these will be our main focus. ==Syllabus== *Introduction to ordinary differential equations. **[[/Ordinary Differential Equations/]] **[[Slope Fields]] *First-order equations **[[/Introduction to First Order Linear Differential Equations/]] **[[/First-order nonlinear equations/]] **[[/Separable differential equations/]] <small>50%</small>[[Image:50%25.svg|20px|]] **[[/Exact differential equations/]] <small>25%</small>[[Image:25%25.svg|20px|]] **[[/Homogeneous differential equations/]] <small>25%</small>[[Image:25%25.svg|20px|]] **[[/Integrating factors/]] <small>25%</small>[[Image:25%25.svg|20px|]] **[[/Change of variables/]] <small>25%</small>[[Image:25%25.svg|20px|]] *Higher-order linear equations **[[/Linear homogeneous differential equations/]] <small>25%</small>[[Image:25%25.svg|20px|]] **[[/Linear inhomogeneous differential equations/]] <small>25%</small>[[Image:25%25.svg|20px|]] **[[/Laplace transforms/]] <small>25%</small>[[Image:25%25.svg|20px|]] **[[/Power series solutions/]] <small>25%</small>[[Image:25%25.svg|20px|]] *Introduction to nonlinear equations **[[/Fixed point classification in 1D|Stability problems in 1D/]] **[[/Fixed point classification in 2D|Stability problems in 2D/]] **[[/Approximate solutions to differential equations/]] [[Category:Differential equations| ]] [[Category:Mathematics courses]] {{subpagesif}} f0dtqwh1kzyfhd75yzah6u563nph0wy 2 106888 2814853 2812769 2026-06-09T12:41:59Z Atcovi 276019 +{{User researcher}} 2814853 wikitext text/x-wiki __NOTOC__ {{userboxtop}} {{User Male}} {{User Muslim}} {{Sri Lankan}} {{User contrib|25000}} {{User SUL Box|2=v}} {{User Wikiversitan For|year=2011|month=1|day=28}} {{User custodian}} {{User admin Wikibooks}} {{User admin MediaWiki}} {{Global rollbacker}} {{User Meta-Wiki}} {{User researcher}} {{User soccer}} {{User psychology}} {{userboxbottom}}{{Introduction}} I'm Atcovi, a member of the English Wikiversity community since January 2011. I currently serve as a [[Wikiversity:Custodianship|custodian]] (since June 2021) and a [[Wikiversity:Bureaucratship|bureaucrat]] (since May 2026). My academic interests mainly lie within [[School:Psychology|psychology]], specifically [[clinical psychology]] (with subfields of interest being [[suicidology|suicide]] and [[General Psychopathology|psychopathology]]). My activity is high at the moment but may fluctuate due to life circumstances. Reach out to my talk page for any inquiries. ===Links=== [[File:Sura Minshawi 2.ogg|thumb|left|[[w:Muhammad_Siddiq_Al-Minshawi|Sheikh Minshawi's]] recitation of Surah Al-Baqara]] [[File:Notifications-Talk-Indicator-OptionG-OBOD -Screenshot-Closeup-05-01-2013.png|thumb|right|I remember when I used to get these notifications... (2013)]] I've left an arrangement of random links for me to easily access if I so desire at any given time. # [[Help:Project boxes]] - For projects/pages. # [[Help:Quiz]] - This is also important. # [[Special:CentralAuth/Atcovi]] # [[:Category:Atcovi's Work]], [[User:Atcovi/Science]] & [[User:Atcovi/History]] # https://tools.wmflabs.org/meta/crossactivity/Atcovi # https://tools.wmflabs.org/topviews/?project=en.wikiversity.org&platform=all-access&date=yesterday&excludes= # <code><nowiki>{{under construction}}</nowiki></code> #[https://en.wikipedia.org/wiki/Category:Psychology_stubs Psychology stubs] and [https://en.wikipedia.org/wiki/Category:Health_stubs Health stubs] #[[User:Atcovi/Essays]] #[[:Category:Featured resources]] #[[Special:BrokenRedirects]] [[User:Atcovi/To merge]]: Pages needing to be merged<br> [https://en.wikiversity.org/w/index.php?title=Special%3APrefixIndex&prefix=User%3AAtcovi%2F&namespace=0 Pages under "User:Atcovi"]<br> {{Languages and skills|en-N|de-2}} {{User:Atcovi/to do}} == Wikiversity's To-do == {{Opentask}} [[File:Flagge Palaestina.jpg|350px|frameless|center]] [[Category:Wikiversity custodians]] [[Category:Wikiversity bureaucrats]] [[Category:User pages]] [[Category:Atcovi's Work]] 8iq0igifwcmn6po7sxgd0v8i4mbxlr4 2814862 2814853 2026-06-09T14:00:40Z Atcovi 276019 /* Links */ +[[Category:Stub templates]] 2814862 wikitext text/x-wiki __NOTOC__ {{userboxtop}} {{User Male}} {{User Muslim}} {{Sri Lankan}} {{User contrib|25000}} {{User SUL Box|2=v}} {{User Wikiversitan For|year=2011|month=1|day=28}} {{User custodian}} {{User admin Wikibooks}} {{User admin MediaWiki}} {{Global rollbacker}} {{User Meta-Wiki}} {{User researcher}} {{User soccer}} {{User psychology}} {{userboxbottom}}{{Introduction}} I'm Atcovi, a member of the English Wikiversity community since January 2011. I currently serve as a [[Wikiversity:Custodianship|custodian]] (since June 2021) and a [[Wikiversity:Bureaucratship|bureaucrat]] (since May 2026). My academic interests mainly lie within [[School:Psychology|psychology]], specifically [[clinical psychology]] (with subfields of interest being [[suicidology|suicide]] and [[General Psychopathology|psychopathology]]). My activity is high at the moment but may fluctuate due to life circumstances. Reach out to my talk page for any inquiries. ===Links=== [[File:Sura Minshawi 2.ogg|thumb|left|[[w:Muhammad_Siddiq_Al-Minshawi|Sheikh Minshawi's]] recitation of Surah Al-Baqara]] [[File:Notifications-Talk-Indicator-OptionG-OBOD -Screenshot-Closeup-05-01-2013.png|thumb|right|I remember when I used to get these notifications... (2013)]] I've left an arrangement of random links for me to easily access if I so desire at any given time. # [[Help:Project boxes]] - For projects/pages. # [[Help:Quiz]] - This is also important. # [[Special:CentralAuth/Atcovi]] # [[:Category:Atcovi's Work]], [[User:Atcovi/Science]] & [[User:Atcovi/History]] # https://tools.wmflabs.org/meta/crossactivity/Atcovi # https://tools.wmflabs.org/topviews/?project=en.wikiversity.org&platform=all-access&date=yesterday&excludes= # <code><nowiki>{{under construction}}</nowiki></code> #[https://en.wikipedia.org/wiki/Category:Psychology_stubs Psychology stubs] and [https://en.wikipedia.org/wiki/Category:Health_stubs Health stubs] #[[User:Atcovi/Essays]] #[[:Category:Featured resources]] #[[Special:BrokenRedirects]] #[[:Category:Stub templates]] [[User:Atcovi/To merge]]: Pages needing to be merged<br> [https://en.wikiversity.org/w/index.php?title=Special%3APrefixIndex&prefix=User%3AAtcovi%2F&namespace=0 Pages under "User:Atcovi"]<br> {{Languages and skills|en-N|de-2}} {{User:Atcovi/to do}} == Wikiversity's To-do == {{Opentask}} [[File:Flagge Palaestina.jpg|350px|frameless|center]] [[Category:Wikiversity custodians]] [[Category:Wikiversity bureaucrats]] [[Category:User pages]] [[Category:Atcovi's Work]] 128ygj1x7m2echuplwrgfgj1nr5z150 3 106891 2814851 2814702 2026-06-09T12:39:02Z Atcovi 276019 /* Question */ re 2814851 wikitext text/x-wiki [[User:Atcovi/Archive 1|/Archive 1 (September 25, 2013 - November 15, 2013)]] • [[User talk:Atcovi/Archive 2|/Archive 2 (November 15, 2013 - November 27, 2013)]] • [[User talk:Atcovi/Archive 3|/Archive 3 (December 3, 2013 - December 25, 2013)]] • [[User talk:Atcovi/Archive 4|/Archive 4 (December 24, 2013 - January 1, 2014)]] • [[User talk:Atcovi/Archive 5|/Archive 5 (January 2, 2014 - January 20, 2014)]] • [[User talk:Atcovi/Archive 6|/Archive 6 (March 24, 2014 - April 14, 2014)]] • [[User talk:Atcovi/Archive 7|/Archive 7 (April 19, 2014 - September 8, 2014)]] • [[User talk:Atcovi/Archive 8|/Archive 8 (September 12, 2014 - November 3, 2014)]] • [[User talk:Atcovi/Archive 9|/Archive 9 (November 6, 2014 - January 26, 2015)]] • [[User talk:Atcovi/Archive 10|/Archive 10 (January 28, 2015 - March 11, 2015)]] • [[User talk:Atcovi/Archive 11|/Archive 11 (March 22, 2015 - June 25, 2016)]] • [[User talk:Atcovi/Archive 12 (June 26, 2016 - January 8, 2018)|/Archive 12 (June 26, 2016 - January 8, 2018)]] • [[User talk:Atcovi/Archive 13 (January 9, 2018 - April 14, 2023)|/Archive 13 (January 9, 2018 - April 14, 2023)]] • [[User talk:Atcovi/Archive 14 (April 15, 2023 - May 5, 2026)|/Archive 14 (April 15, 2023 - May 5, 2026)]] :''Before 2013: [https://en.wikiversity.org/w/index.php?title=User_talk:Atcovi&diff=750617&oldid=740650 see this]'' {{tmbox |small = |image = [[Image:Busy desk.svg|{{#ifeq:|yes|40px|75x50px}}]] |text = This user is busy in [http://en.wikipedia.org/wiki/Real_life Real Life] {{#if:|until&nbsp;{{{end}}}&nbsp;}}{{#if:|due to&nbsp;{{{reason}}}&nbsp;}}and may not respond swiftly to queries.{{#if:|<P>{{{msg}}} }} | style = {{#if:|width: {{{width}}}px;}} {{#ifeq:{{{shadow}}}|yes|{{box-shadow|0px|2px|4px|rgba(0,0,0,0.2)}}|}} }} == Please vote == on Wikinews rebirth possibly on Wikiversity, thanks @[[User:Atcovi|Atcovi]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 22:21, 15 May 2026 (UTC) :Hi BigKrow. I've been watching the discussion on the sidelines. Hopefully I'll have an input soon, I just have other commitments I'm catering to. Best of luck with your projects and welcome to Wikiversity! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:44, 16 May 2026 (UTC) == ''The Signpost'': 22 May 2026 == <div lang="en" dir="ltr" class="mw-content-ltr" style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">[[File:WikipediaSignpostIcon.svg|40px|right]] ''News, reports and features from the English Wikipedia's newspaper''</div> <div style="column-count:2;"> * News and notes: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/News and notes|Offline: Osama Khalid still in prison]] * In the media: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/In the media|Indonesian editors, you shall return!]] * Disinformation report: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Disinformation report|Who is a typical paid editor? Who are their typical clients?]] * Recent research: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Recent research|WikiLambda the Ultimate]] * Traffic report: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Traffic report|This is where I'll be, so heavenly, so come and dance with me Michael!]] * Forum: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Forum|WikiAnnotate: help us build a dataset of article quality evaluations]] * In focus: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/In focus|Demystifying the 2026-27 Annual Plan]] * Opinion: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Opinion|Wikipedia isn't a battleground. So why does it feel like one?]] * Serendipity: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Serendipity|Wikinews: Into the Wikiverse]] * Special report: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Special report|Wikimedia Foundation closes Wikinews after 21 years]] * Community view: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Community view|Wikipedia's traffic drop: more on languages and freshness]] * Gallery: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Gallery|Earth Day and Mother's Day]] * Comix: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Comix|Brother, can you spare a page?]] </div> <div style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">'''[[w:en:Wikipedia:Wikipedia Signpost|Read this Signpost in full]]''' · [[w:en:Wikipedia:Signpost/Single|Single-page]] · [[m:Global message delivery/Targets/Signpost|Unsubscribe]] · [[m:Global message delivery|Global message delivery]] 05:19, 22 May 2026 (UTC) <!-- Sent via script ([[w:en:User:JPxG/SPS]]) --></div> <!-- Message sent by User:Bri@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Signpost&oldid=30513885 --> == Wikiversity:Candidates for Bureaucratship/Atcovi == RE: [[Wikiversity:Candidates for Bureaucratship/Atcovi]] I have closed this as successful. Congrats! See [https://en.wikiversity.org/w/index.php?title=Wikiversity:Candidates_for_Bureaucratship/Atcovi&diff=prev&oldid=2812184] and [https://en.wikiversity.org/w/index.php?title=Special:Log&logid=3549048]. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 19:23, 30 May 2026 (UTC) :Thank you Mike! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 20:55, 30 May 2026 (UTC) ::Congratulations @[[User:Atcovi|Atcovi]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 16:58, 8 June 2026 (UTC) == Question == Hello don't mean to bother and Ik its a silly question, on wikipedia there is a tool that allows for the creation of user boxes does wikiversity have it? Or should I create them myself like {{Userbox | border-c = #000000 | id = [[File:(logo) Gate jieitai kanochi nite, kaku tatakaeri.svg|100x50px]] | id-c = #000000 | id-fc = #000000 | id-s = 14 | info = This user testified in front of the [[National Diet|national diet]] | info-c = #000000 | info-fc = #ffffff | info-s = 8 }} however when i try to display them like on wikipedia i can't {{yytop}} {{yy|User:AUBSTRAWBS/GATE}} {{yyend}} Anyways sorry to bother you with somthing like this but i'm really stumped as to how to share them. Any help would be super apreciated also if you want any user boxes i can make them :). {{unsigned|AUBSTRAWBS}} :Hello {{ping|AUBSTRAWBS}} no need to worry about bothering me, I'm always happy to help. I think for Wikiversity you'd have to manually create them, as I have done so. For example: [[Template:User Sri Lankan]] & [[Template:User soccer]]. If you'd like to bring over templates from Wikipedia, then feel free to just copy them and paste them here - tho it may be better just to manually create them as it could be a lengthy and messy process. I do think the way you've created the "national diet" userbox is perfect and achieves the intended goal. Let me know if you have any more questions! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:38, 9 June 2026 (UTC) adczzr8aoqrkat2fz20q5ylrud8rahf 2814856 2814851 2026-06-09T12:45:26Z Atcovi 276019 /* Question */ Reply 2814856 wikitext text/x-wiki [[User:Atcovi/Archive 1|/Archive 1 (September 25, 2013 - November 15, 2013)]] • [[User talk:Atcovi/Archive 2|/Archive 2 (November 15, 2013 - November 27, 2013)]] • [[User talk:Atcovi/Archive 3|/Archive 3 (December 3, 2013 - December 25, 2013)]] • [[User talk:Atcovi/Archive 4|/Archive 4 (December 24, 2013 - January 1, 2014)]] • [[User talk:Atcovi/Archive 5|/Archive 5 (January 2, 2014 - January 20, 2014)]] • [[User talk:Atcovi/Archive 6|/Archive 6 (March 24, 2014 - April 14, 2014)]] • [[User talk:Atcovi/Archive 7|/Archive 7 (April 19, 2014 - September 8, 2014)]] • [[User talk:Atcovi/Archive 8|/Archive 8 (September 12, 2014 - November 3, 2014)]] • [[User talk:Atcovi/Archive 9|/Archive 9 (November 6, 2014 - January 26, 2015)]] • [[User talk:Atcovi/Archive 10|/Archive 10 (January 28, 2015 - March 11, 2015)]] • [[User talk:Atcovi/Archive 11|/Archive 11 (March 22, 2015 - June 25, 2016)]] • [[User talk:Atcovi/Archive 12 (June 26, 2016 - January 8, 2018)|/Archive 12 (June 26, 2016 - January 8, 2018)]] • [[User talk:Atcovi/Archive 13 (January 9, 2018 - April 14, 2023)|/Archive 13 (January 9, 2018 - April 14, 2023)]] • [[User talk:Atcovi/Archive 14 (April 15, 2023 - May 5, 2026)|/Archive 14 (April 15, 2023 - May 5, 2026)]] :''Before 2013: [https://en.wikiversity.org/w/index.php?title=User_talk:Atcovi&diff=750617&oldid=740650 see this]'' {{tmbox |small = |image = [[Image:Busy desk.svg|{{#ifeq:|yes|40px|75x50px}}]] |text = This user is busy in [http://en.wikipedia.org/wiki/Real_life Real Life] {{#if:|until&nbsp;{{{end}}}&nbsp;}}{{#if:|due to&nbsp;{{{reason}}}&nbsp;}}and may not respond swiftly to queries.{{#if:|<P>{{{msg}}} }} | style = {{#if:|width: {{{width}}}px;}} {{#ifeq:{{{shadow}}}|yes|{{box-shadow|0px|2px|4px|rgba(0,0,0,0.2)}}|}} }} == Please vote == on Wikinews rebirth possibly on Wikiversity, thanks @[[User:Atcovi|Atcovi]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 22:21, 15 May 2026 (UTC) :Hi BigKrow. I've been watching the discussion on the sidelines. Hopefully I'll have an input soon, I just have other commitments I'm catering to. Best of luck with your projects and welcome to Wikiversity! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:44, 16 May 2026 (UTC) == ''The Signpost'': 22 May 2026 == <div lang="en" dir="ltr" class="mw-content-ltr" style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">[[File:WikipediaSignpostIcon.svg|40px|right]] ''News, reports and features from the English Wikipedia's newspaper''</div> <div style="column-count:2;"> * News and notes: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/News and notes|Offline: Osama Khalid still in prison]] * In the media: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/In the media|Indonesian editors, you shall return!]] * Disinformation report: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Disinformation report|Who is a typical paid editor? Who are their typical clients?]] * Recent research: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Recent research|WikiLambda the Ultimate]] * Traffic report: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Traffic report|This is where I'll be, so heavenly, so come and dance with me Michael!]] * Forum: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Forum|WikiAnnotate: help us build a dataset of article quality evaluations]] * In focus: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/In focus|Demystifying the 2026-27 Annual Plan]] * Opinion: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Opinion|Wikipedia isn't a battleground. So why does it feel like one?]] * Serendipity: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Serendipity|Wikinews: Into the Wikiverse]] * Special report: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Special report|Wikimedia Foundation closes Wikinews after 21 years]] * Community view: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Community view|Wikipedia's traffic drop: more on languages and freshness]] * Gallery: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Gallery|Earth Day and Mother's Day]] * Comix: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Comix|Brother, can you spare a page?]] </div> <div style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">'''[[w:en:Wikipedia:Wikipedia Signpost|Read this Signpost in full]]''' · [[w:en:Wikipedia:Signpost/Single|Single-page]] · [[m:Global message delivery/Targets/Signpost|Unsubscribe]] · [[m:Global message delivery|Global message delivery]] 05:19, 22 May 2026 (UTC) <!-- Sent via script ([[w:en:User:JPxG/SPS]]) --></div> <!-- Message sent by User:Bri@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Signpost&oldid=30513885 --> == Wikiversity:Candidates for Bureaucratship/Atcovi == RE: [[Wikiversity:Candidates for Bureaucratship/Atcovi]] I have closed this as successful. Congrats! See [https://en.wikiversity.org/w/index.php?title=Wikiversity:Candidates_for_Bureaucratship/Atcovi&diff=prev&oldid=2812184] and [https://en.wikiversity.org/w/index.php?title=Special:Log&logid=3549048]. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 19:23, 30 May 2026 (UTC) :Thank you Mike! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 20:55, 30 May 2026 (UTC) ::Congratulations @[[User:Atcovi|Atcovi]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 16:58, 8 June 2026 (UTC) == Question == Hello don't mean to bother and Ik its a silly question, on wikipedia there is a tool that allows for the creation of user boxes does wikiversity have it? Or should I create them myself like {{Userbox | border-c = #000000 | id = [[File:(logo) Gate jieitai kanochi nite, kaku tatakaeri.svg|100x50px]] | id-c = #000000 | id-fc = #000000 | id-s = 14 | info = This user testified in front of the [[National Diet|national diet]] | info-c = #000000 | info-fc = #ffffff | info-s = 8 }} however when i try to display them like on wikipedia i can't {{yytop}} {{yy|User:AUBSTRAWBS/GATE}} {{yyend}} Anyways sorry to bother you with somthing like this but i'm really stumped as to how to share them. Any help would be super apreciated also if you want any user boxes i can make them :). {{unsigned|AUBSTRAWBS}} :Hello {{ping|AUBSTRAWBS}} no need to worry about bothering me, I'm always happy to help. I think for Wikiversity you'd have to manually create them, as I have done so. For example: [[Template:User Sri Lankan]] & [[Template:User soccer]]. If you'd like to bring over templates from Wikipedia, then feel free to just copy them and paste them here - tho it may be better just to manually create them as it could be a lengthy and messy process. I do think the way you've created the "national diet" userbox is perfect and achieves the intended goal. Let me know if you have any more questions! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:38, 9 June 2026 (UTC) :Here's one I created just now: [[Template:User university student]]. More templates are listed here: [[:Category:User templates]]. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:45, 9 June 2026 (UTC) 4tvt8c63vsbldwtihb4y2w5pnfi2ryl 2814923 2814856 2026-06-09T23:02:19Z AUBSTRAWBS 3060598 /* Question */ Reply 2814923 wikitext text/x-wiki [[User:Atcovi/Archive 1|/Archive 1 (September 25, 2013 - November 15, 2013)]] • [[User talk:Atcovi/Archive 2|/Archive 2 (November 15, 2013 - November 27, 2013)]] • [[User talk:Atcovi/Archive 3|/Archive 3 (December 3, 2013 - December 25, 2013)]] • [[User talk:Atcovi/Archive 4|/Archive 4 (December 24, 2013 - January 1, 2014)]] • [[User talk:Atcovi/Archive 5|/Archive 5 (January 2, 2014 - January 20, 2014)]] • [[User talk:Atcovi/Archive 6|/Archive 6 (March 24, 2014 - April 14, 2014)]] • [[User talk:Atcovi/Archive 7|/Archive 7 (April 19, 2014 - September 8, 2014)]] • [[User talk:Atcovi/Archive 8|/Archive 8 (September 12, 2014 - November 3, 2014)]] • [[User talk:Atcovi/Archive 9|/Archive 9 (November 6, 2014 - January 26, 2015)]] • [[User talk:Atcovi/Archive 10|/Archive 10 (January 28, 2015 - March 11, 2015)]] • [[User talk:Atcovi/Archive 11|/Archive 11 (March 22, 2015 - June 25, 2016)]] • [[User talk:Atcovi/Archive 12 (June 26, 2016 - January 8, 2018)|/Archive 12 (June 26, 2016 - January 8, 2018)]] • [[User talk:Atcovi/Archive 13 (January 9, 2018 - April 14, 2023)|/Archive 13 (January 9, 2018 - April 14, 2023)]] • [[User talk:Atcovi/Archive 14 (April 15, 2023 - May 5, 2026)|/Archive 14 (April 15, 2023 - May 5, 2026)]] :''Before 2013: [https://en.wikiversity.org/w/index.php?title=User_talk:Atcovi&diff=750617&oldid=740650 see this]'' {{tmbox |small = |image = [[Image:Busy desk.svg|{{#ifeq:|yes|40px|75x50px}}]] |text = This user is busy in [http://en.wikipedia.org/wiki/Real_life Real Life] {{#if:|until&nbsp;{{{end}}}&nbsp;}}{{#if:|due to&nbsp;{{{reason}}}&nbsp;}}and may not respond swiftly to queries.{{#if:|<P>{{{msg}}} }} | style = {{#if:|width: {{{width}}}px;}} {{#ifeq:{{{shadow}}}|yes|{{box-shadow|0px|2px|4px|rgba(0,0,0,0.2)}}|}} }} == Please vote == on Wikinews rebirth possibly on Wikiversity, thanks @[[User:Atcovi|Atcovi]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 22:21, 15 May 2026 (UTC) :Hi BigKrow. I've been watching the discussion on the sidelines. Hopefully I'll have an input soon, I just have other commitments I'm catering to. Best of luck with your projects and welcome to Wikiversity! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:44, 16 May 2026 (UTC) == ''The Signpost'': 22 May 2026 == <div lang="en" dir="ltr" class="mw-content-ltr" style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">[[File:WikipediaSignpostIcon.svg|40px|right]] ''News, reports and features from the English Wikipedia's newspaper''</div> <div style="column-count:2;"> * News and notes: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/News and notes|Offline: Osama Khalid still in prison]] * In the media: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/In the media|Indonesian editors, you shall return!]] * Disinformation report: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Disinformation report|Who is a typical paid editor? Who are their typical clients?]] * Recent research: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Recent research|WikiLambda the Ultimate]] * Traffic report: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Traffic report|This is where I'll be, so heavenly, so come and dance with me Michael!]] * Forum: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Forum|WikiAnnotate: help us build a dataset of article quality evaluations]] * In focus: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/In focus|Demystifying the 2026-27 Annual Plan]] * Opinion: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Opinion|Wikipedia isn't a battleground. So why does it feel like one?]] * Serendipity: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Serendipity|Wikinews: Into the Wikiverse]] * Special report: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Special report|Wikimedia Foundation closes Wikinews after 21 years]] * Community view: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Community view|Wikipedia's traffic drop: more on languages and freshness]] * Gallery: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Gallery|Earth Day and Mother's Day]] * Comix: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Comix|Brother, can you spare a page?]] </div> <div style="margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;">'''[[w:en:Wikipedia:Wikipedia Signpost|Read this Signpost in full]]''' · [[w:en:Wikipedia:Signpost/Single|Single-page]] · [[m:Global message delivery/Targets/Signpost|Unsubscribe]] · [[m:Global message delivery|Global message delivery]] 05:19, 22 May 2026 (UTC) <!-- Sent via script ([[w:en:User:JPxG/SPS]]) --></div> <!-- Message sent by User:Bri@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Signpost&oldid=30513885 --> == Wikiversity:Candidates for Bureaucratship/Atcovi == RE: [[Wikiversity:Candidates for Bureaucratship/Atcovi]] I have closed this as successful. Congrats! See [https://en.wikiversity.org/w/index.php?title=Wikiversity:Candidates_for_Bureaucratship/Atcovi&diff=prev&oldid=2812184] and [https://en.wikiversity.org/w/index.php?title=Special:Log&logid=3549048]. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 19:23, 30 May 2026 (UTC) :Thank you Mike! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 20:55, 30 May 2026 (UTC) ::Congratulations @[[User:Atcovi|Atcovi]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 16:58, 8 June 2026 (UTC) == Question == Hello don't mean to bother and Ik its a silly question, on wikipedia there is a tool that allows for the creation of user boxes does wikiversity have it? Or should I create them myself like {{Userbox | border-c = #000000 | id = [[File:(logo) Gate jieitai kanochi nite, kaku tatakaeri.svg|100x50px]] | id-c = #000000 | id-fc = #000000 | id-s = 14 | info = This user testified in front of the [[National Diet|national diet]] | info-c = #000000 | info-fc = #ffffff | info-s = 8 }} however when i try to display them like on wikipedia i can't {{yytop}} {{yy|User:AUBSTRAWBS/GATE}} {{yyend}} Anyways sorry to bother you with somthing like this but i'm really stumped as to how to share them. Any help would be super apreciated also if you want any user boxes i can make them :). {{unsigned|AUBSTRAWBS}} :Hello {{ping|AUBSTRAWBS}} no need to worry about bothering me, I'm always happy to help. I think for Wikiversity you'd have to manually create them, as I have done so. For example: [[Template:User Sri Lankan]] & [[Template:User soccer]]. If you'd like to bring over templates from Wikipedia, then feel free to just copy them and paste them here - tho it may be better just to manually create them as it could be a lengthy and messy process. I do think the way you've created the "national diet" userbox is perfect and achieves the intended goal. Let me know if you have any more questions! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:38, 9 June 2026 (UTC) :Here's one I created just now: [[Template:User university student]]. More templates are listed here: [[:Category:User templates]]. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:45, 9 June 2026 (UTC) :Thank you very much [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 23:02, 9 June 2026 (UTC) 1jwimsljj2fx3hmo9upg1dacljci2jr 0 118036 2814886 1517437 2026-06-09T17:12:26Z Atcovi 276019 cleanup 2814886 wikitext text/x-wiki {{research}} ==Abstract== A recently new sport with its introduction to the 1998 Winter Olympics, snowboarding has come an exceptionally long way since the first boards were crafted in 1976, aided by advancements in materials, board shapes and super progressive riders. Jake Burton who founded [http://au.burton.com/on/demandware.store/Sites-Burton_AU-Site/default/Home-Show Burton] Snowboards and Tom Sims who founded [http://www.simsnow.com/ Sims] Snowboards a couple of the original manufacturers of snowboards. To lessen the impact this sport has on the environment, companies like [http://arborcollective.com/snow/ Arbor] and [http://k2snowboarding.com/ K2] are using bamboo, [http://au.burton.com/on/demandware.store/Sites-Burton_AU-Site/default/Home-Show Burton] and [http://www.forumsnowboards.com/ Forum] have specialised forests and [http://www.lib-tech.com/ Lib-Tech] use a unique bio plastic as a pose to normal plastics. These companies also pursue different board shapes with camber, reverse camber and boards with both as well as rail designs and board sidewalls. Like [http://www.lib-tech.com/ Lib-Tech’s] [http://www.lib-tech.com/snowboard-technology/ Magne-traction] edge which provides better traction and grip by cutting into the snow, and [http://au.burton.com/on/demandware.store/Sites-Burton_AU-Site/default/Home-Show Burton's] [http://au.burton.com/on/demandware.store/Sites-Burton_AU-Site/default/Search-Show?q=mid+spoon Mid-Spoon] edge which makes the board more catch-free when sliding rails in the park. Since [http://au.burton.com/on/demandware.store/Sites-Burton_AU-Site/default/Home-Show Burton] and [http://www.simsnow.com/ Sims] were founded, most snowboards were shaped like skis, with the tip and tail on contact with the snow and the middle of the board off the snow. In the last few years snowboard shapes have changed dramatically, giving rise to reverse camber, the middle for the board touches to snow and the tip and tail are off and boards with both camber and reverse camber, giving the benefits of both shapes. {{stub}} [[Category:Research]] jsibm37zyxoeoev82cjqztacacxk7be 0 139384 2814863 2814567 2026-06-09T14:03:22Z Young1lim 21186 /* Adder */ 2814863 wikitext text/x-wiki == Adder == * Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]] ) {| class="wikitable" |- ! Adder type !! Overview !! Analysis !! VHDL Level Design !! CMOS Level Design |- | '''1. Ripple Carry Adder''' || [[Media:VLSI.Arith.1A.RCA.20250522.pdf|A]]|| || [[Media:Adder.rca.20140313.pdf|pdf]] || [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]] |- | '''2. Carry Lookahead Adder''' || [[Media:VLSI.Arith.2A.CLA.20260609.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20260609.pdf|B]] || || [[Media:Adder.cla.20140313.pdf|pdf]]|| |- | '''3. Carry Save Adder''' || [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]|| || || |- || '''4. Carry Select Adder''' || [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]|| || || |- || '''5. Carry Skip Adder''' || [[Media:VLSI.Arith.5A.CSkip.20250405.pdf|A]]|| || || [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]] |- || '''6. Carry Chain Adder''' || [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]|| || [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]], [[Media:Adder.cca.20140313.pdf|pdf]] || [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]] |- || '''7. Kogge-Stone Adder''' || [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]|| || [[Media:Adder.ksa.20140409.pdf|pdf]]|| |- || '''8. Prefix Adder''' || [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]|| || || |- || '''9.1 Variable Block Adder''' || [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]], [[Media:VLSI.Arith.1C.VBA.20250218.pdf|D]]|| || || |- || '''9.2 Multi-Level Variable Block Adder''' || [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]|| || || |} </br> === Adder Architectures Suitable for FPGA === * FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]]) * FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]]) * FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]]) * FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]]) * Carry-Skip Adder </br> == Barrel Shifter == * Barrel Shifter Architecture Exploration ([[Media:Bshift.20131105.pdf|bshfit.vhdl]], [[Media:Bshift.makefile.20131109.pdf|bshfit.makefile]]) </br> '''Mux Based Barrel Shifter''' * Analysis ([[Media:Arith.BShfiter.20151207.pdf|pdf]]) * Implementation </br> == Multiplier == === Array Multipliers === * Analysis ([[Media:VLSI.Arith.1.A.Mult.20151209.pdf|pdf]]) </br> === Tree Mulltipliers === * Lattice Multiplication ([[Media:VLSI.Arith.LatticeMult.20170204.pdf|pdf]]) * Wallace Tree ([[Media:VLSI.Arith.WallaceTree.20170204.pdf|pdf]]) * Dadda Tree ([[Media:VLSI.Arith.DaddaTree.20170701.pdf|pdf]]) </br> === Booth Multipliers === * [[Media:RNS4.BoothEncode.20161005.pdf|Booth Encoding Note]] * Booth Multiplier Note ([[Media:BoothMult.20160929.pdf|H1.pdf]]) </br> == Divider == * Binary Divider ([[Media:VLSI.Arith.1.A.Divider.20131217.pdf|pdf]])</br> </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:Digital Circuit Design]] [[Category:FPGA]] ema4oibgrp8d6ebrih42rnz9r5rao7t 10 145749 2814854 1652832 2026-06-09T12:42:31Z Atcovi 276019 change some stuff around 2814854 wikitext text/x-wiki {{userbox | border-c = blue | id = [[File:Soccer ball.JPG|50px]] | id-c = green | id-fc = white | id-s = 14 | info = This user plays '''[[Football (soccer)|football (soccer)]]'''. | info-c = white | info-fc = black | info-s = 8 }}<noinclude> [[Category:Userboxes|Soccer]] </noinclude> c3zcsr1lvrmfxpulv61jhjyg1dms7mb 0 154046 2814921 2803794 2026-06-09T20:23:40Z ~2026-33955-08 3091653 I addded why the Spanish aplhebet has one more letter than the English alphabet 2814921 wikitext text/x-wiki The Spanish alphabet has 27 letters. Unlike the English alphabet which has 26 letters and and Spanish has 27 because of the unique letter ñ. Synonyms: alfabeto (from Ancient Greek''ἄλφα/alfa/alpha'', ''beta''...) and abecedario (from Spanish sounding: a-be-ce-da-rio). [[file:Es-alphabet-Spain.oga]] ---- Notes: #Only vocals may contain an accent (''' ' '''). Only the letter "u" may show an accent or dieresis ('''ü'''). #The letter "'''H'''" or "'''h'''" is silent, unless it is preceded by the letter "'''C'''" or "'''c'''" to form "'''''Ch'''''" (or by the letter '''S''' to form '''''Sh'''''). #The combination of the letters '''P''' or '''p''' and '''h''' used in English is not used in Spanish, particularly to produce the '''''f''''' sound; so words like ''alpha'' (in English) are written as ''alfa'' in Spanish. #The letter "'''Q'''" or "'''q'''" is basically accompanied by the letter "'''u'''", to be followed by a vocal, such as "'''e'''" or "'''i'''" to form the syllable ''que'' and ''qui''. #The letter "'''W'''" or "'''w'''", as the combination "'''Sh'''" or "'''sh'''" (and in some instances the letter "'''x'''" or "'''x'''") present flexibility to accommodate some neologisms, anglicisms, and other terms and names not necessarily original in Spanish. For instance: '''''W'''a'''sh'''ington''. #The letter "'''X'''" or "'''x'''", encompasses different sounds in pronunciation depending on the use of the word and the position of the letter in the word. ---- {|class="wikitable" |- !# !Letter (Uppercase) !Letter (Lowercase) !Pronunciation (Name of the letter) !Sound (Name of the letter) |- |1 |A |a (á) |a |[[file:letter a es es.flac]] |- |2 |B |b |be |[[file:letter b es es.flac]] |- |3 |C |c |ce |[[file:letter c es es.flac]] |- |4 |D |d |de |[[file:letter d es es.flac]] |- |5 |E |e (é) |e |[[file:letter e es es.flac]] |- |6 |F |f |efe |[[file:letter f es es.flac]] |- |7 |G |g |ge |[[file:letter g es es.flac]] |- |8 |H |h |atcheh |[[file:letter h es es.flac]] |- |9 |I |i (í) |i |[[file:letter i es es.flac]] |- |10 |J |j |jota |[[file:letter j es es.flac]] |- |11 |K |k |ka |[[file:letter k es es.flac]] |- |12 |L |l |ele |[[file:letter l es es.flac]] |- |13 |M |m |eme |[[file:letter m es es.flac]] |- |14 |N |n |ene |[[file:letter n es es.flac]] |- |15 |Ñ |ñ |eñe |[[File:Letter ñ es es.flac]] |- |16 |O |o (ó) |o |[[file:letter o es es.flac]] |- |17 |P |p |pe |[[file:letter p es es.flac]] |- |18 |Q |q |coo |[[file:letter q es es.flac]] |- |19 |R |r |ere |[[file:letter r es es.flac]] |- |20 |S |s |ese |[[file:letter s es es.flac]] |- |21 |T |t |te |[[file:letter t es es.flac]] |- |22 |U |u (ú, ü) |u |[[file:letter u es es.flac]] |- |23 |V |v |uve |[[file:letter v es es.flac]] |- |24 |W |w |doble uve |[[file:letter w es es.flac]] |- |25 |X |x |equis |[[file:letter x es es.flac]] |- |26 |Y |y |ye, i griega |[[File:Letter y recommended es es.flac]][[file:letter y es es.flac]] |- |27 |Z |z |zeta |[[file:letter z es es.flac]] |- |} * [[Wikipedia: Spanish alphabet]] * [[Portal: Spanish]] [[Category:Alphabets]] [[Category:Spanish]] [[Category:Spanish One]] covi5tob9bl0q77ig662d10ew8rbx3e 0 168166 2814978 2707407 2026-06-10T07:02:43Z Jtneill 10242 + categorytree 2814978 wikitext text/x-wiki ''A topic page''. A game is structured playing, usually undertaken for enjoyment and sometimes used as an educational tool. Games are distinct from work, which is usually carried out for remuneration, and from art, which is more often an expression of aesthetic or ideological elements.<ref>[[Wikipedia: Game]]</ref> <categorytree mode="pages">Games</categorytree> == Subpages == * [[/Learn to Fly/]] == References == {{Reflist}} {{CourseCat}} 4svdw25yw5w2f3y17mep031p423f9lv 0 170868 2814871 2133621 2026-06-09T14:24:26Z Atcovi 276019 project box(es) 2814871 wikitext text/x-wiki {{LeftTOC}} {{chemistry}} {{secondary}} {{Lesson}} {{complete}} {{clear}} ==Chemical element and Chemicals== A chemical element is a substance that cannot be changed into any other substance by a normal chemical reaction.Chemicals are made of two or more chemical elements. An chemical element is made up of atoms containing electrons, protons and neutrons.Electrons some chemicals can move. Chemicals that electrons are moving ,are called conductor . ==Definition of Electrical current== Electrical current(or current) is movement of electrons in a conductor. [[file:Ampere coulomb.svg|500px|left]] {{clear}} == [[image:Nuvola_apps_bookcase.svg | 24px]] Readings == # Read [[Current|Electrical Current]]. {{clear}} dk2pov7bromfwypgcb4h3kavhsqduj6 2814872 2814871 2026-06-09T14:24:51Z Atcovi 276019 cleanup 2814872 wikitext text/x-wiki {{cleanup|grammar needs work}} {{LeftTOC}} {{chemistry}} {{secondary}} {{Lesson}} {{complete}} {{clear}} ==Chemical element and Chemicals== A chemical element is a substance that cannot be changed into any other substance by a normal chemical reaction.Chemicals are made of two or more chemical elements. An chemical element is made up of atoms containing electrons, protons and neutrons.Electrons some chemicals can move. Chemicals that electrons are moving ,are called conductor . ==Definition of Electrical current== Electrical current(or current) is movement of electrons in a conductor. [[file:Ampere coulomb.svg|500px|left]] {{clear}} == [[image:Nuvola_apps_bookcase.svg | 24px]] Readings == # Read [[Current|Electrical Current]]. {{clear}} gv8sspbaeost10r5b448iup4kcac66v 0 171005 2814868 2813276 2026-06-09T14:21:00Z Young1lim 21186 /* Geometric Series Examples */ 2814868 wikitext text/x-wiki Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers <math>x+iy</math>, where <math>i=\sqrt{-1}</math>, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other <math>L</math>-functions, modular forms, elliptic functions, etc. <blockquote>The shortest path between two truths in the real domain passes through the complex domain. — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]</blockquote>In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}} ==''' Complex Functions '''== * Complex Functions ([[Media:CAnal.1.A.CFunction.20140222.Basic.pdf|1.A.pdf]], [[Media:CAnal.1.B.CFunction.20140111.Octave.pdf|1.B.pdf]], [[Media:CAnal.1.C.CFunction.20140111.Extend.pdf|1.C.pdf]]) * Complex Exponential and Logarithm ([[Media:CAnal.5.A.CLog.20131017.pdf|5.A.pdf]], [[Media:CAnal.5.A.Octave.pdf|5.B.pdf]]) * Complex Trigonometric and Hyperbolic ([[Media:CAnal.7.A.CTrigHyper..pdf|7.A.pdf]], [[Media:CAnal.7.A.Octave..pdf|7.B.pdf]]) '''Complex Function Note''' : 1. Exp and Log Function Note ([[Media:ComplexExp.29160721.pdf|H1.pdf]]) : 2. Trig and TrigH Function Note ([[Media:CAnal.Trig-H.29160901.pdf|H1.pdf]]) : 3. Inverse Trig and TrigH Functions Note ([[Media:CAnal.Hyper.29160829.pdf|H1.pdf]]) ==''' Complex Integrals '''== * Complex Integrals ([[Media:CAnal.2.A.CIntegral.20140224.Basic.pdf|2.A.pdf]], [[Media:CAnal.2.B.CIntegral.20140117.Octave.pdf|2.B.pdf]], [[Media:CAnal.2.C.CIntegral.20140117.Extend.pdf|2.C.pdf]]) ==''' Complex Series '''== * Complex Series ([[Media:CPX.Series.20150226.2.Basic.pdf|3.A.pdf]], [[Media:CAnal.3.B.CSeries.20140121.Octave.pdf|3.B.pdf]], [[Media:CAnal.3.C.CSeries.20140303.Extend.pdf|3.C.pdf]]) ==''' Residue Integrals '''== * Residue Integrals ([[Media:CAnal.4.A.Residue.20140227.Basic.pdf|4.A.pdf]], [[Media:CAnal.4.B.pdf|4.B.pdf]], [[Media:CAnal.4.C.Residue.20140423.Extend.pdf|4.C.pdf]]) ==='''Residue Integrals Note'''=== * Laurent Series with the Residue Theorem Note ([[Media:Laurent.1.Residue.20170713.pdf|H1.pdf]]) * Laurent Series with Applications Note ([[Media:Laurent.2.Applications.20170327.pdf|H1.pdf]]) * Laurent Series and the z-Transform Note ([[Media:Laurent.3.z-Trans.20170831.pdf|H1.pdf]]) * Laurent Series as a Geometric Series Note ([[Media:Laurent.4.GSeries.20170802.pdf|H1.pdf]]) === Laurent Series and the z-Transform Example Note === * Overview ([[Media:Laurent.4.z-Example.20170926.pdf|H1.pdf]]) ====Geometric Series Examples==== * Causality ([[Media:Laurent.5.Causality.1.A.20191026n.pdf|A.pdf]], [[Media:Laurent.5.Causality.1.B.20191026.pdf|B.pdf]]) * Time Shift ([[Media:Laurent.5.TimeShift.2.A.20191028.pdf|A.pdf]], [[Media:Laurent.5.TimeShift.2.B.20191029.pdf|B.pdf]]) * Reciprocity ([[Media:Laurent.5.Reciprocity.3A.20191030.pdf|A.pdf]], [[Media:Laurent.5.Reciprocity.3B.20191031.pdf|B.pdf]]) * Combinations ([[Media:Laurent.5.Combination.4A.20200702.pdf|A.pdf]], [[Media:Laurent.5.Combination.4B.20201002.pdf|B.pdf]]) * Properties ([[Media:Laurent.5.Property.5A.20220105.pdf|A.pdf]], [[Media:Laurent.5.Property.5B.20220126.pdf|B.pdf]]) * Permutations ([[Media:Laurent.6.Permutation.6A.20230711.pdf|A.pdf]], [[Media:Laurent.5.Permutation.6B.20251225.pdf|B.pdf]], [[Media:Laurent.5.Permutation.6C.20260608.pdf|C.pdf]], [[Media:Laurent.5.Permutation.6C.20240528.pdf|D.pdf]]) * Applications ([[Media:Laurent.5.Application.6B.20220723.pdf|A.pdf]]) * Double Pole Case :- Examples ([[Media:Laurent.5.DPoleEx.7A.20220722.pdf|A.pdf]], [[Media:Laurent.5.DPoleEx.7B.20220720.pdf|B.pdf]]) :- Properties ([[Media:Laurent.5.DPoleProp.5A.20190226.pdf|A.pdf]], [[Media:Laurent.5.DPoleProp.5B.20190228.pdf|B.pdf]]) ====The Case Examples==== * Example Overview : ([[Media:Laurent.4.Example.0.A.20171208.pdf|0A.pdf]], [[Media:Laurent.6.CaseExample.0.B.20180205.pdf|0B.pdf]]) * Example Case 1 : ([[Media:Laurent.4.Example.1.A.20171107.pdf|1A.pdf]], [[Media:Laurent.4.Example.1.B.20171227.pdf|1B.pdf]]) * Example Case 2 : ([[Media:Laurent.4.Example.2.A.20171107.pdf|2A.pdf]], [[Media:Laurent.4.Example.2.B.20171227.pdf|2B.pdf]]) * Example Case 3 : ([[Media:Laurent.4.Example.3.A.20171017.pdf|3A.pdf]], [[Media:Laurent.4.Example.3.B.20171226.pdf|3B.pdf]]) * Example Case 4 : ([[Media:Laurent.4.Example.4.A.20171017.pdf|4A.pdf]], [[Media:Laurent.4.Example.4.B.20171228.pdf|4B.pdf]]) * Example Summary : ([[Media:Laurent.4.Example.5.A.20171212.pdf|5A.pdf]], [[Media:Laurent.4.Example.5.B.20171230.pdf|5B.pdf]]) ==''' Conformal Mapping '''== * Conformal Mapping ([[Media:CAnal.6.A.Conformal.20131224.pdf|6.A.pdf]], [[Media:CAnal.6.A.Octave..pdf|6.B.pdf]]) go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:Complex analysis]] a98lia5ipnrlybd6fack1nixx2lbcdd 2814873 2814868 2026-06-09T14:27:02Z Young1lim 21186 /* Geometric Series Examples */ 2814873 wikitext text/x-wiki Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers <math>x+iy</math>, where <math>i=\sqrt{-1}</math>, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other <math>L</math>-functions, modular forms, elliptic functions, etc. <blockquote>The shortest path between two truths in the real domain passes through the complex domain. — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]</blockquote>In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}} ==''' Complex Functions '''== * Complex Functions ([[Media:CAnal.1.A.CFunction.20140222.Basic.pdf|1.A.pdf]], [[Media:CAnal.1.B.CFunction.20140111.Octave.pdf|1.B.pdf]], [[Media:CAnal.1.C.CFunction.20140111.Extend.pdf|1.C.pdf]]) * Complex Exponential and Logarithm ([[Media:CAnal.5.A.CLog.20131017.pdf|5.A.pdf]], [[Media:CAnal.5.A.Octave.pdf|5.B.pdf]]) * Complex Trigonometric and Hyperbolic ([[Media:CAnal.7.A.CTrigHyper..pdf|7.A.pdf]], [[Media:CAnal.7.A.Octave..pdf|7.B.pdf]]) '''Complex Function Note''' : 1. Exp and Log Function Note ([[Media:ComplexExp.29160721.pdf|H1.pdf]]) : 2. Trig and TrigH Function Note ([[Media:CAnal.Trig-H.29160901.pdf|H1.pdf]]) : 3. Inverse Trig and TrigH Functions Note ([[Media:CAnal.Hyper.29160829.pdf|H1.pdf]]) ==''' Complex Integrals '''== * Complex Integrals ([[Media:CAnal.2.A.CIntegral.20140224.Basic.pdf|2.A.pdf]], [[Media:CAnal.2.B.CIntegral.20140117.Octave.pdf|2.B.pdf]], [[Media:CAnal.2.C.CIntegral.20140117.Extend.pdf|2.C.pdf]]) ==''' Complex Series '''== * Complex Series ([[Media:CPX.Series.20150226.2.Basic.pdf|3.A.pdf]], [[Media:CAnal.3.B.CSeries.20140121.Octave.pdf|3.B.pdf]], [[Media:CAnal.3.C.CSeries.20140303.Extend.pdf|3.C.pdf]]) ==''' Residue Integrals '''== * Residue Integrals ([[Media:CAnal.4.A.Residue.20140227.Basic.pdf|4.A.pdf]], [[Media:CAnal.4.B.pdf|4.B.pdf]], [[Media:CAnal.4.C.Residue.20140423.Extend.pdf|4.C.pdf]]) ==='''Residue Integrals Note'''=== * Laurent Series with the Residue Theorem Note ([[Media:Laurent.1.Residue.20170713.pdf|H1.pdf]]) * Laurent Series with Applications Note ([[Media:Laurent.2.Applications.20170327.pdf|H1.pdf]]) * Laurent Series and the z-Transform Note ([[Media:Laurent.3.z-Trans.20170831.pdf|H1.pdf]]) * Laurent Series as a Geometric Series Note ([[Media:Laurent.4.GSeries.20170802.pdf|H1.pdf]]) === Laurent Series and the z-Transform Example Note === * Overview ([[Media:Laurent.4.z-Example.20170926.pdf|H1.pdf]]) ====Geometric Series Examples==== * Causality ([[Media:Laurent.5.Causality.1.A.20191026n.pdf|A.pdf]], [[Media:Laurent.5.Causality.1.B.20191026.pdf|B.pdf]]) * Time Shift ([[Media:Laurent.5.TimeShift.2.A.20191028.pdf|A.pdf]], [[Media:Laurent.5.TimeShift.2.B.20191029.pdf|B.pdf]]) * Reciprocity ([[Media:Laurent.5.Reciprocity.3A.20191030.pdf|A.pdf]], [[Media:Laurent.5.Reciprocity.3B.20191031.pdf|B.pdf]]) * Combinations ([[Media:Laurent.5.Combination.4A.20200702.pdf|A.pdf]], [[Media:Laurent.5.Combination.4B.20201002.pdf|B.pdf]]) * Properties ([[Media:Laurent.5.Property.5A.20220105.pdf|A.pdf]], [[Media:Laurent.5.Property.5B.20220126.pdf|B.pdf]]) * Permutations ([[Media:Laurent.6.Permutation.6A.20230711.pdf|A.pdf]], [[Media:Laurent.5.Permutation.6B.20251225.pdf|B.pdf]], [[Media:Laurent.5.Permutation.6C.20260609.pdf|C.pdf]], [[Media:Laurent.5.Permutation.6C.20240528.pdf|D.pdf]]) * Applications ([[Media:Laurent.5.Application.6B.20220723.pdf|A.pdf]]) * Double Pole Case :- Examples ([[Media:Laurent.5.DPoleEx.7A.20220722.pdf|A.pdf]], [[Media:Laurent.5.DPoleEx.7B.20220720.pdf|B.pdf]]) :- Properties ([[Media:Laurent.5.DPoleProp.5A.20190226.pdf|A.pdf]], [[Media:Laurent.5.DPoleProp.5B.20190228.pdf|B.pdf]]) ====The Case Examples==== * Example Overview : ([[Media:Laurent.4.Example.0.A.20171208.pdf|0A.pdf]], [[Media:Laurent.6.CaseExample.0.B.20180205.pdf|0B.pdf]]) * Example Case 1 : ([[Media:Laurent.4.Example.1.A.20171107.pdf|1A.pdf]], [[Media:Laurent.4.Example.1.B.20171227.pdf|1B.pdf]]) * Example Case 2 : ([[Media:Laurent.4.Example.2.A.20171107.pdf|2A.pdf]], [[Media:Laurent.4.Example.2.B.20171227.pdf|2B.pdf]]) * Example Case 3 : ([[Media:Laurent.4.Example.3.A.20171017.pdf|3A.pdf]], [[Media:Laurent.4.Example.3.B.20171226.pdf|3B.pdf]]) * Example Case 4 : ([[Media:Laurent.4.Example.4.A.20171017.pdf|4A.pdf]], [[Media:Laurent.4.Example.4.B.20171228.pdf|4B.pdf]]) * Example Summary : ([[Media:Laurent.4.Example.5.A.20171212.pdf|5A.pdf]], [[Media:Laurent.4.Example.5.B.20171230.pdf|5B.pdf]]) ==''' Conformal Mapping '''== * Conformal Mapping ([[Media:CAnal.6.A.Conformal.20131224.pdf|6.A.pdf]], [[Media:CAnal.6.A.Octave..pdf|6.B.pdf]]) go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:Complex analysis]] trzasrhcrsimvrogi35710ap7iuiac6 102 206305 2814908 2696939 2026-06-09T19:42:04Z IanVG 2918363 added table of math related resources, collected from the hunter-gatherer page. 2814908 wikitext text/x-wiki * [[{{Titleparts|2}}/Resources|here]] {| class="wikitable sortable" |+ !Resource !Mathematics material !Focus |- |[http://www.bymath.com/studyguide/alg/alg_topics.html ByMath: Algebra] |Algebra lessons and study guides |Mathematics |- |[http://www.bymath.com/studyguide/ari/ari_topics.html ByMath: Basic Arithmetic] |Basic arithmetic lessons |Mathematics |- |[http://www.geogebratube.org/student/m14908?mobile=true GeoGebra: Similar Figures] |Interactive geometry activity about similar figures |Mathematics |- |[https://www.math10.com/ Math10] |Lessons and problems from elementary through introductory university mathematics |Mathematics |- |[https://www.math-drills.com/ Math Drills] |Elementary and middle-school mathematics worksheets |Mathematics |- |[http://www.mathsolutions.com/index.cfm Math Solutions] |Mathematics teaching resources by Marilyn Burns |Mathematics education |- |[http://www.mathvillage.info/ Math Village] |Middle-school mathematics teaching materials |Mathematics education |- |[http://stemsheets.com/ STEM Sheets] |Customizable and printable mathematics and science worksheets |Mathematics and science |- |[http://archives.math.utk.edu/visual.calculus/ Visual Calculus] |Single-variable calculus explanations and exercises |Mathematics |- |[https://visualfractions.com/ Visual Fractions] |Fractions, percentages, calculators, and unit conversions |Mathematics |- |[http://www.ck12.org/ CK-12 Foundation] |Mathematics textbooks, lessons, and exercises |General education |- |[http://www.coursera.org Coursera] |Online mathematics and related courses |General education |- |[[w:EdX|EdX]] |Online mathematics and related courses |General education |- |[http://hippocampus.org/ HippoCampus] |Multimedia mathematics and general-education materials |General education |- |[http://www.khanacademy.org/ Khan Academy] |Mathematics lessons and exercises covering many levels |General education |- |[http://ocw.mit.edu/index.html MIT OpenCourseWare] |University-level mathematics course materials |University courseware |- |[http://cnx.org/ OpenStax] |Open mathematics textbooks and scholarly materials |Open textbooks |- |[http://www.open.ac.uk/openlearn OpenLearn] |Open mathematics and general-education course materials |General education |- |[http://www.saylor.org/ Saylor.org] |Free college-level mathematics and related courses |General education |- |[http://demonstrations.wolfram.com Wolfram Demonstrations Project] |Interactive demonstrations covering mathematics and related subjects |Mathematics and science |- |[https://worksheetgenius.com WorksheetGenius] |Customizable educational worksheet generators, including mathematics |General education |} tdgmncm3mld59l4blqxa3rnv6kl2m01 2814909 2814908 2026-06-09T19:42:56Z IanVG 2918363 2814909 wikitext text/x-wiki {| class="wikitable sortable" |+ !Resource !Mathematics material !Focus |- |[http://www.bymath.com/studyguide/alg/alg_topics.html ByMath: Algebra] |Algebra lessons and study guides |Mathematics |- |[http://www.bymath.com/studyguide/ari/ari_topics.html ByMath: Basic Arithmetic] |Basic arithmetic lessons |Mathematics |- |[http://www.geogebratube.org/student/m14908?mobile=true GeoGebra: Similar Figures] |Interactive geometry activity about similar figures |Mathematics |- |[https://www.math10.com/ Math10] |Lessons and problems from elementary through introductory university mathematics |Mathematics |- |[https://www.math-drills.com/ Math Drills] |Elementary and middle-school mathematics worksheets |Mathematics |- |[http://www.mathsolutions.com/index.cfm Math Solutions] |Mathematics teaching resources by Marilyn Burns |Mathematics education |- |[http://www.mathvillage.info/ Math Village] |Middle-school mathematics teaching materials |Mathematics education |- |[http://stemsheets.com/ STEM Sheets] |Customizable and printable mathematics and science worksheets |Mathematics and science |- |[http://archives.math.utk.edu/visual.calculus/ Visual Calculus] |Single-variable calculus explanations and exercises |Mathematics |- |[https://visualfractions.com/ Visual Fractions] |Fractions, percentages, calculators, and unit conversions |Mathematics |- |[http://www.ck12.org/ CK-12 Foundation] |Mathematics textbooks, lessons, and exercises |General education |- |[http://www.coursera.org Coursera] |Online mathematics and related courses |General education |- |[[w:EdX|EdX]] |Online mathematics and related courses |General education |- |[http://hippocampus.org/ HippoCampus] |Multimedia mathematics and general-education materials |General education |- |[http://www.khanacademy.org/ Khan Academy] |Mathematics lessons and exercises covering many levels |General education |- |[http://ocw.mit.edu/index.html MIT OpenCourseWare] |University-level mathematics course materials |University courseware |- |[http://cnx.org/ OpenStax] |Open mathematics textbooks and scholarly materials |Open textbooks |- |[http://www.open.ac.uk/openlearn OpenLearn] |Open mathematics and general-education course materials |General education |- |[http://www.saylor.org/ Saylor.org] |Free college-level mathematics and related courses |General education |- |[http://demonstrations.wolfram.com Wolfram Demonstrations Project] |Interactive demonstrations covering mathematics and related subjects |Mathematics and science |- |[https://worksheetgenius.com WorksheetGenius] |Customizable educational worksheet generators, including mathematics |General education |} cssm4pa8g55tvt10ispr91cxznctzmb 102 206307 2814901 2804480 2026-06-09T19:00:22Z IanVG 2918363 2814901 wikitext text/x-wiki Participating to help others:)Mecayla Joyce Williams from Kagisho Comprehensive Sec.School in Bloemfontein Free State,Mathematician * Since 2026 --[[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 19:00, 9 June 2026 (UTC) * Since 11 November 2013‎ with [[Draft:Mathematics|Mathematics]] --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 18:23, 9 May 2018 (UTC) * [[User:MathematicalAdam|MathematicalAdam]] ([[User talk:MathematicalAdam|discuss]] • [[Special:Contributions/MathematicalAdam|contribs]]) 01:09, 2 May 2019 (UTC) * Palanie Komala. [[User:Pelanie For Life|Pelanie For Life]] ([[User talk:Pelanie For Life|discuss]] • [[Special:Contributions/Pelanie For Life|contribs]]) * [[User:Sashatookool|Sashatookool]] ([[User talk:Sashatookool|discuss]] •[[Special:Contributions/Sashatookool|contribs]]) 11:20,22 December 2020(UTC) * [[User:Addemf|Addem]] ([[User talk:Addemf|discuss]] • [[Special:Contributions/Addemf|contribs]]) 4:19, 4 July 2021 (UTC) * <span style="border-radius:9em;padding:2px;background-color:darkblue; {{Text color default}};">[[User:MrMeAndMrMe|<span style="color:#FFF;">'''MrMe<i>And</i>MrMe'''</span>]]</span>_{[[User talk:MrMeAndMrMe|<span style="color:darkblue;">'''Let's talk'''</span>]]} 03:30, 26 March 2022 (UTC) * [[User:CamjamPNG|Camjam]] ([[User talk:CamjamPNG|discuss]] • [[Special:Contributions/CamjamPNG|contribs]]) 23:56, 22 September 2022 (UTC) *[[User:AHEJJWILEMAMALIDGED|AHEJJWILEMAMALIDGED]] ([[User talk:AHEJJWILEMAMALIDGED|discuss]] • [[Special:Contributions/AHEJJWILEMAMALIDGED|contribs]]) 03:35, 2 November 2022 (UTC) * [[User:PicoMath|PicoMath]] ([[User talk:PicoMath|discuss]] • [[Special:Contributions/PicoMath|contribs]]) 20:02, 2 December 2023 (UTC) * [[User:Fluxjupyter|Fluxjupyter]] ([[User talk:Fluxjupyter|discuss]] • [[Special:Contributions/Fluxjupyter|contribs]]) , 17:41, 25 June 2024 (UTC) gvryt8dhabp0nqor9exqzvq26a7wg1l 0 207402 2815028 2637172 2026-06-10T11:03:59Z Jessephu 3079828 2815028 wikitext text/x-wiki =A= '''Alignment''' – The arrangement of elements in a design so they line up neatly. '''Aspect Ratio''' – The proportional relationship between an image's width and height (e.g., 16:9). =B= '''Bleed''' - The area outside the trim edge that ensures no unprinted edges appear. '''Brand Identity''' – Visual elements such as logos, colors, and typography that represent a brand. =C= '''CMYK''' - Cyan, Magenta, Yellow, Key (Black). CMYK is the standard colour mode used for printing. =D= '''DPI''' - Dots per Inch on a printed page. =E= =F= =G= =H= '''Hierarchy''' - The arrangement of elements in a way that signifies importance. =I= =J= =K= '''Kerning -''' The spacing between individual characters in a word. =L= =M= '''Mockup -''' A realistic model of a design used for presentation or feedback. =N= '''Negative Space -''' The empty space around and between elements in a design. =O= =P= '''PPI''' - Pixels per Inch; number of pixels per inch in an image. =Q= =R= '''Raster Image''' - an image made up of thousands of pixels. Also known as bitmap image. Photos are an example of a raster image. <ref>Creative Blog. (2015). 6 Key Terms Every Graphic Designer should know. Available: http://www.creativebloq.com/graphic-design/key-terms-to-know-6133210</ref> '''RGB''' - Red, Green, Blue. RGB is the colour mode used for screen output. =S= =T= '''Typography -''' The art of arranging text to make it legible and visually appealing. =U= =V= '''Vector Image''' - an image made up of points, as opposed to raster images which are made from pixels. Each point has a defined X and Y coordinates. Vector images can be resized without loss of quality. <ref>Creative Blog. (2015). 6 Key Terms Every Graphic Designer should know. Available: http://www.creativebloq.com/graphic-design/key-terms-to-know-6133210</ref> =W= '''Wireframe -''' A basic layout of a design that outlines structure without detailed visuals. =X= =Y= =Z= =See Also= [http://www.superdream.co.uk/glossary-of-design-terms/ Superdream Glossary of Design Terms] =References= <div class="references-small"> <references/> </div> [[Category:Graphic design]] jc2taoz47l6ys290vljj99b7cth2jor 2815029 2815028 2026-06-10T11:07:34Z Jessephu 3079828 /* C */ 2815029 wikitext text/x-wiki =A= '''Alignment''' – The arrangement of elements in a design so they line up neatly. '''Aspect Ratio''' – The proportional relationship between an image's width and height (e.g., 16:9). =B= '''Bleed''' - The area outside the trim edge that ensures no unprinted edges appear. '''Brand Identity''' – Visual elements such as logos, colors, and typography that represent a brand. =C= '''CMYK''' - Cyan, Magenta, Yellow, Key (Black). CMYK is the standard colour mode used for printing. '''Cropping''' – Removing unwanted parts of an image. =D= '''DPI''' - Dots per Inch on a printed page. =E= =F= =G= =H= '''Hierarchy''' - The arrangement of elements in a way that signifies importance. =I= =J= =K= '''Kerning -''' The spacing between individual characters in a word. =L= =M= '''Mockup -''' A realistic model of a design used for presentation or feedback. =N= '''Negative Space -''' The empty space around and between elements in a design. =O= =P= '''PPI''' - Pixels per Inch; number of pixels per inch in an image. =Q= =R= '''Raster Image''' - an image made up of thousands of pixels. Also known as bitmap image. Photos are an example of a raster image. <ref>Creative Blog. (2015). 6 Key Terms Every Graphic Designer should know. Available: http://www.creativebloq.com/graphic-design/key-terms-to-know-6133210</ref> '''RGB''' - Red, Green, Blue. RGB is the colour mode used for screen output. =S= =T= '''Typography -''' The art of arranging text to make it legible and visually appealing. =U= =V= '''Vector Image''' - an image made up of points, as opposed to raster images which are made from pixels. Each point has a defined X and Y coordinates. Vector images can be resized without loss of quality. <ref>Creative Blog. (2015). 6 Key Terms Every Graphic Designer should know. Available: http://www.creativebloq.com/graphic-design/key-terms-to-know-6133210</ref> =W= '''Wireframe -''' A basic layout of a design that outlines structure without detailed visuals. =X= =Y= =Z= =See Also= [http://www.superdream.co.uk/glossary-of-design-terms/ Superdream Glossary of Design Terms] =References= <div class="references-small"> <references/> </div> [[Category:Graphic design]] dqsep1do0uozerjh0ww6k5lxr4352d2 2815030 2815029 2026-06-10T11:08:12Z Jessephu 3079828 2815030 wikitext text/x-wiki =A= '''Alignment''' – The arrangement of elements in a design so they line up neatly. '''Aspect Ratio''' – The proportional relationship between an image's width and height (e.g., 16:9). =B= '''Bleed''' - The area outside the trim edge that ensures no unprinted edges appear. '''Brand Identity''' – Visual elements such as logos, colors, and typography that represent a brand. =C= '''CMYK''' - Cyan, Magenta, Yellow, Key (Black). CMYK is the standard colour mode used for printing. '''Cropping''' – Removing unwanted parts of an image. =D= '''DPI''' - Dots per Inch on a printed page. =E= =F= =G= =H= '''Hierarchy''' - The arrangement of elements in a way that signifies importance. =I= =J= =K= '''Kerning -''' The spacing between individual characters in a word. =L= =M= '''Mockup -''' A realistic model of a design used for presentation or feedback. =N= '''Negative Space -''' The empty space around and between elements in a design. =O= =P= '''PPI''' - Pixels per Inch; number of pixels per inch in an image. =Q= =R= '''Raster Image''' - an image made up of thousands of pixels. Also known as bitmap image. Photos are an example of a raster image. <ref>Creative Blog. (2015). 6 Key Terms Every Graphic Designer should know. Available: http://www.creativebloq.com/graphic-design/key-terms-to-know-6133210</ref> '''RGB''' - Red, Green, Blue. RGB is the colour mode used for screen output. =S= =T= '''Typography -''' The art of arranging text to make it legible and visually appealing. '''Tracking''' – Adjusting the spacing across a group of letters. =U= =V= '''Vector Image''' - an image made up of points, as opposed to raster images which are made from pixels. Each point has a defined X and Y coordinates. Vector images can be resized without loss of quality. <ref>Creative Blog. (2015). 6 Key Terms Every Graphic Designer should know. Available: http://www.creativebloq.com/graphic-design/key-terms-to-know-6133210</ref> =W= '''Wireframe -''' A basic layout of a design that outlines structure without detailed visuals. =X= =Y= =Z= =See Also= [http://www.superdream.co.uk/glossary-of-design-terms/ Superdream Glossary of Design Terms] =References= <div class="references-small"> <references/> </div> [[Category:Graphic design]] kwlm8r965c5isqhigm4b6krfp9wve8b 2815031 2815030 2026-06-10T11:08:50Z Jessephu 3079828 /* G */ 2815031 wikitext text/x-wiki =A= '''Alignment''' – The arrangement of elements in a design so they line up neatly. '''Aspect Ratio''' – The proportional relationship between an image's width and height (e.g., 16:9). =B= '''Bleed''' - The area outside the trim edge that ensures no unprinted edges appear. '''Brand Identity''' – Visual elements such as logos, colors, and typography that represent a brand. =C= '''CMYK''' - Cyan, Magenta, Yellow, Key (Black). CMYK is the standard colour mode used for printing. '''Cropping''' – Removing unwanted parts of an image. =D= '''DPI''' - Dots per Inch on a printed page. =E= =F= =G= '''Grid''' – A framework of lines used to organize content in a design. =H= '''Hierarchy''' - The arrangement of elements in a way that signifies importance. =I= =J= =K= '''Kerning -''' The spacing between individual characters in a word. =L= =M= '''Mockup -''' A realistic model of a design used for presentation or feedback. =N= '''Negative Space -''' The empty space around and between elements in a design. =O= =P= '''PPI''' - Pixels per Inch; number of pixels per inch in an image. =Q= =R= '''Raster Image''' - an image made up of thousands of pixels. Also known as bitmap image. Photos are an example of a raster image. <ref>Creative Blog. (2015). 6 Key Terms Every Graphic Designer should know. Available: http://www.creativebloq.com/graphic-design/key-terms-to-know-6133210</ref> '''RGB''' - Red, Green, Blue. RGB is the colour mode used for screen output. =S= =T= '''Typography -''' The art of arranging text to make it legible and visually appealing. '''Tracking''' – Adjusting the spacing across a group of letters. =U= =V= '''Vector Image''' - an image made up of points, as opposed to raster images which are made from pixels. Each point has a defined X and Y coordinates. Vector images can be resized without loss of quality. <ref>Creative Blog. (2015). 6 Key Terms Every Graphic Designer should know. Available: http://www.creativebloq.com/graphic-design/key-terms-to-know-6133210</ref> =W= '''Wireframe -''' A basic layout of a design that outlines structure without detailed visuals. =X= =Y= =Z= =See Also= [http://www.superdream.co.uk/glossary-of-design-terms/ Superdream Glossary of Design Terms] =References= <div class="references-small"> <references/> </div> [[Category:Graphic design]] j2q9a8yubo0ox159q51i9x3gln357jo 0 212802 2814888 2095610 2026-06-09T17:13:19Z IanVG 2918363 Added two example problems. 2814888 wikitext text/x-wiki A differential equation <math> y'=f(t,y(t)) </math> is a first order differential equation. When placed in the form <math> y'+p(t)y=g(t) </math> where <math> p(t) </math> and <math> g(t) </math> are functions defined on an interval <math> a<t<b </math> is called a first order linear differential equation. For example: * <math> y'+3y=t </math> where <math> p(t) = 3 </math> and <math> g(t) = t </math> * For <math> e^{t}y'+3y=sin(t) </math> we divide through by <math> e^{t} </math> to place the equation in the proper form and <math> y'+\frac{3}{e^t}y=\frac{sin(t)}{e^{t}} </math> <math> p(t) = 3e^{-t} </math> and <math> g(t) = sin(t)e^{-t} </math> Method == Problems == Determine the equilibrium solutions of the followint differential equations: # <math>\frac {dy}{dt} = \frac {(t^3-2)(y^2-8)}{y^-2}</math> # <math>\frac {dy}{dt}= \frac {y+ 8}{2-y}</math> {{CourseCat}} 12b5qp55q0577yq2opb9t8gpoaidrmi 0 220855 2814870 2048815 2026-06-09T14:21:43Z Atcovi 276019 project box(es) 2814870 wikitext text/x-wiki {{Languages}} *Introduction: {{0Percent}} [[Russian 1/countries|Countries]], {{0Percent}} [[/linguistic characteristics|Linguistic characteristics]] * {{0Percent}} [[/The Basics/]] — ''Country focus: Russia'' * {{0Percent}} [[/Colours/]] — ''Country focus: Russia'' * {{0Percent}} [[/Nature/]] — ''Country focus: Russia'' * {{0Percent}} [[/Weather/]] — ''Country focus: Russia'' * {{0Percent}} [[/Animals/]] — ''Country focus: Russia'' * {{0Percent}} [[/Activities/]] — ''Country focus: Russia'' * {{0Percent}} [[/Adjectives/]] — ''Country focus: Russia'' * {{0Percent}} [[/School/]] — ''Country focus: Russia'' * {{0Percent}} [[/The Classroom/]] — ''Country focus: Russia'' * {{0Percent}} [[/Food & Drink/]] — ''Country focus: Russia'' * {{0Percent}} [[/Health/]] — ''Country focus: Russia'' * {{0Percent}} [[/Destinations/]] — ''Country focus: Russia'' * {{0Percent}} [[/Recreation & Lifestyle/]] — ''Country focus: Russia'' * {{0Percent}} [[/Family & Celebrations/]] — ''Country focus: Russia'' * {{0Percent}} [[/Adjectives & in a Restaurant/]] — ''Country focus: Russia'' * {{0Percent}} [[/The Bedroom/]] — ''Country focus: Russia'' * {{0Percent}} [[/The Household/]] — ''Country focus: Russia'' * {{0Percent}} [[/Shopping/]] — ''Country focus: Russia'' * {{0Percent}} [[/At the Mall/]] — ''Country focus: Russia'' * {{0Percent}} [[/Vacation/]] — ''Country focus: Russia'' * {{0Percent}} [[/Community/]] — ''Country focus: Russia'' * {{0Percent}} [[/Television/]] — ''Country focus: Russia'' * {{0Percent}} [[/Computers/]] — ''Country focus: Russia'' * {{0Percent}} [[/Asking Questions/]] — ''Country focus: Russia'' [[Category:Russian]] 64guoczrnoziv53h8mfu0mm16evwyo3 0 236722 2814882 2722405 2026-06-09T16:55:42Z ~2026-33927-78 3091576 added citation for: One of the biggest challenges in Artificial Intelligence is to store domain spec… 2814882 wikitext text/x-wiki {{Article info |journal = |last1 = Schröder |first1 = Frank |affiliations = Independent |keywords = Artificial Intelligence |license = {{CC BY SA work}} |abstract = Abstract: For solving complex robotics problems, every planner needs a heuristic to speed up the search process in the game tree. One possibility to formalize domain knowledge is a qualitative physics engine which has different layers. A random generator produces a plan, which is tested against the naive physics engine and the resulting score is used to determine if the plan is valid or not. Such a problem solving technique is an important part of a cognitive architecture which is known as BDI-agent framework. }} == Introduction == === Game-engine which stores Domain specific knowledge === One of the biggest challenges in Artificial Intell<ref>{{cite web|url=https://ai-trove.com/en/codegraph/code-knowledge-graph-for-ai-coding-agents|title=Code knowledge graphs for AI coding agents | AI Trove|access-date=2026-05-19}}</ref>igence is to store domain specific knowledge in a machine readable form. In the literature, many concepts are discussed for example ontologies or semantic networks. Sure, they are machine-readable, in a form that the ontology is a owl-file, which can be parsed automatically, the problem is that the practical usage of hierarchical data are not very high. Because, how exactly can a mind map be used for controlling a robot-agent? Right, it is unsolved yet. A more elaborated form of knowledge storage is to accept only a symbolic game-engine as a knowledge container. A game-engine is per definition a piece of software which can predict the future. For example, if we are sending the command “move-left” to the game-engine, the system will update the internal state of the player, e.g. with “newpos = {x-10,y}”. From a formal perspective a game-engine is programmed in computer code, for example in C. But from a usage scenario a game-engine can be utilized as a prediction machine. That means, for any game-engine the game-tree can be generated, and this is the pathway to Artificial Intelligence. If we can sample a system with a brute-force solver it is possible to solve the game, that means to generate a plan which fulfills certain conditions. Let us investigate how modern sampling planner are working. RRT and other graph-based sampler are using an existing game-engine (called action model) for testing out random moves. They can do so, because the game-engine contains a list of allowed moves, and the game-engine calculates what the follow state will be. Or to make the point more clear. If we do not have a game-engine, we can not generate the game-tree and we can not generate the optimal path through it. Machine readable knowledge is equal to construct a game-engine. That is the only option we have. We can discuss in detail how the game engine can be programmed for example in the Game description language <ref name="bjornsson2009cadiaplayer" />, in PDDL, in plain C or in object-oriented C++, but in any case we need some kind of prediction ready game-engine. Such a system stores the knowledge about the system. Knowledge in a form, that we define which motion primitives are possible and what they will change, after executing them. ==== Grounding ==== In most papers about cognitive architectures the term grounding is introduced for connecting the agent actions with the real world. The idea behind grounding is, that the agent has a synthetic memory (episodic memory) in which he is able to describe the world. This description is not totally wrong, but it ignores the engineering aspect for generating such an “episodic memory”. From the game-programming / C++ perspective there is no software module available called episodic memory or grounded perception. The only tool what software-engineers can implement is a game-engine. The engine has the same obligation, to map actions, perception and environment. But the term has nothing to do with psychology but with game-development. <ref name="lugrin2007making" /> Let us go back to the example with the “move-left” command. If we want to ground the action, we need a new C-function in the form “if (input==”move-left”) do something”. Do something means, to change an internal variable for example “newpos = {x-10,y}”. The action is grounded because we have connected a natural language input “move-left” with C-function in the source code. I would guess, that it is enough to program a layered game-engine; a different kind of grounding module is not needed. The term “game engine” is a bit uncommon for classical Artificial Intelligence research, because it has nothing to do with psychology science or neural networks in real animals. Also, it is not a mathematical model, so there is the suspicion that a “Game engine” is not scientific at all. And yes, the term is used by engineers not by Academia. They want to implement executable code on real computers. They are not interested in philosophical debates about thinking machines or perception in the human brain. ==== Game Engine ==== A game engine is usually used for rendering the graphic and evaluate the user input. The same technique can be used in the context of AI. An early example is the GOAP paradigm which is grouped around a game-engine. The game-engine evaluates a plan and gives a score back. A more sophisticated example are HTN-planners which have as a core a pddl file which acts similar to a game engine. But it is even possible to model story based games with an engine. This is used by text adventures and the engine can be used similar to a HTN-planner: for figuring out what the best plan through the game is. === Knowledge engineering with natural language === About AI planning itself, there is a huge amount of literature available. From the early beginning with the STRIPS planning system in the 1970s, over later developed pddl like HTN-planners upto modern game-oriented GOAP planning goes the history. A sub problem in AI planning is until now open: how does the action model look like for a certain domain? For example, we want to play Robocup and need the domain model in a pddl description. How the path may be look like to develop machine readable knowledge from scratch is discussed under the term GIVE challenge (“Generating Instructions in Virtual Environments”).<ref name="give2012" /> That is not directly an algorithm, but a programming challenge which is grouped around ready-to-run domain knowledge. For all, who are not aware with the original challenge, here the idea in short. The aim is not to guide a robot through a maze but a human. The software generates the output in natural language and the human must act. From the AI history the setup is called decision support system, because the computer only give help to the human but not decides for him. The advantage from the programmer perspective is, that such a system is easier to implement then a full-blown robot control system. A second interesting part is that the output is in natural language. How the challenge can be solved is not very complicated. At first, a pddl domain description is used, then the current situation of the game is read, and a planner is used for producing the next step for the human. The planner can be based on STRIPS, PDDL, HTN or GOAP. He generates the plan, for example “open door”, “walk left”, “push button”. How the human is executing the plan is on his side. The feedback for the AI is only the situation in the game itself. That means, the software has the tasks: * track the actions * plan future actions * with the help of an internal model The GIVE challenge doesn't only generate natural language, but it is a subgoal generator. The idea is not to control a robot with low-level commands, instead the goal is to planning part-goals, for example “open door” and similar actions. == Cognitive architecture == === Agent oriented programming === Object-oriented programming is the classical paradigm: a method contains commands in a linear fashion. The actions are executed from top to bottom. If another program flow is needed, the programmer has to modify the source code. In contrast: agent-oriented programs are using a layer who is controlling the program flow. The layer is called blackboard and is some kind of note-area to rearranging commands. For example, the first draft contains the following commands: print, if-then, print After a modification by an agent, the new program flow is maybe: if-then, print, goto In early years this technique was called self-modifying programs, because in the software is another software which is created from scratch at run-time, called the agent-plan. Agent-oriented programming is not using a fixed program flow, created by the programmer, instead the Artificial Intelligence creates the program flow for the need of a problem. If the plan is ready, it will be executed by the interpreter. The amount of modules and subroutines to realize such behavior is called an agent-framework. It is based on object-oriented programming but has new features. === Understanding SOAR and ACT-R === It is widely known, that cognitive architectures are simulating the human process of thinking. But this is a bad introduction for explaining the principle to beginners who are not familiar with Artificial Intelligence. The better approach is to understand a cognitive architecture as a "hierarchical task network". This term is used in the gaming industry in the domain of GOAP (Goal oriented action planning) and means not to create a behavior tree, but to implement a symbolic game engine. The game engine is used to testing out plans and scoring them. SOAR is an advanced form of a HTN-network. It is basically a hierarchical planner, on the lowest level there are actions, and above the actions are tasks. They are planned hierarchical. Additionally, SOAR has some agent-like features, for example a semantic memory and a blackboard. This is used for evaluating plans.<ref name="wray2005introduction" /> It is true, that SOAR was programmed in LISP, but LISP is not the best programming language for implement an AI system. It makes sense to start with a mainstream language, for example C/C++. It is a mainstream language, because C/C++ can be used for any kind for software development tasks, from hardware drivers, over desktop-application up to Artificial intelligence. Instead, LISP is only used for AI problems and compiler-compiler which makes it's harder to understand. The bottom up approach for teaching cognitive architecture is first to implement a simple HTN-planner in C++, and extends this later into a cognitive architecture which has agents and deep.-learning features. Let us explain the core idea behind HTN-planning, because this results into Artificial Intelligence. On the lowest level, an AI is a planner. An easy to understand example is a path-planner like A* and RRT. At first, random movements are created, and each of the nodes is evaluated with a score. The node with the highest score is executed by the AI. Such a core-feature can be written down in two simple lines of code: for trial calculate score for node The algorithm runs in a for loop and on each iteration he is evaluating the score. Sure, this simple pseudo-code isn't able to control any robot, but it makes it easier to understand what AI is. SOAR, HTN-planners and agent-architecture are about the details of how to implement such a for-loop. The reason, why the term cognitive architecture is used has to do with the importance of natural language for domain modeling. Classical path-planning works not with language, but with numbers. For example, with the distance from point A to B. The problem is, that with numbers alone is not possible to implement complex domain models. And without a domain model the planner can not search for a node. The bottleneck in HTN-planning as well in SOAR is to create complex domain models for games. A domain model is usually a mini-game which has possible actions and an outcome after executing it. All agent architecture have such mini-games in the sleeve. In the literature there are some examples given of how to store a domain model: PDDL, Game description language or ontologies. The holy grail of AI is to generate a domain model on-the-fly. That means, the agent plays a new game, creates the domain model in GDL, and this is used by SOAR to plan the next actions. === AGI is a testbed for Narrow AI === A look backwards in the history of AI is often done with the explanation what the difference between Artificial General Intelligence and Narrow AI is. But perhaps the difference is only artificial? Perhaps both are the same? Let us investigate how the robocup robotics challenge work. It contains of two types of software. At first a testbed, which is able to evaluate which of the teams has won, and secondly the agent itself, which is able to play the game. This distinction makes also sense for other domains, for example chess playing, Micromouse and a line following robot. In any case, two types of software is used: one which has the function to act like a turing-test, and the second which is the core AI. Without any doubt, the second part (the AI) is called in the literature a narrow AI. For example, if somebody has programmed in C++ and with ABL a robocup AI who is able to kick the ball, then this agent can be called intelligent, or at least it es equal to Narrow AI. That means, the software is able to solve a certain problem. On the other hand, the problem which is equal to play robocup has to be monitored by a different type of software. It is uncommon in the literature, to call this part of the overall setup an Artificial General Intelligence, but it would make sense. From a programming point of view, the evaluation software is called a testbed. It is software which was programmed to run other software inside. For example, the Robocode challenge contains the robocode testbed itself (which was programmed in Java) and the agents which can be executed inside Robocode. The same is true for the famous Mario AI challenge. It contains of the Super Mario Game itself, and possible agents who can be run inside the environment. The hypothesis is, that early so called Artificial General Intelligence projects like SOAR and the General problem solver, were never invented as a Narrow AI. That means, SOAR is not able to control a robot or play a game. Instead, SOAR is a testbed and can be used in a robotics challenge in which the teams have the task to program the narrow AI.<ref name="arrabales2009cera" /> ==== Robo-soar ==== For the software “robo-soar” in a paper was described, that this tool acts mainly as a testbed.<ref name="laird1991robo" /> Soar is explained as a integrated AI framework, and on page 5 the general setup is shown. It is mainly a robotarm, which can pick&place objects and also a human-operator is in the loop. The interesting information is, that SOAR is not the agent for controlling the robotarm (a so called weak artificial intelligence), but SOAR is the overall framework. The modern description would be, that it is the software to monitor a robotics challenge. A program which can say, which participant in the competition has written the best agent-software. But there is in the above cited paper something else which is interesting, in the 17 pages long paper there is no heuristics or sourcecode given how to solve the problem. The problem was to pick&place objects with the robotarm. A possible solution would be to program an agent in C or LISP which is planning some actions. But, this strategy is not part of the paper. The paper is only about the competition itself, that means it describes in detail what the task is, not how to program the ai-agent. The same approach is visible in a more recent paper, which was published by the same author in year 2000. The first sentence in the abstract explains very well what the aim is: “We are creating an environment in which to investigate the role of advanced AI in computer games. [...] In this paper we describe our test bed for pursuing research in developing human-level AI characters within computer games.” <ref name="laird2000test" /> The goal is not to program an AI which is able to play the game autonomously, instead the aim is to program a robocode like environment and to delegate the programming of the AI to participants of the competition. == Implementation == === GOAP Planning === If HTN-planning is the academic term, GOAP is the realization in a computer game. Goal and action planning means usually a hierarchical planning system for an Artificial Intelligence. But what exactly is planning? Geometric planning is the search for a path in a maze around the obstacles, while HTN-planning has to do with a trajectory in a symbolic maze.<ref name="porteous2009controlling" /> This maze can't be visualized with x/y coordinates but can be described with tasks and subtasks. If the action gets executed the agent is doing something in a computergame, for example he walks to the door and opens it. What a HTN-planner is doing is to describe the domain with symbolic constraints. They are similar to obstacles in a 2d space, but they are defined with abstract descriptions for example the goal “search for the key” is an obstacle. It is something which can be fulfilled or not and which is used by the planner to generate actions. Because of the abstract nature of HTN-planners i would call these system natural language evaluation systems. Because the action space is defined by English vocabulary and the plan is also a sequence of action-words. What the programmer of a GOAP system is doing, is to formulate the domain with abstract obstacle. He is building a map in which the planner is searching for a trajectory. A trajectory means usually a plan which is a sequence of actions. A HTN-planner is able to evaluate a plan, he can score a given plan and search for a better one. === Inner working of a sampling based HTN-planner === What a HTN-planning system is doing on a theoretical level is well known. It plans something according to an domain model. But how can we implement this into software? The first step is to define a hierarchy of actions. There are some textual commands which can be send to the layered simulator: * layer0: keyboard left, keyboard right, keyboard up, keyboard down * layer1: walk left, jump over box, jump against block * layer2: walk to point a, walk to princess, get key According to the names, this is an example of a side scrolling game in which the player has some problems to solve. On the lowest level the game can be controlled with the arrow keys only, additional there are more layers which are expressing the semantics of a move. The game can be played on all the layers. Either the player can use the arrow keys, or he can enter high-level commands which brings the game-engine directly to the goal state. For example if he enters the command “get the key”, he has without further problems simply the key in his inventory. This bypass all the details which are usually be necessary to get the key, it is some kind of trainer mode. The planner works in a way, that random actions are send against all layers of the game-engine to see what will happen. If only the lowest layer is selected it will take a long time because the state space is high. If the planner is using all layers, the game will solved much faster. I want to a give a example. According to the random generator the following actions are selected: “get key”, “jump over box”, “keyboard left”, “walk left” That means, we have run the random generator 4 times and these are the random commands. How can we execute the commands against the game engine? Execution of the lowlevel command on layer0 “keyboard left” is easy, we are sending simply the command to the game engine and see what will happen. It is a normal search in the game tree. Executing a mid-level action like “walk left” is a bit harder. According to the table, this command is from layer1, that means it can't be executed directly. The original game has no command with such a name. Instead the command is a subgoal and the planner is forced to solve on the level below. That means, he is sending keyboard commands to the engine, until the player “walks left”. In general the inner working is equal to a HTN-planning system. We have lowlevel and high-level actions and must generate partial order plans to satisfy certain conditions. Only commands on the lowest layer can be executed directly, the other commands are only symbolic goals, which guides the search procedure. Usually HTN-planning system are implemented in agents. The agent has an abstract plan and a blackboard <ref name="rudenko2007overview" /> and he generates for the plan the lowlevel actions to execute them against the game engine. If some plans are failing during runtime, the agent detects the problem and replans. ==== BDI agent ==== If a HTN-planner is able to solve a game, what is the purpose of an BDI agent? The idea is to layer the overall system into submodules: the game itself, ontop a hierarchical simulator, then a HTN-planner and on top of all the BDI-agent. The BDI agent is only the upper way to communicate with the overall system. It is implemented in a C++ class, which has commands like “start-agent”, “show plan”. The class also have a solver integrated, which is based on the module defined on a lower levels. The solver for example is using a HTN-planner for generating a new plan. But additionally to planning, the BDI agent has some other features, for example to search for new goals. {{fig|1 |align=right |Artificial-intelligence-BDI-agent-framework.png |Artificial intelligence, BDI agent framework}} Sometimes a BDI-agent framework is associated with a concrete programming language for example Agentspeak. But it is not a piece of software, it is a software-engineering technique like object-oriented programming. It is a certain way to structure the sourcecode. That means, a BDI agent can be programmed in normal C++ without the need of inventing a new programming language for Artificial Intelligence. In the easiest form a BDI-agent is a class inside of a robot-control-system which has high-level-methods like “start”, “Stop”, “ask for a goal”, “plan”. Usually, it is interacting directly with the GUI. That means, the BDI agent is a high-level description of the Artificial Intelligence. The user starts the program code with “./a.out”, and then he sees a GUI, and in the gui he can enter the goals and sees the plans of the agent to reach the goals. A BDI-agent is doing the same like a HTN-planner or a layered game-engine. The idea is to play a game autonomously. The difference is, that such a piece of software is very complex and one of the submodule is called BDI-agent. === Programming a HTN-planner === The most demanding task in HTN-planning is not the solver itself. This is in most cases a simple brute-force search algorithm which takes the hierarchical domain model and searches in the graph. No, the bottleneck is to create the hierarchical task network, which is equal to a domain model. Let us go into the details to recognize potential pitfalls. Suppose, we have a game like Mario AI. The interaction works in a way, that the user is pressing the left/right arrow and navigates Mario through the level. So every game has to do with pressing keys: left and right. The sad news is, that this description is correct but there is missing some important detail. Mario AI is more then only pressing left and right, Mario AI has to do with jumping over a wall, finding the princess and avoid the enemies. Describing a game as a hierarchical task network has to do with finding out the inner working of the game, known as the semantic description. This inner working differs from game to game: a car racing game has a different meaning then a real time strategy game. A HTN-network is about the inner working, it divides the overall control task into modules. The general idea is to transform a monolithic game into smaller subgames which can be solved on an abstract level. The aim is to increase the interaction level into the direction of natural language. What the user of Mario AI want's is no longer pressing left/right keys, he want's to enter “jump over wall”, “avoid enemy”, “jump with trajectory #2”. The only problem is, that these kinds of commands are not available in the original game, they have to be invented from scratch. It is like designing a new game, which wasn't there before. So the logical question is: how does the new abstract game will look like? This depends heavily from the domain. The subgames of a racing game have to do with car physics, racing and trajectories for curve ride. The subgames in a RTS-game have to do with build order, long term strategy and position of units on the map. The more general term for describing the subgames is “Domain knowledge”. Domain knowledge is something which is important to win a game and this has to be transferred into an abstract subgame inside the original game. There is an important reason why domain knowledge has to be formulated as a subgame. Because this is the only way to evaluate the knowledge and search for a plan. A newly invented game engine is the perfect choice for predicting future states of the game, and this is the fundamental idea behind a HTN-planner. What a htn-planner is doing, is using one of the actions from a layer and testing out a move, then he scores the result. That means, the HTN-planner is utilizing a subgame for predicting the future. The usual idea in the literature is to find the subgames automatically by genetic programming or neural networks. This attempts fails. A game engine for real purposes has to be created by hand, that means inside a software-engineering process. Domain modeling is nothing which can be done automatically, it is mainly a task for the programmer. He has to define, how the HTN-network will look like, what possible high-level-commands are there and what the scoring value of an outcome is. The programming task is similar to programming any other game. Usually the programmer has to implement the game rules in many edit-compile-run iterations until the game works. Or to make the point clear: if manual C++ programming in an IDE together with git and some question on stackoverflow are the only option to program a Mario game from scratch, then the more demanding task of programming a subgame inside Mario AI for predicting the jumping behavior is also a manual task. The infrastructure for building a HTN-network can be described with: * object oriented programming language like C++ * a bugtracker for note down the issues * an online forum for asking other programmer for help * some examples from the past written down in sourcecode and academic papers * a git-like version control system to modify the sourcecode I want to elaborate the overall process for developing subgames for a HTN-planner. What the programmer usually has, is the monolithic game and a walkthrough tutorial in natural language. What he doesn't have is machine readable domain knowledge. The task is to convert the instruction manual written in English into subgames which can be run by the HTN-planner for solving the game. The result will look like a layered physics engine written from scratch. It works from the outside view like this: action(layer,command), e.g. action(layer2, “jump over wall”) How this physics engine works internally is up to the programmer. The general idea behind abstract subgames is to provide subgoals and scoring rewards. These are used by the HTN-solver for figuring out the best move. Or to be more specific: the monolithic game is split into smaller, layered subgames which can each solved in a small amount of time. ==== Simulating a plan ==== Hierarchical task networks were invented in the year 1975 under the term “procedural net / NOAH”. According to the original paper <ref name="sacerdoti1975nonlinear" />, the general idea is to create a hierarchical plan and simulate it with a procedural net. Let us make a short example. At first we need a hierarchical plan, this plan is executed in a simulator. The simulator gives feedback for the plan. Is the plan valid? How long does it take? What is the overall score? go to room B go forward (physics engine #1) leg1 down leg2 left leg1 up take the key (physics engine #2) open door (physics engine #3) go forward (physics engine #1) leg1 up leg2 up leg1 down Running a hierarchical plan is straightforward compared to running a simple plan in a simulator. Because the procedural net is not only a single physics engine, but each node is it's own physics engine. In the figure the plan and the matching physics engine are given as an example. What the overall HTN-planner is doing is to take a subplan, recognizing which physics engine is the right one, and testing the plan against the engine. Finding the right plan is surprising simple. In the easiest implementation, random plans are generated, tested against the procedural net and the aim is to identify the plan with the highest score. == Hierarchical task networks == === Hierarchical Backtracking: a less known problem solving technique === With the advent of behavior trees for ingame-AI most software developers are aware of how to build an artificial agent. The idea is, that a designer specify the behavior of the bot and after the script is created it will get executed like a normal computer program. But scripting-AI is only the beginning, there are more sophisticated AI techniques available like HTN-planning and especially HTN-backtracking. The difference is, that they are not recognized broadly and most programmers doesn't know how to implement such strategies. The basic idea behind HTN planning is, not to specify a script but program a symbolic game engine which is able to predict the future. Such a game-engine is equal to a textadventure, it is some kind of software modul which takes a command like “open door” and reacts to the command. One possible option to realize the engine is the PDDL programming language, but it is also possible to use normal C++ or Python code to program a textadventure. And now the things become complicated. A game-engine itself is not enough, what HTN-planning is about is a hierarchy of game-engines. On the lower level the agent moves around, on the second level he gets strategy commands like “move to point A” and on the high-level he gets very general commands like “search for the door”. Suppose we have implemented a layered textadventure, then the next step is to use a technique called backtracking. Backtracking is usually known for game-tree search in computerchess. It means, to try out different paths in the tree. But, computerchess contains only of one tree, no hierarchy is used. If a multi-layer-gameengine is available it is available to backtrack over many layers, This saves lot of cpu-time. It means, that the solver is used for each game-engine separately. Let us make a simple example. The plan which was created is: search for a door --move to point A --open door It is not a flat plan, but a hierarchical one. There are three layers available, and each layer is a separate game. Suppose in one of the layer some action is blocked, because of external reasons. Then the planer can search for a bypass. He uses the game-engine to predict future states and searches for a subplan which fulfills the conditions. The most important feature is, that the solver is really fast. Hierarchical planning means to leave huge parts of the overall plan unchanged. It is possible to replan only a subpart of the plan. In most cases, the lowest layer of the HTN-planner is connected to a physics engine. Running a physics engine is very CPU-intensive. So the goal is to reduce the number of iterations. For example, if it is known, that the command “open door” will result into a certain situation it is not necessary to start the physics engine on each planning step to check if the agent really is able to open the door. Instead the result can be assumed as successful and this situation is used by the planners in a higher hierarchy. Now, let us suppose, the lowest physics engine layer has created a bug; search for a door --move to point A --open door -> not working search for a door --move to point B --open door -> success The planner can react to the bug with replanning everything. He is trying to reach the same goal, but with different task sequence. How can we imagine the inner working of such a hierarchical planning tool? The easiest way is to see a HTN-planner as a generator for “natural language”. His primary function is to output sentences which are guiding the agent to a goal. It is similar to street-guidance system. Such software is usually separated in different layers, it can plan for the highway and also for small streets. The user have to enter the starting point and the goal and the system will calculate the steps between them. Backtracking search is used for figuring paths even if one street is blocked, so at the end the user can reach the goal. The difference is only, that this time the game is more complex. It doesn't only contains streets but behaviors of an agent. For example, the robot navigates through the kitchen, opens the shelf and is using tools for making food. === Bottleneck in HTN-planning === Programming the solver itself to search for a plan is simple. In most cases, it will not take more then 100 lines of code because planning means only to testing out random actions until a condition is true. If the PDDL file for a domain is known, it is very easy to prove the theorem, which means to find a plan. The bottleneck in HTN-planning is somewhere else. It is the creation of the domain model, that means a machine readable prediction of how the world will change after a certain command. Let's take in look into a formal STRIPS and PDDL file. It is mainly a syntax for describing a game-engine. We have a command for example “open door”, a precondition and the result. It is possible to construct with PDDL a game-engine for playing pacman or a textadventure. But what defines, how a pddl for a certain domain will look like? Usually only the low level game-engine is given for example the box2d simulation of a robot-grasping task. If the human-user is playing around with the physics engine he will generate some example, of how the game works. He is able to generate an internal model how to play this game. But, what he doesn't know is how the matching PDDL domainfile will look like. And here is the bottleneck. How much effort is necessary to create for given domain the PDDL file? Can this done automatically, for example with neural networks? Let us imagine the PDDL file (or a similar specification in an object-oriented language) is given. That means, the possible actions, their outcome and sub-actions are known. The consequence is, that the game can be called solve. Because if the domain model is hierarchical organized and well implemented it is very easy for every planner to bring the system into a goal state. The current game-situation is know, also the goal of the game, and with a bit brute-force searching and backtracking the solver will find the path through it. It is simply not possible that the planner will fail. That means, the result depends only from the domain model. If we have no domain model, or a flat model without hierarchy, then the planner will fail. If the only available domain model is the pure low-level box2d engine it makes no sense to search for plans, because the state-space is way to huge. The only bottleneck is the question of how to construct the domain model. Answering this question will result into a working robot-control-system. In general there are two options a programmer has. At first, he can type in the PDDL file from scratch in his texteditor and see it as a software-engineering process. He can formulate requirements to the domain model to express which features are needed. Then he can track the changes with a version control system and test the model in a simulator. Such technique will result into a working domain model, but it is time consuming. The other option is to use some kind of data-driven learning method. Which means to record a gametrace in a logfile and generate the domain model automatically. This method needs very little human effort but it is unclear what the result will be. A more easier task is not generate a domain model from scratch but testing out a given domain model if his predicted actions are correct. We are taken a PDDL file and running it against a low-level game-engine and at the end both models must be synchronous. According to previous published literature it is very hard to generate or evaluate a domain model automatically. The reason is, that a high-quality domain model not only contains Finite state machines, but also the action names in natural language, for example the “open door” task is called in the PDDL not “task1” but “open door”. It seems that automatically generate the Finite state machine plus the appropriate English-description by an algorithm is very hard and that a manual software-engineering process is the only way to get a functional domain model. Surprisingly the amount of effort isn't very high to do so. Suppose the PDDL file is 200 lines of code long. According to the average productivity such a domain model can be created by hand in 200/10=20 days by a single programmer. If he finds on github the needed PDDL file with a fulltext search it is possible to reduce the time further. Perhaps it is time to explain how a domain model looks formally. Mainly it is a computerfile in the language STRIPS, PDDL, Prolog, Golog or even normal C++ code. It is equal to a layered game engine to express possible behaviors of an agent. The value of the domain model is equal to the used lines of code, a pddl file with 10 lines of code is inferior to a domain model which needs 1000 lines of code. A working real life domain model is similar to computercode. That means, more complex actions can not express in PDDL but needs a description in C++ object-oriented programming. Nevertheless, there was some research in the direction of using LSTM neural networks to predict the action model with natural language. The idea is use a corpus of actions vocabulary and match this with the result in the game for predicting the future with a neural network: “We developed a symbolic simulator based on the domain knowledge (i.e. Planning Domain Definition Language (PDDL) file) in the Tell Me Dave corpus to train the RL agent.” <ref name="zamanideep2017" />, page 3 The same idea is described by <ref name="lin2017knowledge" />. The figure on page 4 topleft explains the pipeline. At first a plan is presented: “1. move to cup, 2. grasp cup, 3. stop stop”. The plan gets executed in the game and results into some states. The LSTM neural networks learns the mapping between the actions and the consequences, so it can predict future commands: “Our model can correctly predict complete action sequences with a probability of 92% on average.” <ref name="lin2017knowledge" /> === Hierarchical plan simulation === A plan in computerchess contains only one layer. The moves are numbered and are executed from top to bottom: 1. f2f4 e7e5 2. f4xe5 d7d6 Such a notation can be executed against a chess-engine. It is executing the moves and can print out the board situation at the end. A less known way of executing plans is a hierarchy. The original plan contains more then one layer, the example has three layer, because of the maximum depth. Such a plan can also be executed against a simulator. But the simulator must be more sophisticated, because it reacts to an action on layer0 different then to an action on layer1. task1 subtask1 action1 action2 subtask2 action3 action4 task2 subtask3 action5 The main idea behind HTN-planning is to construct hierarchical plans and testing them in an simulator. Like in the introductionary chess-example the idea is to calculate a score for a plan. It is a measurement what will happen if the plan is executed. Like a chess plan there are many possible outcomes available. The plan can run smoothly and result into the output “player1 has won”. Or the simulator can detect, that a move is illegal, then the execution will stop with an error message. But the most important feature is, that it is possible to generate a plan by a random generator and search the system backwards for a plan. That means, at the beginning we have no idea which actions are needed, and we must testing out some alternatives. This is discussed in the domain of PDDL and other hierarchical planning systems. The question is, how to construct a solver which will generate a high-score plan from scratch. The PDDL syntax was mainly invented to increase the performance. If a high-level-action has a clear precondition, postcondition statement it is able to calculate a plan in high-speed. The PDDL environment acts like a super-fast game engine. Another technique (partial order planning) was also invented to increase the plan-search-speed. Here is the idea to plan only parts and aggregate this into much broader plans. A not so advanced planner is not using any kind of optimization and simply generates the complete plan from scratch with a random generator for testing it out against the simulator. === Hierarchical reinforcement learning === Under the term Reinforcement learning usually a data-driven is approach is discussed. The algorithm gets a database, learns from the data and is able to reproduce the movement. Because the missing programming effort, reinforcement learning is attractive in beginners of Artificial Intelligence. There are some main problems in implementing such an algorithm. The main issue what all beginners and experts in machine learning will recognize is the topic of “my algorithm doesn't learn”. That means, they have constructed some kind of learning policy which is adapting to the data, but the error rate didn't fall under a certain level. What is wrong with the algorithm? This is very easy to answer, it is has to do with the reward function. If no reward function is available the best LSTM network will not learn. But what is a reward-function? A game like “Mario AI” has a time period in which the game is running. For example it takes 5 minutes until a level is solved. In this time-period the agent is doing some actions, which are controlled by a neural network. The problem is, that the algorithm doesn't know if an action was right or wrong, because it is unclear what an action taken on timeframe 0:04 will has as a future consequence. Missing rewards during the game results into a huge state-space and this is equal to a not-learning reinforcement learning algorithm. Overcome the problem is simple, and it is discussed in the literature under the term hierarchical reinforcement learning. A hierarchy is a set of actions which are grouped in layers. On the lowlevel layer primitive actions like “press button left”, “press button right” are used. On a higher level a command like “move to object” is given. A hierarchy of actions is equal to subgoals and this is equal to a reward function. For example, if the mid-level command “move to object” was entered then the rewards depends on if Mario AI has reached the object or not. In my opinion, a hierarchical reinforcement learning system is possible to implement. The systems learns from the data all the tasks, subtasks and actions and is able to simulate the tasks in the agent's memory. The problem is, that from a bootstrapping perspective such an agent-architecture is very complex, because it is an HTN-planner and a action model learning tool at the same time. That means, after starting the software no PDDL domain is available, it is has to be learned from scratch. And if the model is ready, it will be used for planning the next action. It is possible to explain the algorithm a bit in detail. In a normal HTN-planner a given action model is used, the pddl-file. This file is used for running a simulation. That means, the agent knows what will happen if he executes the command “move to object”. The pddl file is created by hand. In reinforcement learning the idea is to not program something but using existing data and generate the algorithm from the data. That means, Hierarchical reinforcement learning needs no initial PDDL file, it extracts the file from the data. Programming such an algorithm is technical possible but it is a very hard task. I would suggest it is easier to implement a normal HTN-planner which is filled with a manual pddl file. The reason is, that Hierarchical planning in general is very complicated and with reinforcement learning the things will get more complicated. === Expand of high-level actions === A hierarchical plan contains of actions, only the lowlevel actions can be executed directly, the mid- and high-level actions are symbolic labels: task1 action1 action2 lowlevelaction1 lowlevelaction2 action3 But how exactly can we expand a high-level action? In the domain of compiler and interpreter programming a label is resolved with a grammar to a executable action. In the case of HTN-planning this idea will fail, because the high-level actions are resulting in a different sequence of lowlevel actions. It is a problem called “Answer set programming”, that means the high-level-label is equal to a goal which has to solved. A HTN-plan is not a program which can be executed from start to end, it is more a list of goals. Some of the goals (the lowlevel actions) can be fulfilled easily, but the most goals not. Let us assume an example from Mario AI. If the goal is “jump with Mario over the wall”, then it is maybe clear what the purpose is and how the result will look like, but it is hard to transform this goal into a sequence of keyboard inputs. The good news is, that there is help. A so called brute-force solver can transform a goal into actions. It is some kind of backward search: we have possible random actions, a game engine and now we can search for the exact path in the gametree. Apropos gametree. In classical game-theory the game tree is a graph of possible movements, it structures the decision making. In the area of HTN-planning there are many sub-game trees and the goal are equal to a desired state in the gametree. That means, the goal “jump with Mario over the wall” is pointing to a node in the gametree. On this node, the goal is fulfilled. The question from the perspective of implementing such a system is how we formulate this problem. In the PDDL community the answer is to create a pddl domain model and search in the model for a plan. But I'm in doubt if this is the best way, because the PDDL language is limited and is not able to model object-oriented relations. From the use-case an goal-resolution system should be work that the user can give a goal and a game-engine while the system gives back a plan: solver(“jump with Mario over the wall”, gameengine2) -> plan=”right, right, jump, right”. The gametree of the game engine is calculated, and the textual goal is used to identify a node. Then the solver calculates the plan to reach the goal-node. Such a system must work with different game-engines and different goals: solver(“find the princess”, gameengine4) -> plan=”jump over wall, jump on enemy, walk right”. I know, from a certain point of view, this workflow is trivial, because it is the description of a HTN-planner. But it seems, that even the first HTN-planner was developed in the mid 1970s the concept is not very often implemented. Let us describe the complete workflow for expanding a plan into actions: task1 action1 -> solver("action1", gameengine1) action2 -> solver("action2", gameengine2) plan: lowlevelaction1 plan: lowlevelaction2 action3 -> solver("action3", gameengine3) It is basically the same plan like in the beginning of the chapter, except that the high-level-goals are converted into requests to a solver. === Goal-Oriented Action Planning for realizing neural networks === Most beginners in neural networks will run into trouble because their agent have no reward function. A reward is equal to a goal, for example reaching a place on the map. In most cases the goal of the game itself is clear: win the game. But with this simple reward function it is impossible to train an agent. Suppose, the agent is doing random action ingame, is that an improvement of his situation or not? And because the algorithm can't answer the question, the agent will not learn, that means, he will not reach the goal nor he will improve his policy. The answer to that problem is called “Goal-Oriented Action Planning”, which is basically the known “Hierarchical task network” principle adapted for computergames. GOAP means, to define a goal hiearchy which is equal to a reward function. A typical agent has a plan like “1. open door”, “2. move forward”, “3. take the key”, “4. open the door” and so forth. He has structured the game in smaller chunks, called symbolic goals. The list of all goals are stored in the goal tree and a planner is adjusting the plans in realtime. For example, if we are taking the agent into the air and placing him to another location, he replans everything. It is possible to combine a neural network with the GOAP architecture? Yes, and it is a very good idea. The neural network is able to learn from example without programming to much manually while the GOAP principle is used for generating subgoals which guides the policy search of the agent. The strengths and weakness of GOAP and neural networks are asymmetrical. The GOAP architecture is very strong in finding subgoals and planning. His weakness is, that the domain model has to be programmed manually. In contrast, neural networks have no subgoals and can't plan anything, but they need no manual programming and can adapt to given data. It is possible to combine both ideas into a powerful neural network cognitive architecture? == Example == === Conflict resolution in HTN-planning === The assumption for most HTN-planning domains is, that a working domain model is available. That means, the prediction of the model is accurate. The problem is, that this assumption is not realistic. In the figure a blue robotarm should push the object to the left. According to the HTN-domain model a push against an object moves the object. For the planner, the push action against the object will solve the problem. But in reality, a push against the object will be blocked by a second object. This case is not stored in the domain model so we have a gap between prediction and reality. {{fig|2 |align=right |Robotics-push-grasp-with-conflict-resolution.png |Robotics push grasp with conflict resolution}} What will happen, if the HTN planner is trying to create a plan? According to his domain model he will create the plan “1. push object to left”, and the planner believes this will work. After executing the plan in the simulator there is an error. That means the plan is executed but the result is different from the expectation. This is a tragedy, but it is useful information. Now the planner knows this his plan is wrong. And here comes the magic into the game. The new request to the HTN-planner is: “bring the object to the left but don't use the plan 'push object to the left', because this plan doesn't work”. That means, the planner is forced to solve the problem with a different plan. He can push the other object first and then push the original one. It is acceptable, if the domain model isn't perfect and can't predict the reality. The HTN-planner will produce plans who doesn't work and this plans are marked as fail. In a second step, an alternative plan is generated which bypass the problem. The feature of detecting failed plans and bypass them is called “Backtracking”: “If the decomposition is not possible (e.g. because of colliding restrictions), the planner backtracks and creates a different decomposition.” <ref name="lekavy2007expressivity" />, page 4 === Lowlevel HTN-planning === Understanding a HTN-planner and a BDI agent is complicated, because the system contains of many submoduls. Perhaps it would help to describe the system from the bottom up? On the low-level the player can press left-key or the right-key for move his character on the screen. And the aim is to solve the game. The HTN-planner has the obligation to provide subgoals, for example to bring Mario first to the wall and then Mario should jump. Every of the subgoals can be fulfilled with pressing the left/right key. And the planner provides some more goals on the high-level-layer for example, to master level 1, or avoid an enemy. What the HTN-planner is doing is pressing right-left key to fulfill subgoals and high-level-goals. The process works in a realtime-fashion, that means the overall BDI architecture monitors the systems, calculates new subgoals and determines lowlevel actions to fulfill the goals. Perhaps a small example. If the game takes 2 minutes, the normal player will have pressed 100 times on of the buttons, either left or right. Over the timespan he has navigated Mario through the level, pickups objects and jumped on top of enemies. A HTN-planner structures the 100 lowlevel actions together with the BDI-architecture in a formal way.. == References == <references> {{reflist}} <ref name="arrabales2009cera"> {{cite journal | title = CERA-CRANIUM: A test bed for machine consciousness research | author = Arrabales, Raul | author2 = Ledezma, Agapito | author3 = Sanchis, Araceli | year = 2009 }} </ref> <ref name="bjornsson2009cadiaplayer"> {{cite journal | title = Cadiaplayer: A simulation-based general game player | author = Bjornsson, Yngvi | author2 = Finnsson, Hilmar | year = 2009 | publisher = IEEE | journal = IEEE Transactions on Computational Intelligence and AI in Games | pages = 4--15 | volume = 1 | number = 1 }} </ref> <ref name="give2012"> {{citation | last = Garoufi | first = Konstantina | title = Homepage of the Challenge on Generating Instructions in Virtual Environments (GIVE) | url = | year = 2012 | publisher = Saarland University }} </ref> <ref name="laird1991robo"> {{cite journal | title = Robo-Soar: An integration of external interaction, planning, and learning using Soar | author = Laird, John E | author2 = Yager, Eric S | author3 = Hucka, Michael | author4 = Tuck, Christopher M | year = 1991 | publisher = Elsevier | journal = Robotics and Autonomous Systems | pages = 113--129 | volume = 8 | number = 1-2 }} </ref> <ref name="laird2000test"> {{cite journal | title = A test bed for developing intelligent synthetic characters | author = Laird, John E | author2 = Assanie, Mazin | author3 = Bachelor, Benjamin | author4 = Benninghoff, Nathan | author5 = Enam, Syed | author6 = Jones, Bradley | author7 = Kerfoot, Alex | author8 = Lauver, Colin | author9 = Magerko, Brian | author10 = Sheiman, Jeff | author11 = et al. | year = 2000 | journal = Ann Arbor | pages = 48109--2110 | volume = 1001 }} </ref> <ref name="lekavy2007expressivity"> {{cite conference | title = Expressivity of STRIPS-like and HTN-like planning | author = Lekavy, Marian | author2 = Navrat, Pavol | year = 2007 | conference = KES International Symposium on Agent and Multi-Agent Systems: Technologies and Applications | pages = 121--130 | publisher = Springer }} </ref> <ref name="lin2017knowledge"> {{cite conference | title = Knowledge-guided recurrent neural network learning for task-oriented action prediction | author = Lin, Liang | author2 = Huang, Lili | author3 = Chen, Tianshui | author4 = Gan, Yukang | author5 = Cheng, Hui | year = 2017 | conference = Multimedia and Expo (ICME), 2017 IEEE International Conference on | pages = 625--630 | publisher = IEEE }} </ref> <ref name="lugrin2007making"> {{cite conference | title = Making sense of virtual environments: action representation, grounding and common sense | author = Lugrin, Jean-Luc | author2 = Cavazza, Marc | year = 2007 | conference = Proceedings of the 12th international conference on Intelligent user interfaces | pages = 225--234 | publisher = ACM }} </ref> <ref name="porteous2009controlling"> {{cite conference | title = Controlling narrative generation with planning trajectories: the role of constraints | author = Porteous, Julie | author2 = Cavazza, Marc | year = 2009 | conference = Joint International Conference on Interactive Digital Storytelling | pages = 234--245 | publisher = Springer }} </ref> <ref name="rudenko2007overview"> {{cite conference | title = An overview of blackboard architecture application for real tasks | author = Rudenko, D | author2 = Borisov, A | year = 2007 | conference = Scientific Proceedings Of Riga Technical University, Ser | pages = 50--57 | publisher = | volume = 5 }} </ref> <ref name="sacerdoti1975nonlinear"> {{cite conference | title = The Nonlinear Nature of Plans | author = Sacerdoti, Earl D | year = 1975 | conference = Proceedings of the 4th International Joint Conference on Artificial Intelligence IJCAI'75 | pages = 206--214 | publisher = Morgan Kaufmann Publishers Inc. | volume = 1 }} </ref> <ref name="wray2005introduction"> {{cite journal | title = An introduction to Soar as an agent architecture | author = Wray, Robert E | author2 = Jones, Randolph M | year = 2005 | publisher = Cambridge University Press: Cambridge, UK | journal = Cognition and multi-agent interaction: From cognitive modeling to social simulation | pages = 53--78 }} </ref> <ref name="zamanideep2017"> {{cite conference | title = Deep Reinforcement Learning using Symbolic Representation for Performing Spoken Language Instructions | author = Zamani, Mohammad Ali | author2 = Magg, Sven | author3 = Weber, Cornelius | author4 = Wermter, Stefan | year = 2017 | conference = 2nd Workshop on Behavior Adaptation, Interaction and Learning for Assistive Robotics (BAILAR) on Robot and Human Interactive Communication (RO-MAN), 26th IEEE International Symposium on }} </ref> </references> h2ie7j21zw35cfkfdrl6e4albl9g7xu 103 258614 2814902 2651878 2026-06-09T19:02:51Z IanVG 2918363 /* ExampleProblems.com no longer available */ new section 2814902 wikitext text/x-wiki Thanks I am creating a new site here let us discuss == ExampleProblems.com no longer available == It looks like https://www.exampleproblems.com/wiki/index.php/Main_Page and [https://www.exampleproblems.com/wiki/index.php/Main_Page https://www.exampleproblems.com] are both down, which is a bummer. Another good reason to independently develop resources on Wikiversity! [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 19:02, 9 June 2026 (UTC) 7cd9p9oofqrraxcgrbc5kfi84fpos7p 0 264042 2814922 2596971 2026-06-09T22:28:38Z Scogdill 1331941 2814922 wikitext text/x-wiki == Overview == Charles Duncombe became Viscount Helmsley in 1881, when his father died and when he was 2 years old, and he did not marry until 1904. His father was not the later Earl of Feversham, and so Charles Duncombe was not the heir apparent to the earldom. His mother would still have been called Lady Helmsley or Viscountess Helmsley. == Also Known As == *Family name: Duncombe *Viscount Helmsley was a courtesy title for the eldest son and heir apparent of the [[Social Victorians/People/Feversham | Earl of Feversham]] (at the end of the 19th century).<ref>{{Cite journal|date=2020-10-14|title=Baron Feversham|url=https://en.wikipedia.org/w/index.php?title=Baron_Feversham&oldid=983534946|journal=Wikipedia|language=en}}</ref> *Viscount Helmsley **William Reginald Duncombe ( – 24 December 1881) **Charles William Reginald Duncombe (24 December 1881 –1915)<ref>{{Cite journal|date=2020-09-12|title=Charles Duncombe, 2nd Earl of Feversham|url=https://en.wikipedia.org/w/index.php?title=Charles_Duncombe,_2nd_Earl_of_Feversham&oldid=978075739|journal=Wikipedia|language=en}}</ref> *Viscountess Helmsley **Muriel Frances Louisa Chetwynd-Talbot Duncombe (23 December 1876 – 19 January 1904 **Marjorie Blanche Eva Greville Duncombe (19 January 1904 – ) *Dowager Viscountess Helmesley **Muriel Frances Louisa Chetwynd-Talbot Duncombe Owen (19 January 1904 – 2 March 1925) == Acquaintances, Friends and Enemies == == Timeline == '''1876 December 23''', Muriel Frances Louisa Chetwynd-Talbot and William Reginald Duncombe married.<ref name=":0">"Lady Muriel Frances Louisa Talbot." {{Cite web|url=https://thepeerage.com/p1278.htm#i12776|title=Person Page|website=thepeerage.com|access-date=2020-11-24}}</ref> '''1885 June 6''', Muriel Frances Louisa Chetwynd-Talbot Duncombe and Hugh Darby Annesley Owen married.<ref name=":0" /> '''1896 February 12''', Mabel Theresa Duncombe and Sir William Gervase Beckett married.<ref>"Hon. Mabel Theresa Duncombe." {{Cite web|url=https://thepeerage.com/p1727.htm#i17261|title=Person Page|website=thepeerage.com|access-date=2020-11-24}}</ref> '''1897 July 2''', Lord and Lady Helmsley attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House. (Lord Charles Duncombe, Viscount Helmsley is #353 on the [[Social Victorians/1897 Fancy Dress Ball#List of People Who Attended|list of people who were present]]; Muriel Duncombe, Lady Helmsley is #354.) '''1904 January 19''', Charles Duncombe and Marjorie Blanche Eva Greville married.<ref name=":1">"Lady Marjorie Blanche Eva Greville." {{Cite web|url=https://thepeerage.com/p2289.htm#i22881|title=Person Page|website=thepeerage.com|access-date=2020-11-24}}</ref> [[File:Martin van Meytens 003.jpg|alt=Old painting of a 9-year-old boy dressed very formally and richly, seated at a table with a crown nearby and holding a book.|thumb|Archduke Charles Joseph of Austria, c. 1747–1749.]] == Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball == At the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]], Lord Charles Duncombe, Viscount Helmsley was dressed as Archduke Charles in the Archduchess Marie-Karoline and Emperor Joseph II section of the Austrian Court of Maria Theresa Quadrille.<ref name=":2">"Ball at Devonshire House." The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.</ref><ref name=":3">"Fancy Dress Ball at Devonshire House." ''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.</ref> * He was in "Court costume."<ref>“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.</ref> * "V<small>ISCOUNT</small> H<small>ELMSLEY</small> was in a Court costume."<ref>“Additional Costumes Worn at the Duchess of Devonshire’s Fancy Ball.” The ''Queen, The Lady’s Newspaper''17 July 1897, Saturday: 63 [of 97 BNA; p. 138 on the print page], Col. 2a–3a [3 of 3 cols.]. ''British Newspaper Archive''  https://www.britishnewspaperarchive.co.uk/viewer/BL/0002627/18970717/283/0064.</ref>{{rp|Col. 3a}} Muriel Duncombe, Lady Helmsley was dressed as Princess Charlotte of Lorraine, also in the Austrian Court of Maria Theresa Quadrille.<ref name=":2" /><ref name=":3" /> Muriel Duncombe was not Charles's wife but his mother. No photographs of their costumes exist at this time. === Who They Were Dressed As === If Charles Duncombe, Viscount Helmsley was dressed as Archduke Charles Joseph of Austria, second son of Maria Therese and Francis I, then Marie-Karoline and Emperor Joseph II were his historical siblings. Charles Duncombe was 18 at the time of the ball, and Archduke Charles Joseph was nearly 16 when he died. He is shown at perhaps 9 years old in a portrait by Martin an Meytens (right). Who Muriel, Lady Helmsley was dressed as is more difficult to determine. An Anne Charlotte of Lorraine-Brionne, known as Mademoiselle de Brionne, was at the court of Marie Antoinette and would more likely have been in that procession, although perhaps she wanted to be nearer her husband. A few other Princess Charlottes or Princess Anne Charlottes of Lorraine existed, but they are all at least one generation older and associated with the French court. So it is not clear who she was dressed as. == Demographics == *Nationality: English == Family == * William Ernest Duncombe, 1st Earl Feversham of Ryedale (28 January 1829 – 13 January 1915)<ref>{{Cite web|url=https://www.thepeerage.com/p1873.htm#i18721|title="William Ernest Duncombe, 1st Earl Feversham of Ryedale." Person Page 1872|website=www.thepeerage.com|access-date=2026-06-09}}</ref> * Mabel Violet Graham (15 February 1833 – 28 August 1915) *# Lady Ulrica Duncombe ( – 27 April 1935) *# '''William Reginald Duncombe, Viscount Helmsley''' (1 August 1852 – 24 December 1881) *# Hon. James Henry Duncombe (20 October 1853 – 10 January 1886) *# Hon. Hubert Ernest Valentine Duncombe (14 February 1862 – 21 October 1918) *# Lady Hermione Wilhelmina Duncombe (30 March 1864 – 19 March 1895) *# Lady Helen Venetia Duncombe (1866 – 16 May 1954) *# Lady Mabel Cynthia Duncombe (1869 – 25 April 1926) *Muriel Frances Louisa Chetwynd-Talbot Duncombe Owen (c. 1860 – 2 March 1925)<ref name=":0" /> *William Reginald Duncombe, Viscount Helmsley (1 August 1852 – 24 December 1881)<ref>{{Cite journal|date=2019-08-11|title=William Duncombe, Viscount Helmsley|url=https://en.wikipedia.org/w/index.php?title=William_Duncombe,_Viscount_Helmsley&oldid=910373349|journal=Wikipedia|language=en}}</ref> *#'''Mabel Theresa Duncombe''' (1877–1913) *#'''Charles William Reginald Duncombe''', 2nd [[Social Victorians/People/Feversham | Earl of Feversham]] (1879–1916) *Hugh Darby Annesley Owen ( – 12 March 1908)<ref>"Hugh Darby Annesley Owen." {{Cite web|url=https://thepeerage.com/p1873.htm#i18722|title=Person Page|website=thepeerage.com|access-date=2020-11-24}}</ref> *Charles William Reginald Duncombe, 2nd Earl of Feversham (8 May 1879 – 15 September 1916)<ref>"Charles William Reginald Duncombe, 2nd Earl of Feversham of Ryedale." {{Cite web|url=https://thepeerage.com/p2288.htm#i22880|title=Person Page|website=thepeerage.com|access-date=2020-11-24}}</ref> *Marjorie Blanche Eva Greville Duncombe (25 October 1884 – 25 July 1964)<ref name=":1" /> #Lady Mary Diana Duncombe (19 March 1905 – October 1943) #Charles William Slingsby Duncombe, 3rd Earl of Feversham (2 November 1906 – 4 September 1963) #Hon. David William Ernest Duncombe (8 February 1910 – September 1927) == Notes and Questions == #Muriel Duncombe Owen may have been Viscountess Helmsley, or Lady Helmsley. Her son Charles William Reginald Duncombe was Viscount Helmsley by this time, but he did not marry until 1904, so no other Lady Helmsely seems likely. She married Hugh Darby Annesley Owen, however, in 1885, but if she retained her title, then she would still be eligible to use it. == Footnotes == {{reflist}} hyz8zll343nqfzrv05fj4ra8hg3lrhw 10 270298 2814852 2242909 2026-06-09T12:41:31Z Atcovi 276019 change some stuff around 2814852 wikitext text/x-wiki {{userbox | border-c = {{{1|#FFCCAA}}} | id = {{{3|[[File:Noun Project research icon 4479310.svg|40px|Science]]}}} | id-c = {{{5|{{{id-s|14}}}}}} | info = {{{4|This user is a '''[[Research|researcher]]'''.}}} | info-c = #61C397 | info-fc = {{{info-fc|black}}} }}<noinclude> [[Category:Occupation user templates|researcher]] </noinclude> ms45izjbijnv2tq8joi8cbyw0y6sfrd 0 275987 2814950 2438363 2026-06-10T00:24:28Z Jtneill 10242 Combine two short sections 2814950 wikitext text/x-wiki {{title|Safety as a psychological need:<br>What is safety and what are its implications as a psychological need?}} {{MECR3|1=https://youtu.be/ayudIkO5CxA}} __TOC__ ==Overview== Safety is a perception that one "feels safe" and is "confident that they will not be harmed" (Gorman, 2010). {{expand}} == Theories == === Maslow's Hierarchy of Needs === [[File:Maslow's Hierarchy of Needs2.svg|thumb|302x302px|Figure 1. Five-tier model of Maslow's Hierarchy of Needs]] Maslow's Hierarchy of Needs was originally proposed by Abraham Maslow in his 1943 article, A Theory of Human Motivation, and was continually refined throughout his career (Maslow, 1943; Maslow & Frager, 1987). The model examines human motivation and suggests an individual's basic needs must be met before they can begin to grow and develop. A pyramid is often used to represent the model and the identified needs at the base of the pyramid (deficiency needs) must be satisfied before higher needs (growth needs, which are physically higher on the pyramid) can be addressed (Figure 1; Figure 2). ==== Five-tier model ==== Maslow identified five key human needs influencing motivation (Figure 1): # Physiological needs (e.g. food, water, air, shelter); # Safety needs (e.g. safey{{sp}} and security); # Social belonging and love needs (e.g. meaningful social, family, and community connections); # Esteem needs (i.e. self-esteem and being held in esteem by others); and # Self-actualisation needs (i.e. seeking and achieving growth and potential) (Malow{{sp}}, 1943). Each need must be met for any further needs to be addressed. ==== Eight-tier model ==== [[File:Expanded Maslow's Needs.webp|thumb|305x305px|Figure 2. Eight-tier model of Maslow's Hierarchy of Needs]] Maslow later developed the model and added three additional needs (Figure 2): # Cognitive needs (curiosity, knowledge, understanding, purpose, and predictability); # Aesthetic needs (appreciation of aesthetic qualities in the environment); and # Transcendence (motivation influenced by values that transcend oneself, e.g. religion, experiencing nature, and caring for others) (Maslow, 1970). Maslow's Hierarchy of Needs reflects safety and security as a psychological need by identifying the concepts as a foundational necessity for psychological functioning and higher achievement. === Attachment Theories === Attachment theory was first developed by John Bowlby (1950), who defined attachment as "lasting psychological connectedness between human beings" (Bowlby, 1969). He suggested a child's attachment to their parent is an evolutionary protective mechanism to ensure they receive safety and security, increasing the child's survival chances. Adult attachment styles are often consistent with attachment styles developed during childhood and adolescence (Kim et al., 2021). A systematic review and meta-analysis from 2021 found anxiety-related and avoidance-related attachment to parental figures were particularly persistent into adulthood, with external social relationships found to positively impact attachment styles that were avoidant but not anxious (Kim et al., 2021). The level of parental involvement in childhood and adolescence can impact a child's attachment style to parents, caregivers, and peers. Overprotective parenting behaviours and unattainable expectations that restrict the autonomy and self-determination of the child can contribute to more anxious and/or avoidant attachment styles. This pattern presents commonly in families living in Asia or who have immigrated from Asian countries (Kim et al., 2012). Kim and colleagues (2012) studied families of South Korean descent living in the US, parenting styles, and attachment styles of the children. They found some cultural aspects of parentaing{{sp}}, including expectations of achievement for academic and extracurricular activities and of choice of university, career, and spouse, triggered anxiety, negatively impacted interpersonal relationships, and was associated with a negative self-concept. Anxious and avoidant attachment styles were more likely to be developed in childhood and continued into adult life (Kim et al., 2012). In contrast, underprotective and uninvolved parenting such as neglect has been associated with the same outcomes as overprotective parenting with high expectations (Bowlby, 1969). Attachment styles can affect an individual's functioning throughout adulthood. ==== Types of Attachment ==== John Bowlby (1960) identified four styles of attachment: secure, avoidant, ambivalent, and disorganised. A child's attachment style shapes their adult attachment style, although this can change over time. A secure attachment is formed when the caregiver appropriately and quickly responds to the child's needs. Avoidant attachments are formed when the child avoids interaction with the caregiver, likely due to previously unmet needs. Ambivalent attachments are formed in infants who are 'anxious' about being separate from their caregiver and 'clingy'. Disorganised attachment is often observed in infants with attachment disorders, as there is a confusing relationship between wanting their needs met and being fearful of their caregiver. ==== Stages of Attachment ==== {{expand}} ==== Schaffer & Emerson ==== In 1964, Schaffer & Emerson (1964) devised stages through observations to describe how infants form attachments to caregivers: # Asocial; # Indiscriminate attachment; # Specific attachment; and # Multiple attachment The stages were based on measures of stranger anxiety, separation anxiety, and social referencing. According to their model, an infant aged 0-6 weeks will not necessarily respond to social stimuli and will produce age-appropriate facial expressions, such as a smile (asocial stage). After reaching 6 weeks of age, an infant will respond positively to any caregiver and will become distressed if interaction stops (indiscriminate attachment stage). Once the infant is 3 months of age, they will begin to recognise familiar caregivers. From 7-9 months of age, the infant will begin to prefer a single caregiver and fear strangers (specific period). During this time they will experience separation anxiety, such as if held by another caregiver. Infants aged 10 months and older begin to form multiple attachments with family members and familiar caregivers other than the primary caregiver (multiple attachment). Many infants will have developed attachments to multiple caregivers by 18 months of age, which are often hierarchical in nature. Stronger attachments were had with caregivers who quickly and correctly met the infant's needs, rather than who spends the most time with the infant.{{fact}} ==== Ainsworth ==== Ainsworth developed four stages of attachment (Ainsworth, 1970; Ainsworth 2014): # Pre-attachment; # Attachment in the making; # Clear-cut attachment; and # Goal-corrected partnership. Hierarchical Attachment Representations An improved hierarchical structure was proposed in 2016 to further describe attachment models that include attachment elements specific to relationships (Gillath et al., 2016). They posited that attachment-specific behaviours are not constant and are influenced by specific interpersonal relationships, the context, and episodic factors. Episodic factors are temporary influences on attachment and may include goals and behavioural strategies. ==== The Self Regulation Model of Attachment Trauma and Addiction ==== [[File:Self-Regulation Model of Attachment Trauma and Addition.jpg|thumb|545x545px|Figure 3. The self-regulation model of attachment trauma and addition.]] The self-regulation model of attachment trauma and addiction (SRM) related trauma-related attachment disorder with self-regulation difficulties (Padykula & Conklin, 2009). It was developed to provide a clinical perspective of addiction and trauma, and how they are impacted by attachment, particularly disordered attachment. The model suggests trauma can "injure" an individual's attachment style and ability to self-regulate, and suggests sustance{{sp}} use is an effort to self-regulate to cope with the injured attachment system and experienced trauma. The benefits of this model are that multiple contributing factors are considered when working with individual and it informs a method to measure and identify addictive behaviours and various presentations of self-regulation. The SRM suggests the concept of dialectical poles of self-regulation (Figure 3). Every individual falls along a multi-faceted spectrum of over or under-regulation regarding: * Behaviour (pathological caretaking ↔ harm to self and/or others); * Physiology (hyperarousal ↔ dissassociation); * Affect (over-emote ↔ alexithymia); * Interpersonal domains (pathological dependence ↔ counterdependence); * Cognition (negative self-concept ↔ positive self-concept); and * Self-pathological (grandiosity ↔ self-loathing). For individuals suffering with addiction and a history of trauma, such as in this model, chronic dysregulation is often experienced at extreme ends of the various dialectical poles. Disrupted and disordered attachment resulting from trauma results from feeling unsafe and insecure. Safety is foundational psychological need to assist with functional self-regulation.{{fact}} == Childhood Experiences of Trauma == Trauma is an emotional response to an individual's experience that they perceive as stressful and are unable to manage the associated emotions. Trauma can result in trauma disorders such as post-traumatic stress disorder (PTSD), acute stress disorder (ASD), reactive attachment disorder (RAD), disinhibited social engagement disorder (DSED), adjustment disorders, and other related disorders. While the impact of trauma is caused by a psychological response, the responses and possibly disorders that follow can have overwhelmingly physical, as well as psychological, symptoms (Wamser-Nanny & Vandenberg, 2013). === Common types of childhood trauma === * Abuse and neglect * Unstable family home * Exposure to family violence * Natural disasters * Perception. === Out of home care === There are three main forms of out of home care for children in Australia - kinship, foster, and residential care. '''Kinship Care''' When a child is first removed from their family, child protection bodies first seek to find an appropriate placement with kin, known as a kinship care placement. Kin can include immediate and extended family, friends, and other people in the child 's community, such as ex-step-parents. For children who are Aboriginal and Torres Strait Islander, kinship placements are deemed the most appropriate placements to maintain vital connections to culture and community, which is a key feature of their culture. By keeping connection to community and culture, displaced children can reduce the number of changes presented by a new placement and the loss of stability and security. Along with case plans developed with a social worker or case manager to maintain and build capacity around care domains such as stability and security, cultural plans are common practice to help children from culturally and linguistically diverse backgrounds continue to feel connected and continue positively learning and experiencing their culture. Kinship care placements have the most favourable outcomes for children in out of home care, compared to other types of care. Children in kinship care will typically experience fewer behavioural issues and mental illnesses, improved well-being, and fewer placement moves than children in other forms of out-of-home care (Winokur et al., 2014). They are also more likely to successfully achieve permanency, such as Enduring Parental Responsibility (in Australian Capital Territory), compared to foster placements. This may be due to difficulties faced by blood relatives securing adoption rather than other forms of permanency, so as not to alter the family tree and the child's sense of family and belonging. Kinship care placements are generally more stable and are more likely to end after placement restoration with the birth parents, rather than by relationship or placement breakdown (Perry et al., 2021). This may be due to fewer lifestyle changes and increased feelings of stability and security, resulting in less frequent and less severe behavioural issues, which are often the reason for placement breakdown (Font, 2015). Alternatively, this may be due to behavioural issues that present prior to placement impacting a child 's suitability for a kinship placement and the kinship carer's willingness and capacity to provide care (Font, 2015). Adults who have transitioned to independence from kinship care placements report strong relationships with their kinships carers, very similar to the bond usually held by a child and their biological parent (Dolbin-Macnab et al., 2009). The increased stability, safety, and security from the caregiving relationships between children and their kinship carers can have significant psychological benefits for the child. In Australia, kinship carers receive payments and can access further funds to cover the additional costs borne by caring for children, such as food, shelter, and medical needs. This is not the case for all countries or for informal kinship placements where child protective services are not officially involved. The unexpected placement of the child can lead to lack of money and financial insecurity, which becomes stressful for the carer and for the child. Research conducted in Denmark and USA found worse outcomes for kinship carers compared to foster carers - kinship carers receive less financial support, reduced access to respite care, reduced eligibility to access government financial support, and reduced access to professional and peer support (Sakai et al., 2011). ====== Foster Care ====== If no appropriate kinship care placements are available, a child may be placed with foster carers. This may be in a short-term placement initially, or planned to be long-term. Children who were initially placed in foster care at age 3 or less on average remained in their placements for 5 years longer than children who were placed at older than 3 (Oosterman et al., 2007). Neglect, abuse, family issues, and abandonment as reasons for entering the out of home care system were associated with a greater number of foster placements than were children voluntarily placed into care and behavioural issues. While a child becomes more independent with age, they still require a sense of psychological security and safety, and this need is also reflected in children in the out of home care system (Kungl et al., 2019). Foster children however require greater and more frequent proximity and reassurance to feel the same sense of security and safety. This may be due to the relationship with the caregiver being newer than with a birth parent, meaning the relationship may still be progressing through the stages of attachment and the child is still forming a schema of their relationship with the caregiver (Kungl et al., 2019). When a child experiences their first and likely susequent placement breakdowns, their experiences of psychological safety are negatively impacted, feeling as though they cannot develop safe attachments with caregivers due to their own self-esteem, fear of being judged or not accepted, and they do not expect they will benefit from the developing an attachment long-term. This is mirrored across all types of out of home care. Placement stability is a necessity to ensure safety, and in turn, psychological wellbeing. ====== Residential Care ====== Children placed in residential care often do not feel safe in their placement and experience difficulties developing safe and secure relationships with caregivers and peers. Attachment disorders are common in children in residential care. This is often resulting from the trauma they experienced with their birth families prior to removal from home, the trauma of the removal, the trauma of multiple placement breakdowns, and the inability to form secure attachments with residential carers due to high staff turnover, casual staff rosters, and the rotating roster of staff. Trauma-informed case management can protect from further psychological harm and promote psychological healing of children, young people, carers, and families who have experienced trauma (Topitzes et al., 2019). Ongoing connection to their birth family, where safe and appropriate, can maintain a sense of belonging and psychological safety for the child (Collings et al., 2021). This is particularly important for children Aboriginal and Torres Strait Islander descent, given the cultural importance of family and community and the very recent Stolen Generation. ===== Dyadic Developmental Psychotherapy ===== Dyadic Developmental Psychotherapy (DDP) was developed by Dan Hughes and is a form of family therapy used to support primarily with attachments and feelings of safety and security within relationships between children and their caregivers (Becker‐Weidman & Hughes, 2008). DDP focuses on developing the understanding and abilities of the child, caregiver, and therapist to attune to each other's needs, with the therapist removed from the therapeutic relationship after the capacity is sufficiently developed. DDP is funded by some public health systems, such as through Canberra, Australia's public Child at Risk Health Unit (CARHU). By increasing the child and caregiver's attunement to needs, they can each exist in the relationship with healthier attachments and more psychological safety and security. ===== Cognitive Behavioural Therapy ===== Cognitive Behavioural Therapy (CBT) can be an effective therapy for assisting with healing from various forms of trauma, including domestic abuse and childhood trauma. CBT is a very common form of therapy in Australia and is publicly funded under Medicare. ==== Therapeutic Life Story Work ==== A child's understanding of where they come from and their journey, or life story, to their present experience is a crucial element of healing from the trauma and lack of safety experienced by the child. Life story work promotes trust, feelings of safety, and ability to adapt to changes (Coman & Rainey, 2016). == Domestic Abuse and Wellbeing == Domestic abuse, otherwise known as domestic violence or intimate partner violence, is defined by Wong and Bouchard (2021) as "any act of physical violence (e.g., punching, choking, slapping) or psychological abuse (e.g., manipulation, verbal assault, threatening) that occurs between current or former partners/spouses [and] can also include sexual harm (e.g., coerced sex, sexual assault) and economic abuse (e.g., financial control/deprivation)". Domestic abuse is illegal in Australia and many countries across the world. Survivors of domestic violence are facing overwhelmingly unsafe environments, which negatively impacts psychological wellbeing. Help-seeking behaviours for domestic abuse vary across different groups. Survivors who are Black Muslim women in America may first seek help first from social and community, religious, professional services, and finally law enforcement (Oyewuwo-Gassikia, 2020). Female survivors are more likely to suffer abuse for many years before seeking help, compared to male survivors who on average seek help after six months of suffering abuse. Male survivors are more likely to suffer psychological abuse and less likely to report the abuse, while female survivors more likely to suffer physical, financial, and a combination of abuse (Peraica et al., 2021). Help-seeking may be inhibited by lack of knowledge of options, accommodation insecurity, financial insecurity, lack of social support, fear of losing one's children, fear of the abusive partner, and the complex psychological mechanisms involved in domestic abuse (Weatherall & Tennent, 2021). Increasing and maintaining social and community support, engagement in education and employment, capacity-building, and formal support services for domestic violence and other community services can serve as protective factors for domestic abuse and support survivors to break free of the abuse (Sinko et al., 2021). Perpetrator interventions can have varying success with reduction in and prevention of future abuse (Karakurt et al., 2019). == Workplace Safety and Security == {{expand}} === Leadership and Performance === An employee's feelings of psychological safety at work and with the people they work with can be positively influenced by organisational leadership. Leaders have the power to enhance and improve employee psychological safety in the workplace to the benefit of the employee and the organisation. Leadership style and values play an important role in the psychological safety of employees. Consultation in the development and/or evaluation of processes or projects regarding the organisation and regarding decisions that will impact employees can assist employees to feel valued and feel a sense of control, which is important for psychological safety. Modelling the values a leader wishes to see in their employees such as participation, contribution, productivity, and support and acceptance of others can influence others to demonstrate and take on the same behaviours and values in the workplace (Wong et al., 2010).The same applies inversely. Psychologically unsafe workplace cultures, which can be fostered by leaders, can lead to burnout and poor performance (Hoprekstad et al., 2019). Aggression and bullying in the workplace is very common in the healthcare sector, resulting in a psychologically unsafe working environment. Europe's healthcare sector has one of the highest rates of workplace violence, with up to 90% of staff suffering verbal abuse, 64% verbal threats, and 32% physical violence (Pompeii et al., 2013). The negative environment can cause employees to experience sadness, anger, and fear, resulting in counter-productive work behaviour (Fida et al., 2018). Psychological safety is needed to improve and maintain psychological wellbeing. Psychological safety is a necessity for effective employee workplace performance. Employees need to feel they can suggest ideas, work with others, and attempt creative solutions to problem-solving. Psychological safety is needed to mediate the risk of being viewed as a social outcast for breaking the social norms and organisational traditions, offending another employee or leader for voicing contrary beliefs, or being viewed as incompetent and a failure if the employee's input is not appropriate or successful (Newman et al., 2017). The employee alone is risking their social and professional reputation, their self-esteem and self-efficacy, and potentially future employment facilitated by maintaining positive professional networks. Without psychological safety, it is easy for the risks to simply not be taken, preventing future growth and development of the individual employee and the organisation. Fostering a culture where employees are comfortable expressing themselves, seeking and providing honest and constructive feedback, working with others, and take measured risks can increase psychological safety and mediate the aforementioned risks (Edmondson, 1999). Increased feelings of psychological safety can improve the quality and frequency of information sharing. This can be influenced by feeling safe with organisational leaders such as an employee's direct supervisor, and the sense of trust, perceived approachability, and whether the employee believes they will receive judgement or support from their leader. It is important to note that organisations in collectivist societies are more likely to willingly share information with less prompting as it is seen as for the value of the organisational community to proactively share information (Yin et al., 2019). Reflection on past experiences can support future learning and development, which can be extremely beneficial for developing new and old sought-after skills, knowledge, and values. Organisational teams who are open to learning can more easily practice reflection, develop skills, knowledge and values, and alter their practice to better suit their own needs and organisational needs (Bunderson & Sutcliffe, 2003). A culture of acceptance surrounding reflection can increase psychological safety, creating an environment where employees feel comfortable taking risks and asking for help, knowing their skills and experiences will be valued (Wong et al., 2010). This willingness to reflect and adapt can increase employee and organisational ability to adapt to new challenges, which can increase individual self-efficacy and control, improve the strength of the team, and increase workplace social support (Košir et al., 2015). These benefits are associated with preventing or reducing burnout. Increased psychological safety allows for individuals and teams to grow and change through reflection, resulting in improved organisational and individual outcomes. === Burnout === Burnout, or occupational burnout, is a syndrome resulting from a combination of cumulative and long-term occupational stress and individual differences (Salminen et al., 2017). The World Health Organization updated its definition of burnout in the 11th revision of the International Classification of Diseases (ICD-11) to be, "1) feelings of energy depletion or exhaustion; 2) increased mental distance from one’s job, or feelings of negativism or cynicism related to one's job; and 3) a sense of ineffectiveness and lack of accomplishment" (World Health Organization, 2018). Burnout can be measured with tools such as the Maslach Burnout Inventory (MBI), which utilises twenty-two questions to measure emotional exhaustion, depersonalisation, personal accomplishment, cynicism, and professional effifacy{{sp}}. Variations of the inventory exist for employees in human services, health professions, education, students, and a general inventory (Maslach et al., 2016). While burnout results from chronic stress, it does not have to remain a chronic experience. Individuals recover from burnout and with effective self-care and environmental strategies, they can assist in the prevention of future burnout. Circumstantial burnout, resulting from environment and things the individual can control, can be treated and managed by problem-solving any pre-existing workplace challenges, enriching personal lives such as through hobbies and social supports, and taking a period of leave from work (Abedini et al., 2018). Existential burnout, resulting from questioning the meaning of the role and professional efficacy, can be treated and managed by the employee and others (e.g. supervisor) acknowledging the presence of the burnout, developing and strengthening meaningful workpace{{sp}} connections, rediscovering meaningful purpose within the work, and re-examining professional identity (Abedini et al., 2018). == Conclusion == Maslow's hierarchy of needs identifies safety and security as a basic physiological and psychological requirement to reach the growth and development needed for positive psychological wellbeing. Attachment theories help to explain the underlying mechanisms of safe attachments to thrive in one's environment. Childhood trauma and involvement with the child protection system and out of home care is associated with a loss of safety and has many negative impacts psychologically, although many impacts can be mediated by a restoration of safety. Domestic abuse is another example of a loss of safety that through an increase of safety, the psychological wellbeing of both the survivor and the perpetrator can be improved. Safety in the workplace can be created and fostered by leaders and employees alike, and is necessary to recover from and treat burnout, and maintain psychological wellbeing. Safety is a psychological need that can be fostered and developed to meet the best possible outcomes. == See also == {{ic|Use alphabetical order. Rename links so they are more user-friendly, as per Tutorial 1.}} * [[wikipedia:Maslow's_hierarchy_of_needs|Maslow's hierarchy of needs]] (Wikipedia) * [[Motivation and emotion/Book/2020/Transgenerational trauma]] * [[Motivation and emotion/Book/2015/Stolen Generations and emotion]] * [[Motivation and emotion/Book/2019/Indigenous Australian transgenerational trauma and psychological well-being]] * [[Motivation and emotion/Book/2018/Childhood trauma and emotion]] * [[Motivation and emotion/Book/2021/Trauma-informed therapy]] * [[Motivation and emotion/Book/2018/Childhood neglect and emotion]] * [[Motivation and emotion/Book/2016/Vicarious trauma effects on the emotionality of mental health workers]] * [[Motivation and emotion/Book/2013/Burnout]] == References == {{Hanging indent|1= Abedini, N. C., Stack, S. W., Goodman, J. L., & Steinberg, K. P. (2018). “It's not just time off”: a framework for understanding factors promoting recovery from burnout among internal medicine residents. 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C., McLean, L., & Buratti, S. (2021). Trauma-informed family contact practice for children in out-of-home care. ''The British Journal of Social Work''. Coman, W., Dickson, S., McGill, L., & Rainey, M. (2016). Why am I in care? A model for communicating with children about entry to care that promotes psychological safety and adjustment. ''Adoption & Fostering'', ''40''(1), 49-59. Edmondson, A. (1999). Psychological safety and learning behavior in work teams. ''Administrative science quarterly'', ''44''(2), 350-383. Fida, R., Tramontano, C., Paciello, M., Guglielmetti, C., Gilardi, S., Probst, T. M., & Barbaranelli, C. (2018). ‘First, Do No Harm’: The Role of Negative Emotions and Moral Disengagement in Understanding the Relationship Between Workplace Aggression and Misbehavior. ''Frontiers in psychology'', ''9'', 671. Gillath, O., Karantzas, G. C., & Fraley, R. C. (2016). ''Adult attachment: A concise introduction to theory and research''. Academic Press. Gorman, D. (2010). Maslow's hierarchy and social and emotional wellbeing. ''Aboriginal and Islander Health Worker Journal'', ''34''(1), 27-29. Hoprekstad, Ø. L., Hetland, J., Bakker, A. B., Olsen, O. K., Espevik, R., Wessel, M., & Einarsen, S. V. (2019). How long does it last? Prior victimization from workplace bullying moderates the relationship between daily exposure to negative acts and subsequent depressed mood. ''European Journal of Work and Organizational Psychology'', ''28''(2), 164-178. Karakurt, G., Koç, E., Çetinsaya, E. E., Ayluçtarhan, Z., & Bolen, S. (2019). Meta-analysis and systematic review for the treatment of perpetrators of intimate partner violence. ''Neuroscience & Biobehavioral Reviews'', ''105'', 220-230. Kim, E., Im, H., Nahm, E., & Hong, S. (2012). Korean American parents’ reconstruction of immigrant parenting in the United States. ''Journal of cultural diversity'', ''19''(4) 124. Kim, S., Baek, M., & Park, S. (2021). 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A theory of human motivation. ''Psychological review'', ''50''(4), 370. Maslow, A. H. (1970). New introduction: Religions, values, and peak-experiences. ''Journal of Transpersonal Psychology'', ''2''(2), 83-90. Maslow, A., & Frager, R. (1987). ''Motivation and personality''. Addison Wesley Longman. Newman, A., Donohue, R., & Eva, N. (2017). Psychological safety: A systematic review of the literature. ''Human Resource Management Review'', ''27''(3), 521-535. Oosterman, M., Schuengel, C., Wim Slot, N., Bullens, R., & Doreleijers, T. (2007). Disruptions in foster care: A review and meta-analysis. ''Children And Youth Services Review'', ''29''(1), 53-76. <nowiki>https://doi.org/10.1016/j.childyouth.2006.07.003</nowiki> Oyewuwo-Gassikia, O. B. (2020). Black Muslim women’s domestic violence help-seeking strategies: Types, motivations, and outcomes. ''Journal of Aggression, Maltreatment & Trauma'', ''29''(7), 856-875. Padykula, N., & Conklin, P. (2009). 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''Archives of Pediatrics & Adolescent Medicine'', ''165''(2), 159-165. Salminen, S., Andreou, E., Holma, J., Pekkonen, M., & Mäkikangas, A. (2017). Narratives of burnout and recovery from an agency perspective: A two-year longitudinal study. ''Burnout Research'', ''7'', 1-9. Schaffer, H. R., & Emerson, P. E. (1964). The development of social attachments in infancy. ''Monographs of the society for research in child development'', 1-77. Sinko, L., James, R., & Hughesdon, K. (2021). Healing after gender-based violence: a qualitative metasynthesis using meta-ethnography. ''Trauma, Violence, & Abuse'', 1524838021991305. Topitzes, J., Grove, T., Meyer, E., Pangratz, S., & Sprague, C. (2019). Trauma-responsive child welfare services: A mixed methods study assessing safety, stability, and permanency. ''Journal Of Child Custody'', ''16''(3), 291-312. <nowiki>https://doi.org/10.1080/15379418.2019.1607796</nowiki> Wamser‐Nanney, R., & Vandenberg, B. R. (2013). Empirical support for the definition of a complex trauma event in children and adolescents. ''Journal of traumatic stress'', ''26''(6), 671-678. Weatherall, A., & Tennent, E. (2021). “I don’t have an address”: Housing instability and domestic violence in help-seeking calls to a support service. ''Feminism & Psychology'', ''31''(3), 424-445. Winokur, M., Holtan, A., & Batchelder, K. (2014). Kinship care for the safety, permanency, and well-being of children removed from the home for maltreatment. ''Cochrane Database Of Systematic Reviews''. <nowiki>https://doi.org/10.1002/14651858.cd006546.pub3</nowiki> Wong, A., Tjosvold, D., & Lu, J. (2010). Leadership values and learning in China: The mediating role of psychological safety. ''Asia Pacific Journal of Human Resources'', ''48''(1), 86-107. World Health Organization. (2018). ICD-11 for mortality and morbidity statistics (2018). Yin, J., Ma, Z., Yu, H., Jia, M., & Liao, G. (2019). Transformational leadership and employee knowledge sharing: explore the mediating roles of psychological safety and team efficacy. ''Journal of Knowledge Management''. }} == External links == Maslow's hierarchy of needs - [https://www.simplypsychology.org/maslow.html Simply Psychology] [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Needs/Psychological]] bv9t4pmtcx8eipu8nqfgfycvjwpo0fo 0 276238 2815026 2345580 2026-06-10T10:39:06Z Jtneill 10242 /* The self & self-actualisation */ 2815026 wikitext text/x-wiki {{title|Meaning in life: <br>What contributes to a sense of meaning in life and what are the consequences?}}{{MECR3|1=https://youtu.be/s0HtSFmxTcU}} __TOC__ ==Overview== What is meaning in life (MIL)? This is an age-old question. [[Socrates, Plato and Aristotle|Socrates]] (470-399BCE) was one of the first philosophers to question an individual's unique drive and search for meaning in their own lives. However, Frankls{{gr}} (1946) theory on meaning and logotherapy heavily influenced the search for this answer. Meaning can derive from one's sense of importance, purpose, direction, goals, value, cohesion, and coherence in their own life experiences. Although,{{gr}} finding MIL is typically useful and promotes psychological well-being and life-satisfaction, it may not always produce a positive effect, and may have serious negative consequences{{example}}. This chapter explains how MIL is individually crafted, conceptualised, and experienced. {{RoundBoxTop|theme=3}} '''Focus questions:''' * How can meaning in life be defined? * What contributors are there? * What are the consequences? * Is meaning in life universal? {{RoundBoxBottom}} ==What is MIL? == The search for MIL has been suggested as an innate drive,{{gr}} that all humans possess (Harlow, 1993, as cited in Steger, 2012). Meaning has been considered crucial to the ability to thrive and grow, and is associated with authentic, cohesive and happy living. It is suggested that meaning can help nurture conditions to foster happiness (Steger, 2012), decrease psychological distress, and maximise one's potential, thus increasing an individual's overall life satisfaction. === Case study === <blockquote>'''Bob the counsellor (figure 1)''' {{RoundBoxTop|theme=3}} [[File:Ready for a business trip.jpg|thumb|''Figure 1''. Bob, a counsellor, a family man, who travels for work|200px]] Bob is a counsellor who travels for work. Bob values his work, as he is able to help others and feels a sense of purpose and importance in what he does. Bob also values spending weekends with his family when he is not travelling for work, and visiting his elderly parents twice a week. Bob's job is has recently given him a promotion that he is looking forward to.{{RoundBoxBottom}} </blockquote> === Case study: finding meaning === <blockquote>Bob finds meaning in his family life, and within his work. Bob has a feeling of purpose and direction, in the form of a job promotion, which creates cohesion, coherence and a sense of importance. He holds meaning within the relationship with his parents, and his family, implying the significant value, importance and purpose these relationships have for providing meaning in Bobs{{gr}} life. Because Bob identifies meaning in multiple aspects of his life, he has a relatively strong feeling of MIL. The current situation {{missing}} a positive consequence of Bobs{{gr}} identified MIL.</blockquote> ==How can you define MIL? == MIL has many inter-related definitions; however they still have some distinctions from one another. Definitions of MIL can span from sense of purposefulness and coherence (Park & George, 2013), the attainment of meaningfulness by completing personal goals or life narrative (Kenyon, 2000, as cited in Steger & Frazier, 2006). However, the most applicable to this chapter is an individual's capability to act according to their own morals, values and beliefs, whilst maintaining direction for individual goals and the future (Steger & Frazier, 2006). MIL has three main concepts that can be agreed upon by most theorists, including purpose, cohesion and coherence, and importance. These interact with each other to produce meaning within relationships, friendships, occupations, and other life events. === Purpose and direction === Purpose refers to an individual's perceived direction within their own life, including current and future goals (Park & George, 2013; Frankls, 1946 as cited in Steger, 2012). An individual may have more than one purpose in their life. Whilst direction,{{gr}} refers to where an individual thinks their life will go (Steger, 2012), whether that be a owning a dream home, or becoming a pro-basketball player. Purpose and direction interact to produce motivation to keep moving towards goals, and to become more resilient to negative situations (Newman, Nezleck & Thrash, 2018). A study (Dezutter, et al., 2013) investigated the MIL of chronically ill patients, and reported patients who identified meaning, or were actively searching for meaning, had better overall psychological well-being, in comparison to patients who identified limited, or no meaning. Patients who had MIL, saw a sense of purpose and direction in their lives, even though they were chronically ill. This provides insight about how searching for meaning can help improve the psychological well-being, no matter the circumstance, build resilience against adverse life events, and act as a protective factor. === Coherence and cohesion === [[File:Yingyang.svg|thumb|''Figure 2''. Yin-Yang. Represents cohesion, or 'oneness'.]] Coherence and cohesion can assist making sense of personal experience and promote a sense of meaning. Coherence can help an individual make sense of their experiences in the world (Steger, 2012), and is considered the cognitive component of meaning (Martela & Steger, 2016). Cohesion typically refers to a feeling of unity or "oneness" (see Figure 2), which can be explained as an individual's [[Motivation and emotion/Book/2014/Belongingness motivation|sense of belonging]]. An individual may feel a sense of coherence and cohesion through occupations, friendship groups and intimate relationships (Newman, Nezleck & Thrash, 2013). The more coherent and cohesive an experience is, the less risk of encountering psychological distress and conflict between personal values (Martela & Steger, 2016). By increasing coherence and cohesion, one is able to better protect from harm, and contribute to the development of meaning. === Sense of importance === The Ancient Greeks philosophised about the idea of [[Happiness|eudaimonia]], which seeks happiness, personal growth, success and responsibility for the present and future (Steger, 2012). This concept is closely related to Frankls{{gr}} (1946 as cited in Steger, 20120) work of meaning and logotherapy, where an individual's sense of importance,{{gr}} focuses on personal values, self-worth, expectations and standards. Through adverse life events, it is essential for individual's{{gr}} to feel valued, important and worthy, to help protect and strengthen psychological well-being. This sense of importance,{{gr}} increases self-awareness and promotes healthy relationships (Frankl, 1946 as cited in Steger, 2012). An individual may find a sense of importance through values such as family, or work, and when expectations and standards are met, it can increase self-worth. Without a sense of importance, individual's may experience conflict with personal values, resulting in psychological distress, or discomfort and low self-worth. ==What contributes to MIL? == Contributors to MIL,{{gr}} vary from one individual to another, but some (Krok., 2015; Lin & Chan, 2021; Martela & Steger, 2016) suggest three main factors: religion and culture, social factors, and self-awareness. These typically work together to help develop and strengthen individually crafted meanings, and act as a buffer to distressing situations. [[File:Families Belong Together - 6.30.18.jpg|thumb|''Figure 3''. Example of a social contributor. individual's rally together to protest, displaying their sense of belonging, inter-personal relationships and support within the group to help voice their opinions]] ===Social contributors=== Social aspects that contribute to a sense of MIL can include religious and cultural views and support, the number and significance of inter-personal relationships, and perceived sense of belonging (Martela & Steger, 2016), connections and support. Studies (Krock, 2015; Lin & Chan, 2021; Pan, et al., 2010) measuring MIL in international students, found that counselling sessions that target goals and seek meaning within relationships within others and host societies,{{gr}} were effective at protecting against feelings of loneliness, stress and isolation, and increase life satisfaction (Itzick, Kagan & Ben-Ezra, 2018; Pan, et al., 2010). Other studies (Pan, et al., 2010; Steen, Berghuis & Braam, 2019),{{gr}} suggest that common identities that arise from similarities or goals, can also contribute to a sense of meaning, as individual's{{gr}} feel a strong group identity, purpose, and a sense of belongingness. For example, rally's{{gr}} or protests typically consist of people from different walks of life, coming together for a common goal or purpose (see figure 3). ===The self and self-actualisation=== [[File:Maslow's Hierarchy of Needs2.svg|thumb|429x429px|''Figure 4''. Maslow's hierarchy of needs.]] Meaning in one's life is interpreted, and uniquely created by every individual, by their own personal values, morals and beliefs. Psychologists who take a counselling stance, actively use techniques such as outlining personal goals, morals and values, to raise self-awareness, self-esteem and outlook on life (Steger, et al., 2008). Some individual's{{gr}} even look to reach [[Motivation and emotion/Textbook/Motivation/Self-actualisation|self-actualisation]] (see figure 4), defined as an individual's 'highest level of psychological development, where personal potential is fully realized after basic bodily and ego needs have been fulfilled' (Maslow, 1943, as cited in, Book chapter, 2020). Evidence suggests that individuals who use self-help tools,{{gr}} can better strive towards self-actualisation, and are more likely to interpret their lives as meaningful, and find more meaning within the 'little' things in life (Itzick, Kagan & Ben-Ezra., 2018). This suggests that self-improvement, by utilising tools such as meditation, therapy or journaling can help individual's{{gr}} find and understand their own meaning, which can in turn improve well-being, quality of life and increase life satisfaction. Furthermore, studies (Itzick, Kagan & Ben-Ezra., 2018; Steen, Berghuis & Braam, 2019; Steger, 2012) found that people rated basic and psychological needs such as belongingness, autonomy and purpose as more important to fulfilling life satisfaction, than money. Thus, implying individual's must first satisfy their basic and psychological needs to feel accomplished, and rely on these factors over material items and wealth{{gr}}. Increasing self-awareness, and self-help may help an individual find meaning, and work toward self-actualisation. ===Religion and cultural contributors === [[File:Holy Trinity Sloane Street Church Nave 2 - Diliff.jpg|thumb|''Figure 5''. Church. Religion may contribute to an individual's perception of MIL. |279x279px]] Religion and culture are often described as factors that can influence one's sense of MIL, either positively, or negatively (Krok, 2015; Martela & Steger., 2016; Steger, et al., 2008; Pan, et al., 2010). Meaning is individually experienced and perceived, so therefore it cannot be universal. Studies (Krok, 2015; Lin & Chan, 2021; Steger, et al., 2008) suggest that psychological coping and well-being, and religion, go hand-in-hand in developing meaning. Once meaning is established, it can act as a buffer against negative life events, and create an internal structure, strengthening the concept of the self. Supported by numerous studies (Steger, et al., 2008, 2021), one's perception of MIL, is increased when they attribute meaning to multiple factors, which can lead to heightened life satisfaction and happiness. For example, if an individual feels a sense of belongingness, happiness, safety and purpose when attending church, this can develop into a meaning in their lives, as they become to value and hold religion as part of their identity (see figure 5). Additionally, Steger, et al. (2008) found that MIL is perhaps defined and perceived differently across cultures, such that American participants tended to search for meaning after a more negative life event, whilst Japanese participants searched following a positive life event. This implies that culture may have an influence on the way meaning is interpreted and understood by the individual. Overall, religious and cultural factors can contribute (Pan, et al., 2010) to a sense of MIL, by providing belongingness, happiness, life satisfaction and protect against negative life events, and additionally, can even construe how it is searched and interpreted cross-culturally. ==What are consequences of searching for MIL?== Some psychologists (e,g Maslow, 1943; Rogers, 1946) and researchers (Battista & Almond, 1973; Baumeister, 1991, as cited in Steger & Frazier, 2006) suggest that searching for MIL often occurs subsequent to a negative, unexpected or traumatic event in one's personal life (Newman, Nezlek & Thrash, 2018). However, lack of present meaning or not, individuals can seek meaning through many avenues, and develop multiple meanings, to create a strong, protective barrier (Steger, 2012). === Negative consequences === [[File:Aircraft Rescue Firefighting training.jpg|thumb|285x285px|''Figure 6.'' Catastrophic bushfire. If an individual loses a loved one, or home, they may find it hard to find meaning, if meaning is lost.]] Search for meaning or identifying meaning is not always positive, and can seriously hinder and individual's psychological well-being when they struggle to identify, or lose meaning (Newman, Nezlek & Thrash, 2018; Steger & Frazier, 2006; Steger, 2012). The loss or struggle to identify meaning may be a result of lack of awareness of the self, or others, or a catastrophic life event (e.g. illness or death of loved one). Although there are strategies and tools to help searching for MIL, sometimes a negative life event (see figure 6), such the death of a loved one, can be detrimental to MIL, especially if the individual held significant meaning within the relationship. Consequences of losing or having a lack of MIL, can include suicide ideation, increased and worsened anxiety and depression and suicidal attempt (Harlow, 1986; Debates, 1993, as cited in Steger & Frazier, 2006). Steger & Frazier (2006), reported that individuals who found their lives meaningless, also lacked a sense of belonging, importance, and purpose, which heightens the risk of developing or magnifying mental disorders, and more over increased the need for intervention and therapy. Itzick, Kagan and Ben-Ezra (2018),{{gr}} also report lack of meaning can decrease job satisfaction and engagement, increase fatigue, loneliness and burnout, reduce quality of life, and impact the ability to provide adequate services in a workplace. Another factor that can negatively influence searching for meaning,{{gr}} is acculturative stress, which typically occurs when an individual is in an unfamiliar country, where they may not speak the native language, or understand the culture, which can cause frustration, loneliness, and meaninglessness (Itzick, Kagan & Ben-Ezra, 2018). This is often experienced by international students,{{gr}} who are studying in a foreign country,{{gr}} and may have increased troubles finding secure, meaningful relationships in their host country. === Positive consequences === [[File:FC Botev Plovdiv.jpg|thumb|323x323px|''Figure 7.'' Meaning found through soccer: individual's gain enjoyment, friendship, a sense of belonging and group membership.]] For example, meaning-based therapy may assist finding meaning and creating links, which can increase self-confidence, self-esteem, and happiness, which can influence awareness of the self and others (Steger & Frazier, 2006). Although therapy proves to be beneficial to self-improvement and finding meaning, individual's{{gr}} may also find their own meaning through pleasurable activities (e.g. sports, watching movies, exercise), or through establishing friendships (see figure 7) and intimate relationships (Park & George, 2013). Other studies (Dezutter, et al., 2013; Newman, Nezlek & Thrash, 2018; Russo-Netzer, Sinai & Zeevi, 2020) suggest that individual's{{gr}} who feel a sense of meaning, may have increased levels of enjoyment at their place of employment, happiness and satisfaction within. Additionally, meaning can act as a partial mediator when faced with adverse stressors, such as subjection to violence, loneliness, fatigue, burnout (Russo-Netzer, Sanai & Zeevi, 2020), and even acculturative stressors (Pan, et al., 2010). === Combatting consequences === The search of MIL can be assisted through multiple measures,{{gr}} such as through therapy, seeking out new hobbies, friendships, or discarding negatively impacting factors, such as a bad job, or a bad friendship. Searching for additional meanings in one's life,{{gr}} is strongly associated with increased life satisfaction, and overall happiness (Pan, et al., 2010; Steen, Berghuis & Braam, 2019). One method to help develop and find meaning can be done through therapy, by utilising techniques that actively search for meaning, such as outlining personal goals step by step, self-reflection, self-discovery, and trying to see other perspectives (Russo-Netzer, Sanai & Zeevi, 2020). These techniques look to increase self-awareness, teaching the individual how to search for their own meaning. Alternatively through activities, meeting new friends, or even terminating relationships (Steger, 2012) that are negatively impacting an individual's worth, can assist MIL. Some individual's{{gr}} may need extra help to develop techniques to help combat negative events, and help them seek meaning within their lives, whilst some may find it in their own ways. === Case study === Using the same study mentioned above, we will explore the consequences, both negative and positive of MIL. <blockquote>'''Case study - Bob the counsellor''' {{RoundBoxTop|theme=3}} Due to the promotion, Bob had to travel to a foreign country for five weeks. Bob has an increased feeling of coherence and cohesion, importance, purpose and direction from his new position. After a few months, Bob notices that the promotion is not what he had hoped, and his boss has given him significantly increased workload, leading to long days, and three weeks longer stay than planned. Bob is experiencing burnout and fatigue and has not seen his family, or his parents in two months. To add to Bobs distress, he is having a hard time communicating with locals for directions to hotels and restaurants. {{RoundBoxBottom}}</blockquote> ==== Case study: an in-depth look ==== <blockquote>. {{RoundBoxTop|theme=3}} Bob is experiencing conflicts in his values, that is his family and his work. Although he valued his job and promotion to begin with, and felt a sense of purpose, direction, coherence and cohesion, it quickly dwindled when he took on long working days and excess workload. More so, because he has not seen his family, he is likely feeling psychological discomfort as he identifies his family as meaningful to his life. This conflict has lead Bob to feel burnout and fatigued. He is also experiencing acculturative stress, causing loss of meaning, loneliness and purpose, which can increase his risk for anxiety, depression and other mental disorders. {{RoundBoxBottom}} </blockquote> ==== Case study: possible solution ==== <blockquote> {{RoundBoxTop|theme=3}} Bob may be able to relieve his psychological distress he is feeling from being away from his family,{{gr}} by discussing ground rules with his boss. For example, to work for only one month at a time, or have a weekend where he can fly down to see his family. Alternatively, he could ask for more local work, to reduce this conflict with his values even more. Additionally, to help his burnout and fatigue, Bob should speak to his boss to discuss his concerns, and ask for a demotion to his prior position, or reduced hours and time away from his family. This will help Bob recover from his burnout and fatigue, and help prevent against its reoccurrence. He may benefit from staying at a hotel that has English speaking workers, or alternatively asking for a local translator and guide. By having a person to help address his concerns whilst in a foreign country, it may elevate his acculturative stress. {{RoundBoxBottom}} </blockquote> ===End of chapter quiz=== To test what you have learned this chapter, take this quick quiz! <quiz display="simple"> {Contributors to ones{{gr}} meaning in life are universal} - True + False {Meaning in life is individually crafted, to suit an individuals beliefs, values and perceived meaning and purpose in life} + True - False {Ones{{gr}} sense of meaning is only a cognitive process} - True + False {Counselling psychologists use techniques such as; outlining goals and values, self-discovery, and changing outlook on life, to assist individuals to find their own meaning in life} + True - False {A protest is a way that individuals can utilise social factors to add meaning to their lives} + True - False </quiz> ==Conclusion== Overall, MIL varies across culture and is individually crafted, to fit personal beliefs and values, and can act as a protective barrier against negative life events. All humans go through a process that is suggested to be innate, to help explain, understand and give meaning to their lives. It can typically be described as the feeling of importance, sense of belonging, coherence, cohesion, direction and purpose. Purpose and direction explain the motivation an individual has towards a current or future goal, while coherence and cohesion can help elevate psychological distress, and promote the search for meaning. Furthermore, a sense of importance, ensures that an individual views their lives as significant, valued and worth living. These are often influenced by contributors such as social, religious, cultural and aspects of the self, to help develop meaning. Social contributors can be explained by inter-personal relationships, connections and support to other identifying groups, which is similar to religion and cultural factors, that focus on a central shared identity and beliefs, however is important to note that interpretation of meaning varies cross-culturally. Additionally, the self focuses on how self-work can improve awareness of both the self, and others, increase belongingness, happiness and move towards self-actualisation. Contributors such as social, the self and self actualisation, and cultural and religious factors can both hinder or help an individual's perception of meaning. Negative factors, according to research, suggest it is typical for individuals to lack meaning after a negative life event, such as the death of a family member, and can have adverse effects on psychological health, increasing the need for intervention or therapy to help combat and prevent isolation, suicidal attempt, or suicide ideation. However, positive effects of MIL, can include increased psychological health, happiness, feelings of purpose, self-worth and foster self-growth in the form of self actualisation, better job performance and buffers against stressors. Individual's{{gr}} who struggle to find meaning, may benefit from meaning-based therapy, engagement in activities, and in new friendships. Overall, this chapter described and outlined the MIL across culture, settings and how it is influence by both internal and external factors. It is important to note that MIL is individually defined, and proves to be a significant predictor in psychological well-being and life satisfaction. ==See also== * [[Motivation and emotion/Book/2011/Growth through adversity|Growth through adversity]] (Book chapter, 2011) * [[Motivation and emotion/Book/2017/Meaning and happiness|Meaning and Happiness]] (Book chapter, 2017) *[[Motivation and emotion/Textbook/Motivation/Self-actualisation|Self-actualisation]] (Book chapter, 2011) * [[Book Reviews/The Power of Meaning: Crafting a Life That Matters|The power of meaning: Crafting a life that matters]] (Book reviews, 2020) ==References== {{Hanging indent|1= Dezutter, J., Casalin, S., Wachholtz, A., Luyckx, K., Hekking, J., & Vandewiele, W. (2013). Meaning in life: An important factor for the psychogical well-being of chronically ill patients? Rehabilitation Psychology, 58(4), 334-341. https://doi.org/10.1037/a0034393 Itzick, M., Kagan, M., & Ben-Ezra, M. (2018). Social worker characteristics associated with perceived meaning in life. Journal of Social Work 18(3), 326-347. https://doi.org/10.1177/1468017316654345 Krok, D. (2015). The role of meaning in life within the relations of religious coping and psychological well-being. Journal of religion and health, 54(1), 2292-2308. https://doi.org/10.1007/s10943-014-9983-3 Martela, F., & Steger, M. F. (2016). The three meanings of meaning in life: Distinguishing coherence, purpose and significance. The Journal of Positive Psychology, 11(5), 531-545. https://doi.org/10.1080/17439760.2015.1137623 Lin, L., & Chan, H.-W. (2021). When is search for meaning in life beneficial for well-being? A cross-national study. International Journal of Psychology 56(1), 75-84. https://doi.org/10.1002/ijop.12696 Newman, D. B., Nezlek, J. B., & Thrash, T. M. (2018). The dynamics of searching for meaning and presence of meaning in daily life. Journal of Personality 86, 369-379. https://doi.org/10.1111/jopy.12321 Pan, J.-Y., Wong, D. F. K., Joubert, L., & Chan, C. L. W. (2010). The protective function of meaning of life on life satisfaction among chinese students in Australia and Hong Kong: A cross-cultural comparitive study. Journal of American College Health, 57(2), 221-232. https://doi.org/10.3200/JACH.57.2.221-232 Park, C. L., & George, L. S. (2013). Assessing meaning and meaning making in the context of stressful life events: Measurement tools and approaches. Journal of Positive Psychology, 8(6), 483-504. https://doi.org/10.1080/17439760.2013.830762 Russo-Netzer, P., Sinai, M., & Zeevi, M. (2020). Meaning in life and work among counsellors: A qualitative exploration. British Journal of Guidance and Counselling, 48(2), 209-226. https://doi.org/10.1080/03069885.2019.1625026 Steen, A., Berghuis, H., & Braam, A. W. (2019). Lack of meaning, purpose and direction in life in personality disorder: A comparative quantitative approach using Livesleys General Assessment of Personality Disorder. Personality and Mental Health, 13(1), 144-154. https://doi.org/10.1002/pmh.1446 Steger, M., & Frazier, P. (2006). The Meaning in Life Questionaire: Assessing the Presence of and Search of Meaning in Life. Journal of Counselling Psychology, 53(1), 80-93. https://doi.org/10.1037/0022-0167.53.1.80 Steger, M. F., Kashdan, T. B., Sullivan, B. A., & Lorentz, D. (2008). Understanding the search for meaning in life: Personality, cognitive style and the dynamic between seeking and experiencing meaning. Journal of Personality, 76(2), 200-229. https://doi.org/10.1111/j.1467.2007.00484 Steger, M. F. (2012). Making Meaning in Life. An International Journal for the Advancement of Psychological Theory, 23(4), 381-385. https://doi.org/10.1080/1047840X.2012.720832 }} ==External links== *[[doi:10.1080/00332747.1973.11023774|The development of MIL]], (Battista & Almond, 1973). [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Meaning]] [[Category:Motivation and emotion/Book/Self]] tmtbijd5j9m7jro08bth15cs6q3ixeu 0 282186 2814983 2814815 2026-06-10T07:18:19Z Jtneill 10242 2814983 wikitext text/x-wiki {{Title|Chapters by year}} <div p align = "center"> {| class="wikitable" ! style="vertical-align: top;" | Year ! style="vertical-align: top;" | Motivation<br><i>n</i> ! style="vertical-align: top;" | Emotion<br><i>n</i> ! style="vertical-align: top;" | M & E<br><i>n</i> ! style="vertical-align: top;" | Total |- |[[Motivation and emotion/Book/2026|2026]] |TBA |TBA |TBA |TBA |- |[[Motivation and emotion/Book/2025|2025]] | style="text-align: right;" | 59 | style="text-align: right;" | 66 | style="text-align: right;" | 5 | style="text-align: right;" | 130 |- |[[Motivation and emotion/Book/2024|2024]] | style="text-align: right;" | 74 | style="text-align: right;" | 65 | style="text-align: right;" | 11 | style="text-align: right;" | 150 |- |[[Motivation and emotion/Book/2023|2023]] | style="text-align: right;" | 60 | style="text-align: right;" | 67 | style="text-align: right;" | 6 | style="text-align: right;" | 133 |- |[[Motivation and emotion/Book/2022|2022]] | style="text-align: right;" | 63 | style="text-align: right;" | 70 | style="text-align: right;" | 6 | style="text-align: right;" | 140 |- |[[Motivation and emotion/Book/2021|2021]] | style="text-align: right;" | 77 | style="text-align: right;" | 65 | style="text-align: right;" | 1 | style="text-align: right;" | 143 |- |[[Motivation and emotion/Book/2020|2020]] | style="text-align: right;" | 58 | style="text-align: right;" | 79 | style="text-align: right;" | 6 | style="text-align: right;" | 138 |- |[[Motivation and emotion/Book/2019|2019]] | style="text-align: right;" | 66 | style="text-align: right;" | 81 | style="text-align: right;" | 2 | style="text-align: right;" | 149 |- |[[Motivation and emotion/Book/2018|2018]] | style="text-align: right;" | 42 | style="text-align: right;" | 52 | style="text-align: right;" | 1 | style="text-align: right;" | 95 |- |[[Motivation and emotion/Book/2017|2017]] | style="text-align: right;" | 38 | style="text-align: right;" | 63 | style="text-align: right;" | 0 | style="text-align: right;" | 101 |- |[[Motivation and emotion/Book/2016|2016]] | style="text-align: right;" | 62 | style="text-align: right;" | 54 | style="text-align: right;" | 0 | style="text-align: right;" | 116 |- |[[Motivation and emotion/Book/2015|2015]] | style="text-align: right;" | 69 | style="text-align: right;" | 72 | style="text-align: right;" | 0 | style="text-align: right;" | 141 |- |[[Motivation and emotion/Book/2014|2014]] | style="text-align: right;" | 70 | style="text-align: right;" | 63 | style="text-align: right;" | 0 | style="text-align: right;" | 133 |- |[[Motivation and emotion/Book/2013|2013]] | style="text-align: right;" | 48 | style="text-align: right;" | 75 | style="text-align: right;" | 0 | style="text-align: right;" | 123 |- |[[Motivation and emotion/Book/2011|2011]] | style="text-align: right;" | 44 | style="text-align: right;" | 46 | style="text-align: right;" | 0 | style="text-align: right;" | 90 |- |[[Motivation and emotion/Textbook|2010]] | style="text-align: right;" | 42 | style="text-align: right;" | 23 | style="text-align: right;" | 0 | style="text-align: right;" | 65 |- |Total | style="text-align: right;" | 870 | style="text-align: right;" | 940 | style="text-align: right;" | 38 | style="text-align: right;" | 1849 |} </div> [[Category:Motivation and emotion/Book/Year]] abx4qae5e9nhnsb4c51yrcpfsw4h4ew 2815013 2814983 2026-06-10T07:53:34Z Jtneill 10242 2815013 wikitext text/x-wiki {{Title|Chapters by year}} <div p align = "center"> {| class="wikitable" ! style="vertical-align: top;" | Year ! style="vertical-align: top;" | Motivation<br><i>n</i> ! style="vertical-align: top;" | Emotion<br><i>n</i> ! style="vertical-align: top;" | M & E<br><i>n</i> ! style="vertical-align: top;" | Total |- |[[Motivation and emotion/Book/2026|2026]] |TBA |TBA |TBA |TBA |- |[[Motivation and emotion/Book/2025|2025]] | style="text-align: right;" | 59 | style="text-align: right;" | 66 | style="text-align: right;" | 5 | style="text-align: right;" | 130 |- |[[Motivation and emotion/Book/2024|2024]] | style="text-align: right;" | 71 | style="text-align: right;" | 64 | style="text-align: right;" | 11 | style="text-align: right;" | 145 |- |[[Motivation and emotion/Book/2023|2023]] | style="text-align: right;" | 59 | style="text-align: right;" | 63 | style="text-align: right;" | 6 | style="text-align: right;" | 122 |- |[[Motivation and emotion/Book/2022|2022]] | style="text-align: right;" | 63 | style="text-align: right;" | 70 | style="text-align: right;" | 6 | style="text-align: right;" | 140 |- |[[Motivation and emotion/Book/2021|2021]] | style="text-align: right;" | 77 | style="text-align: right;" | 65 | style="text-align: right;" | 1 | style="text-align: right;" | 143 |- |[[Motivation and emotion/Book/2020|2020]] | style="text-align: right;" | 58 | style="text-align: right;" | 79 | style="text-align: right;" | 6 | style="text-align: right;" | 138 |- |[[Motivation and emotion/Book/2019|2019]] | style="text-align: right;" | 66 | style="text-align: right;" | 81 | style="text-align: right;" | 2 | style="text-align: right;" | 149 |- |[[Motivation and emotion/Book/2018|2018]] | style="text-align: right;" | 42 | style="text-align: right;" | 52 | style="text-align: right;" | 1 | style="text-align: right;" | 95 |- |[[Motivation and emotion/Book/2017|2017]] | style="text-align: right;" | 38 | style="text-align: right;" | 63 | style="text-align: right;" | 0 | style="text-align: right;" | 101 |- |[[Motivation and emotion/Book/2016|2016]] | style="text-align: right;" | 62 | style="text-align: right;" | 54 | style="text-align: right;" | 0 | style="text-align: right;" | 116 |- |[[Motivation and emotion/Book/2015|2015]] | style="text-align: right;" | 69 | style="text-align: right;" | 72 | style="text-align: right;" | 0 | style="text-align: right;" | 141 |- |[[Motivation and emotion/Book/2014|2014]] | style="text-align: right;" | 70 | style="text-align: right;" | 63 | style="text-align: right;" | 0 | style="text-align: right;" | 133 |- |[[Motivation and emotion/Book/2013|2013]] | style="text-align: right;" | 48 | style="text-align: right;" | 75 | style="text-align: right;" | 0 | style="text-align: right;" | 123 |- |[[Motivation and emotion/Book/2011|2011]] | style="text-align: right;" | 44 | style="text-align: right;" | 46 | style="text-align: right;" | 0 | style="text-align: right;" | 90 |- |[[Motivation and emotion/Textbook|2010]] | style="text-align: right;" | 42 | style="text-align: right;" | 23 | style="text-align: right;" | 0 | style="text-align: right;" | 65 |- |Total | style="text-align: right;" | 866 | style="text-align: right;" | 934 | style="text-align: right;" | 38 | style="text-align: right;" | 1840 |} </div> [[Category:Motivation and emotion/Book/Year]] h4fn5loj4y4x9a0m7pnm1204x7915q2 0 285380 2814866 2814574 2026-06-09T14:12:01Z Young1lim 21186 /* Applications */ 2814866 wikitext text/x-wiki === Introduction === * Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]]) * Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]]) * Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]]) === Handling Repetition === * Control ([[Media:C02.Repeat1.Control.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat1.Control.1.B.20170918.pdf |B.pdf]], [[Media:C02.Repeat1.Control.1.C.20170926.pdf |C.pdf]]) * Loop ([[Media:C02.Repeat2.Loop.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat2.Loop.1.B.20170918.pdf |B.pdf]]) === Handling a Big Work === * Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]]) * Functions & Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]]) * Functions & Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]]) * Functions & Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]]) === Handling Series of Data === ==== Background ==== * Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]]) ==== Basics ==== * Pointers ([[Media:C04.S1.Pointer.1A.20240524.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]]) * Arrays ([[Media:C04.S2.Array.1A.20240514.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]]) * Array Pointers ([[Media:C04.S3.ArrayPointer.1A.20240208.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]]) * Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20221130.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.1111.pdf |B.pdf]]) * Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]]) * Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]]) ==== Examples ==== * Spreadsheet Example Programs :: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]]) :: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]]) :: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]]) :: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]]) ==== Applications ==== * Address-of and de-reference operators ([[Media:C04.SA0.PtrOperator.1A.20260609.pdf |A.pdf]]) * Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20241121.pdf |A.pdf]]) * Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240715.pdf |A.pdf]]) * Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.pdf |A.pdf]]) * Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]]) * Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]]) * Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]]) === Handling Various Kinds of Data === * Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]]) * Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]]) * Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]]) * Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]]) === Handling Low Level Operations === * Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]]) * Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]]) * Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]]) * Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]]) === Declarations === * Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]]) * Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]]) * Scope === Class Notes === * TOC ([[Media:TOC.20171007.pdf |TOC.pdf]]) * Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library * Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements * Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers * Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts * Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops * Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control * Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions * Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope * Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion * Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions * Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications * Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions * Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications * Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) * Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2) * Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO * Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions * Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications * Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum * Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List * Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing * Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing <!----------------------------------------------------------------------> </br> See also https://cprogramex.wordpress.com/ == '''Old Materials '''== until 201201 * Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]]) * Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]]) * Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]]) * Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]]) * Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]]) * Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]]) * Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]]) * Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]]) * Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]]) * Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]]) * Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]]) * Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]]) * Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]]) <br> until 201107 * Intro.1.A ([[Media:Intro.1.A.pdf |pdf]]) * Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]]) * Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]]) * Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]]) * Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]]) * Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]]) * Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]]) * Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]]) * Array.1.A ([[Media:Array.1.A.pdf |pdf]]) * Type.1.A ([[Media:Type.1.A.pdf |pdf]]) * Structure.1.A ([[Media:Structure.1.A.pdf |pdf]]) go to [ [[C programming in plain view]] ] [[Category:C programming language]] </br> 74jpxaletw3g7xg3kv9kfidhg5yp9x9 0 286856 2814930 2566874 2026-06-10T00:02:07Z Jtneill 10242 /* Overview */ Replace external with interwiki link 2814930 wikitext text/x-wiki {{title|Environmental volunteering motivation:<br>What motivates environmental volunteering?}} {{MECR3|1=https://youtu.be/-2-LWXMM29Q}} __TOC__ == Overview == [[File:People volunteering to help the environment.png|thumb|304x304px|''Figure 1.'' Environmental volunteers.]] [[w:Environmental volunteering|Environmental volunteering]] is important as if people are not [[wikipedia:Motivation#cite_note-Ryan_2000-3|motivated]] to participate in environmental volunteering, the environment will continue to deteriorate at a rapid rate (Adamo et al., 2021). This chapter will discuss environmental volunteering and the what, why and how people can be motivated to volunteer. This book chapter will be focusing on two key [[Motivation and emotion/Book/2021/Goal-framing theory and pro-environmental behaviour|pro-environmental]] motivators and they are personal norms and subjective norms. Personal norms using underlying theory of the norm activation model (NAM) and subjective norms using [[extrinsic motivation]] and value-belief-norms theory (VBN). These theories and concepts can be applied to individuals, groups and communities by using targeted motivational techniques. By understanding motivational theory, individuals, groups or organisations can better approach people to encourage pro-environmental behaviour. Furthermore, environmental volunteering has co-benefits of social connectivity and health promotion in which can be used as techniques to appeal to people and also help motivate them (the target audience) to participate in environmental volunteering. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is environmental volunteering? * Why is environmental volunteering important? * What motivates environmental volunteering? * What benefits are there to environmental volunteering? {{RoundBoxBottom}} ==Environmental volunteering== {{expand|example=Volunteering is the act of working with the expectation that you will not be paid (source). Often times volunteering is working to support goodwill causes such as raising awareness/ money or helping people in need.}} === What is environmental volunteering? === Volunteering is the act of choosing to participate in unpaid work for a movement, group or organisation and environmental volunteering refers to the action of volunteering for an environmentally focused cause to help maintain and safeguard the environment {{citation needed}}{{fact}}. There are two key types of volunteering: formal and informal, both of which are unpaid work. Formal volunteering can be described as contributing help to an organisation whereas informal volunteering is often described as just “helping out” someone typically of closer acquaintance (e.g., picking up rubbish from your neighbour's yard) (Wilson & Musick, 1997; Rochester, 2006). === Why is environmental volunteering important? === A healthy, sustainable environment is fundamental for humans and all living species to survive. According to Adamo et al. (2021) the majority of environmental hazards, damages and problems which have occurred humans have been responsible for. This includes the primary contribution to pollution, greenhouse gases, and deforestation which are major causes of global warming, climate change and the expanding hole in the earth’s ozone layer and this has a significant impact on wildlife, humans and the ecosystem{{explain}}. Environmental volunteering overall promotes social connectivity and health and it is further important for nature conservation and increasing environmental knowledge (Sloane & Pröbstl-Haider, 2019). === Pro-environmental behaviours and examples === Pro-environmental behaviours are actions an individual can make to contribute towards safeguarding the environment or reducing environmental harm and they may be private or public actions (Balunde et al., 2019). Examples of pro-environmental behaviours include recycling, composting, reducing waste and electricity consumption or participation/volunteering in environmental movements. Environmental volunteering also involves participating in actions which benefit the environment and therefore, pro-environmental behaviours are reflected within the core of environmental volunteering.{{fact}} ==How to motivate environmental volunteering== It is important to understand the underlying theories and concepts of motivation such as personal norms, norm activation model, subjective norms and extrinsic motivation and value-belief-norm theory. Identifying the motivations of environmental volunteering is helpful as organisations, groups or individuals can consider how to best interact and develop appropriate programs to be able to better increase the number of volunteers recruited and retaining volunteers over time (Bruyere & Rappe, 2007). === Personal norms === [[File:Norm activation model.png|thumb|451x451px|''Figure 2.'' Norm activation model.]] Personal norms are feelings which motivate a person's intentions and behaviours and can be used to encourage environmentally friendly behaviour and volunteering (Schwartz & Howard, 1981). The norm activation model (NAM) shown in ''Figure 2'' explains that pro-environmental and altruistic behaviour is an associated reaction from three key components including ascription of responsibility (AR), adverse consequence (AC) and personal norms (PN) (Schwartz, 1977). {{RoundBoxTop|theme=3}} '''For example:''' Gary cares about living in a healthy environment and future. Gary see’s{{gr}} an advertisement from a pro-environmental organisation looking for environmental volunteers. The advertisement stated, ‘Your ecosystem and local wildlife is suffering from litter! Can you help us preserve the environment?’ and used personal norms to encourage environmental volunteering. See the table below presenting how this fits in with the NAM. {{RoundBoxBottom}} {| class="wikitable" |+ !AC !AR !PN !Prosocial intentions and behaviour |- |Not picking up litter will damage the ecosystem and harm wildlife. |Some people have been littering. |Gary wants to Preserve the ecosystem and protect wildlife. |Gary's intention is to help preserve the ecosystem and protect wildlife by picking up litter. |} A study conducted by Jansson & Dorrepaal (2015) investigated whether personal norms, problem awareness and [[wikipedia:Social_norm|social norms]] were connected and how the factors attributed towards personal climate change norms. Jansson & Dorrepaal (2015) found that personal climate change norms are significantly influenced by problem awareness and social norms which suggests they are important factors for the engagement of prosocial/environmental behaviour to occur. Moreover, results demonstrated that moral attributes of care, fairness and authority are key elements with the norms for climate change. Limitations identified in the study include the potential of participants' responses having self-selection bias and due to the study design being cross-sectional and correlational, results/conclusions can only demonstrate correlation. Overall, the findings of this study demonstrate the importance of two key attributes (problem awareness and social norms) which can be used as strategies to increase ecological behaviour particularly towards climate change and can be related to further environmental issues. For example, an organisation wants volunteers to plant trees to help reduce global warming. To promote personal norms they use the techniques of increasing awareness of the impacts of deforestation through targeted campaigns and additionally, they create a social norm by mentioning the large number of previous and current volunteers. [[File:Muhammad Intrinsic vs Extrinsic Motivation.png|thumb|''Figure 3.'' Intrinsic versus extrinsic motivation.]] === Subjective norms === People's behaviour towards the environment can also be influenced by subjective norms or people who are the closest to them. Some of whom according to are peers, family, teachers parents and local communities{{gr}} (Law et al., 2013). Furthermore, subjective norms are extrinsic factors that constitute the rules or standards which tell members of a group or society how to behave based on values-belief-norms theory (VBN) (Stern, 2000). VBN theory of environmentalism explains that values, beliefs and norms impact environmentally friendly behaviour (Stern, 2000). Extrinsic norms in particular can be applied to influence someone’s participation in volunteering and thus, pro-environmental behaviour is influenced by the social roles and requirements of social environments and culture. According to past research, adolescents are more likely to volunteer if their best friend and parents volunteer (Van Goethem et al., 2014). The influence of parents on volunteering is greater than that of friends (Smentana, 2006). In addition to reinforcing social norms and informing individuals regarding their perceptions, effectiveness and behaviours, social support and role models can also promote pro-environmental activities such as environmental volunteering. == Quiz == Choose the correct answers and click "Submit": <quiz display="simple"> {Environmental volunteering does not help to safeguard or maintain the environment: |type="()"} - True + False {Extrinsic motivation is doing something because you want to with no external pressure: |type="()"} - True + False {Both personal norms and subjective norms can motivate environmental volunteering: |type="()"} + True - False </quiz> == Benefits of environmental volunteering == [[File:Social connectivity and teamwork.png|thumb|224x224px|''Figure 4.'' Social connectivity promotes teamwork.]] Environmental volunteering not only benefits the environment however there are many co-benefits such as social connectivity and health promotion in which benefit people on an individual level, group or community level as well as future generations. ===Social connectivity=== Environmental volunteering promotes social connectivity (Seymour et al., 2020). This can be observed as volunteers typically work in a small or large groups where they can connect with new people and create new friends as they work together to help the environment. ===Health promotion=== Spending time in nature can co-benefit the environment as well as people's physical and mental health (patrick et al., 2022). For example, an individual participates in environmental volunteering where they walk through nature for a few hours on a sunny day picking up litter. In this example, the individual exercise has been undertaken, vitamin D from the sun has been received and the individual feels a rewarding feeling making them happy. ==Conclusion== Environmental volunteering involves people choosing to do unpaid work which benefits the environment and is important for creating a sustainable, safe environment for current and future generations. Key underlying theories of what motivates environmental volunteering includes the norm activation model, moral framework theory and value-belief-norm theory. Environmental volunteering can be motivated by both personal and social norms which can lead towards positive effects on the environment, social connectivity and mental and physical health.{{tip|Safeguarding and maintaining the environment is crucial for human survival and has personal and external benefits. Take home reflection question Have you previously or are you currently engaging in pro-environmental behaviours and/or environmental volunteering? If not, consider how you motivate yourself or someone else to engage in these behaviours.}} ==See also== * [[Motivation and emotion/Book/2013/Environmental behaviour|Environmental behaviours: How can people be motivated to adopt more environmentally friendly behaviours?]] (Book chapter, 2013) * [[Motivation and emotion/Book/2013/Volunteer motivation|Volunteer motivation: What motivates volunteering?]](Book chapter, 2013) * [[Motivation and emotion/Book/2016/Volunteering motivation: Altruism or egoism?|Volunteering motivation: Altruism or egoism?]] (Book chapter, 2016) ==References== {{Hanging indent|1= Adamo, A. N., & Sissakian, V. (2021). Review of Climate Change Impacts on Human Environment: Past, Present and Future Projections. "Engineering, 13"(11), pp. 605–. https://doi.org/10.4236/eng.2021.1311044 Balunde, A., Perlaviciute, G., & Steg, L. (2019). The relationship between people’s environmental considerations and pro-environmental behaviour in Lithuania. "Frontiers in Psychology". https://doi.org/10.3389/fpsyg.2019.02319 Bruyere, B., & Rappe, S. (2007). Identifying the motivations of environmental volunteers. ''Journal of Environmental Planning and Management'', ''50''(4), pp. 503-516. https://doi.org/10.1080/09640560701402034 Güntert, S. T., Strubel, I. T., Kals, E., & Wehner, T. (2016). The quality of volunteers’ motives: Integrating the functional approach and self-determination theory. ''The Journal of Social Psychology, 156''(3), pp. 310–327. https://doi.org/10.1080/00224545.2015.1135864 Jansson, J. & Dorrepaal, E. (2015). Personal Norms for Dealing with Climate Change: Results from a Survey using Moral Foundations Theory. "Sustainable Development, 23"(6), pp. 381-395. https://doi.org/10.1002/sd.1598 Law, B. M. F., Shek, D. T. L., & Ma, C. M. S. (2013). Validation of Family, School, and Peer Influence on Volunteerism Scale among Adolescents. "Research on Social Work Practice, 23", pp. 458-466. https://doi.org/10.1177/1049731513476144 Oudeyers, P., & Kaplan, F. (2009). What is intrinsic motivation? A typology of computational approaches. ''Frontiers in Neurorobotics.''{{ic|Volume?}} https://doi.org/10.3389/neuro.12.006.2007 Patrick, Henderson‐Wilson, C., Ebden, M., & Smith, J. (2022). Exploring the co-benefits of environmental volunteering for human and planetary health promotion. ''Health Promotion Journal of Australia'', ''33''(1), pp. 57–67. https://doi.org/10.1002/hpja.460 Rochester, C. (2006). Making sense of volunteering. "The Commission of the Future of Volunteering", 1-39. Schwartz, S. H. (1977). Normative Influence on Altruism. In L. Berkowitz (Ed.), Advances in Experimental Social Psychology (pp. 221-279). "Academic Press". http://dx.doi.org/10.1016/s0065-2601(08)60358-5 Schwartz, S. H. & Howard, J. A. (1981). A Normative Decision-Making Model of Altruism. In: P. J. Rushton & R. M. Sorrentino (Eds.), "Altruism and Helping Behavior: Social, Personality, and Developmental Perspectives" (pp. 189-211). Lawrence Erlbaum. Seymour, V., King, M., & Antonaci, R. (2020). Exploring those characteristics which may help to foster and support people’s social-ecological resilience: an environmental volunteering case study. ''Journal of Environmental Studies and Sciences, 10''(4), pp. 438–456. https://doi.org/10.1007/s13412-020-00590-0 Sloane, G., & Pröbstl-Haider, U. (2019). Motivation for environmental volunteering - A comparison between Austria and Great Britain. ''Journal of Outdoor Recreation and Tourism'', ''25'', pp. 158–168. https://doi.org/10.1016/j.jort.2019.01.002 Smetana, J. G., Campione-Barr, N., & Metzger, A. (2006). Adolescent Development in Interpersonal and Societal Contexts. "Annual Review of Psychology, 57", pp. 255-284. https://doi.org/10.1146/annurev.psych.57.102904.190124 Stern, P. C. (2000). Toward a Coherent Theory of Environmentally Significant Behavior. "Journal of Social Issues, 56", pp. 407-424. https://doi.org/10.1111/0022-4537.00175 Udo, G., Bagchi, K. & Maity, M. (2016). Exploring Factors Affecting Digital Piracy Using the Norm Activation and UTAUT Models: The Role of National Culture. "J Bus Ethics, 135", pp. 517–541. https://doi.org/10.1007/s10551-014-2484-1 Van Goethem, A. A. J., Van Hoof, A., Van Aken, M. A. G., De Castro, B. O., & Raaijmakers, Q. A. W. (2014). Socialising Adolescent Volunteering: How Important Are Parents and Friends? Age Dependent Effects of Parents and Friends on Adolescents’ Volunteering Behaviours. "Journal of Applied Developmental Psychology, 35", pp. 94-101. https://doi.org/10.1016/j.appdev.2013.12.003 Wilson, J., & Musick, M. (1997). Who cares? Toward an integrated theory of volunteer work. "American Sociological Review, 62", 694-713. }} ==External links== * [https://www.environment.vic.gov.au/environmental-volunteering/volunteering Environmental volunteering] (Victoria State Government) * [https://studentenergy.org/influencer/climate-change/?gclid=Cj0KCQjw166aBhDEARIsAMEyZh7YMeifMhctzXxr9eZLaGsShvlAxM9JnGGlZdBrbOHBDUK3HEdVY_AaAuubEALw_wcB Climate Change] (Student Energy) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Environment]] [[Category:Motivation and emotion/Book/Volunteering]] 92ieaw9snakuna1p6u4yb8go73mxp4o 0 289771 2814981 2718467 2026-06-10T07:15:57Z Jtneill 10242 /* Emotion */ Move incomplete topic to 2026 - # [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? 2814981 wikitext text/x-wiki {{/Banner}} == Motivation == # [[/Abusive supervision/]] - What is abusive supervision, what motivates it, and what can be done about it?{{Motivation and emotion/Book/Low}} {{ME-By|Lara.Begg}} # [[/Active transport motivation/]] - What motivates use of active transport and how can people be encouraged to use it?{{Motivation and emotion/Book/Low}} {{ME-By|U3214564}} # [[/Actively open-minded thinking/]] - How can AOT be used to improve human performance? {{ME-By|Msherrybarr}} # [[/Ageing and motivation/]] - What is the relationship between ageing and motivation? {{ME-By|U3217153}} # [[/Anomie and motivation/]] - How does anomie affect motivation?{{Motivation and emotion/Book/High}} {{ME-By|U3223909}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour?{{Motivation and emotion/Book/Low}} {{ME-By|U3216706}} # [[/Bedtime procrastination/]] - What causes it, what are the consequences, and what can be done about it?{{Motivation and emotion/Book/High}} {{ME-By|U3227684}} # [[/Brief motivational interviewing as a health intervention/]] - How can brief motivational interviewing be used as a health intervention?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|JasnaM}} # [[/Childhood trauma and criminal behaviour/]] - How does childhood trauma influence subsequent criminal behaviour?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3218323}} # [[/Comprehensive action determination model|Comprehensive action determination model]] - What is the CADM and how can it be applied to understanding human motivation?{{Motivation and emotion/Book/Low}} {{ME-By|Rocxie}} # [[/Conspiracy theory motivation/]] - What motivates people to believe in conspiracy theories?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3223114}} # [[/Construal level theory/]] - What is construal level theory and how can it be applied? {{ME-By|U3217803|U3217803}} # [[/Dark tetrad and motivation/]] - What is the dark tetrad and how does it relate to motivation? {{ME-By|U3223849}} # [[/Death drive/]] - What is the death drive and how can it be negotiated? {{ME-By|U3197031}} # [[/Dopamine and extraversion-introversion/]] - How does dopamine affect extraversion-introversion? {{ME-By|SaucyTuRkLeBiRd}} # [[/Egosystem and ecosystem motivation/]] - What are egosystem and ecosystem motivations and what are their consequences? {{ME-By|U3216125}} # [[/Effective leadership/]] - What does it take to be an effective leader? {{ME-By|U103189}} # [[/Episodic future thinking and delay discounting/]] - What is the relationship between between EFT and DD?{{Motivation and emotion/Book/High}} {{ME-By|LightDragonflies}} # [[/Episodic future thinking and delay discounting 2/]] - What is the relationship between episodic future thinking and delay discounting? {{ME-By|U3224582}} # [[/Episodic memory and planning/]] - What role does episodic memory play in planning? {{ME-By|Maheenusman}} # [[/Environmental volunteering motivation/]] - What motivates environmental volunteering?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U891419}} # [[/Freedom and motivation/]] - What is the effect of freedom on motivation?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3217955}} # [[/Gamer motivation/]] - What motivates game playing?{{Motivation and emotion/Multimedia/High}} {{ME-By|Bellehines}} # [[/Grit and achievement/]] - How does grit affect achievement?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} - [[User:AnaStuart|AnaStuart]] # [[/Growth mindset and mental health/]] - What is the relationship between growth mindset and mental health? {{ME-By|Mia Pearse }} # [[/Health belief model/]] - What is the HBM and how can it be used to enhance motivation for health-promoting behaviour?{{Motivation and emotion/Book/High}} {{ME-By|U3226487}} # [[/Identity fusion and motivation/]] - What are the motivational implications of identity fusion?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3235369}} # [[/IKEA effect/]] - What is the IKEA effect and how can it be applied?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3209567}} # [[/Impact bias/]] - What is impact bias, what causes it, what are its consequences, and how can it be avoided?{{Motivation and emotion/Book/Low}} {{ME-By|De2023}} # [[/Indigenous Australian engagement in mental health services/]] - What helps Indigenous Australians to engage with mental health services? {{ME-By|CaitlinFisher01}} # [[/Intertemporal choice/]] - What are intertemporal choices and how can they be effectively negotiated?{{Motivation and emotion/Book/Low}} {{ME-By|TD3223808}} # [[/Irrational belief motivation/]] - What motivates irrational beliefs?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|Concettazicc}} # [[/Köhler effect and motivation/]] - What is the Köhler effect and how can it be applied? {{ME-By|Declan Robinson}} # [[/Learning motivation and chatbots/]] - How can student motivation to learn be enhanced by using artificial intelligence chatbots? {{ME-By|Ashhstarr}} # [[/Life history theory and human reproductive strategies/]] - How does life history theory explain human reproductive strategies? {{ME-By|U3212234}} # [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|Casey Coulton}} # [[/Mobile phone use motivation/]] - What are the motivations for mobile phone use?{{Motivation and emotion/Book/Low}} {{ME-By|Yaruki91}} # [[/Morning routine and motivation/]] - How can a morning routine be used to facilitate motivation, productivity, and well-being?{{Motivation and emotion/Book/Low}} {{ME-By|U3216585}} # [[/Motivational technologies/]] - How can motivation be fostered through technology? {{ME-By|U3196624}} # [[/Novelty-variety as a psychological need/]] - What is novelty-variety and what are its implications as a psychological need? {{ME-By|U3161239}} # [[/Nudge theory and sedentary behaviour/]] - How can nudge theory can be used reduce sedentary behaviour?{{Motivation and emotion/Book/High}} {{ME-By|U3145851}} # [[/Overchoice and motivation/]] - What is the overchoice effect, why does it occur, and who are most susceptible to it?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3230003}} # [[/Physical activity tracking and exercise motivation/]] - What role can physical activity tracking play in exercise motivation?{{Motivation and emotion/Book/Low}} {{ME-By|U3230383}} # [[/Physiological needs/]] - How do human's physiological needs affect motivation? {{ME-By|Superghostfresh}} # [[/Placebo effect and motivation/]] - What is the placebo effect and how can it affect motivation? {{ME-By|U3203797}} # [[/Protection motivation theory and COVID-19/]] - How does PMT apply to managing COVID-19?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3227759}} # [[/Sadism motivation/]] - What motivates sadism and what can be done about it? {{ME-By|BM3062384}} # [[/Signature strengths/]] - What are signature strengths and how can they be applied? {{ME-By|Rumesa.Aria}} # [[/Sport injury recovery motivation/]] - How can motivation be sustained when recovering from a sport injury? {{ME-By|U3210285}} # [[/SSRIs and motivation/]] - What are the effects of SSRIs on motivation?{{Motivation and emotion/Book/High}} {{ME-By|Josh.Cattle}} # [[/Stretch goals/]] - What are stretch goals? How do they work?{{Motivation and emotion/Book/Low}} {{ME-By|U3231448UC}} # [[/System justification theory/]] - What is SJT, how does it affect our lives, and what can be done about it? {{ME-By|Ishi}} # [[/Testosterone and dominance/]] - What is the relationship between testosterone and dominance? [[User:Giovannihbartlett|Giovannihbartlett]] # [[/Testosterone and violent crime/]] - What is the relationship between testosterone and violent crime?{{Motivation and emotion/Book/Low}} {{ME-By|U3200902}} # [[/Testosterone, winning, and losing/]] - What is the relationship between winning, losing, and testosterone? {{ME-By|Hugo DL}} # [[/Time management/]] - How can one's time be managed effectively? {{ME-By|Madison.Mayo}} # [[/To-do lists/]] - Are to-do lists a good idea? What are their pros and cons? How can they be used effectively?{{Motivation and emotion/Multimedia/High}} {{ME-By|U3162169T}} # [[/Treatment motivation in substance use disorder/]] - What is the role of treatment motivation in substance abuse disorder and how can it be enhanced?{{Motivation and emotion/Multimedia/High}} {{ME-By|KATE HERLIHY}} # [[/Violent extremism motivation/]] - What motivates people to engage in violent extremism? {{ME-By|U3216313}} # [[/Vocational identity/]] - What is vocational identity and how does it develop? {{ME-By|U3171379}} == Emotion == # [[/Artificial emotion/]] - To what extent can artificial intelligence experience emotion and how can this be applied? {{ME-By|KylieGiddings}} # [[/Attributions and emotion/]] - How do attributions affect emotion? {{ME-By|Carol Chariarse}} # [[/Autonomous sensory meridian response and emotion/]] - What emotions are involved in ASMR experiences and why do they occur? {{ME-By|Zmelmoth02}} # [[/Bewilderment/]] - What is bewilderment and how can it be dealt with? {{ME-By|Samanthagrebert}} # [[/Body functionality and body satisfaction/]] - What is the relationship between body functionality and body satisfaction? {{ME-By|Fatima2617}} # [[/Burn injury and body image/]] - What is the impact of burn injury on body image and what can be done? {{ME-By|Biancanappo}} # [[/Burnout/]] - What is burnout and how can be it be managed and prevented? {{ME-By|Grace (Tram) Chu}} # [[/Choice and emotion/]] - How does emotion affect choice? {{ME-By|Jingying Chen}} # [[/Climate change emotion/]] - What is the relationship between climate change and emotion?- [[User:Sireyes|Sireyes]] # [[/Commitment bias/]] - What motivates escalation of commitment even it does not lead to desirable outcomes? {{ME-By|U3203936}} # [[/Community resilience/]] - What is community resilience and how can it be fostered?{{Motivation and emotion/Book/High}} {{ME-By|U3213682}} # [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|Rhys Harmer}} # [[/Core emotions/]] - What are the core emotions and what is their function?{{Motivation and emotion/Book/High}} {{ME-By|U3229619}} # [[/Dark tetrad and emotion/]] - What is the dark tetrad and how does it relate to emotion? {{ME-By|U3214333}} # [[/Death and meaning in life/]] - How does confronting mortality influence the sense of meaning in life? {{ME-By|U3222680}} # [[/Default mode network and the self/]] - What is the relationship between the DMN and the self? {{ME-By|U3190506}} # [[/Deference emotion system/]] - What is the deference emotion system, how does it work, and what are its implications? {{ME-By|U3216127}} # [[/Eco-anxiety/]] - What is eco-anxiety, what are its consequences, and what can be done about it? {{ME-By|MT200107}} # [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|Emkatev}} # [[/Ecopsychology and stress/]] - How can ecopsychology help to explain and deal with stress?{{Motivation and emotion/Multimedia/High}} {{ME-By|Dnamynts}} # [[/Ego death/]] - What is ego death, what are its consequences, and how can it be facilitated?{{Motivation and emotion/Book/Low}} {{ME-By|Gaissa}} # [[/Emotion across the lifespan/]] - How does emotion develop across the lifespan? {{ME-By|Taliastefanoski}} # [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|Adf001}} # [[/Emotional intelligence training/]] - How can emotional intelligence be trained? {{ME-By|RolandC2023}} # [[/Empathy and psychopathy/]] - What is the relationship between empathy and psychopathy? {{ME-By|U3218292}} # [[/Empathy training/]] - How can empathy be enhanced? {{ME-By|U3224924}} # [[/Equine therapy and emotion/]] - What is the role of emotion in equine therapy? {{ME-By|SoSilverLibby}} # [[/Fear of apocalypse/]] - What is apocalyptic fear and how can it be dealt with? {{ME-By|AliciaEdwards1212}} # [[/Flourishing in the elderly/]] - How can psychological flourishing be supported in the elderly?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|JorjaFive}} # [[/Guided meditation and emotion regulation/]] - How can guided meditation be used to regulate emotion?{{Motivation and emotion/Book/Low}} {{ME-By|U3215755}} # [[/Hedonic adaptation prevention model/]] - What is the HAP model and how can it be applied? {{ME-By|Siobhanu3216316}} # [[/Hijack hypothesis of drug addiction/]] - What is the hijack hypothesis and how does it help to understand drug addiction? {{ME-By|U3189442 - K.Ryan}} # [[/Humility/]] - What is humility, what causes it, and is it desirable?{{Motivation and emotion/Book/Low}} {{ME-By|Inandonit365}} # [[/Interoception and mental health/]] - What is the role of interoception in mental health? {{ME-By|U3229576}} # [[/Indigenous Australian mindfulness/]] - How has Indigenous Australian culture traditionally conceived of, and practiced, mindfulness?{{Motivation and emotion/Book/High}} {{ME-By|U3168525}} # [[/Long-term injury in high performance field athletes and emotion/]] - How do long-term injuries impact the emotions of high performance field athletes? {{ME-By|U3216619}} # [[/Meditation and mindfulness/]] - How can meditation contribute to mindfulness? {{ME-By|APMonte}} # [[/Mental health literacy/]] - What is mental health literacy, why does it matter, and how can it be fostered? {{ME-By|U3215387}} # [[/Mental imagery and emotion/]] - How can mental imagery be used to regulate emotion? {{ME-By|MaryamNageeb}} # [[/Mental toughness in the workplace/]] - How can mental toughness be useful in the workplace and how can it be developed? {{ME-By|KubeChu}} # [[/Moral outrage/]] - What is moral outrage, what causes it, what are its consequences, and how can it be managed? {{ME-By|U3215934}} # [[/Narrative therapy and emotion/]] - What is the role of emotion in narrative therapy?{{Motivation and emotion/Book/Low}} {{ME-By|Alyssa.marin}} # [[/Placebo effect and emotion/]] - What is the placebo effect and how can it affect emotion? {{ME-By|U3213568}} # [[/Positive body image development/]] - What is a positive body image and how can it be developed? {{ME-By|U3224053}} # [[/Psilocybin assisted therapy and anxiety/]] - How can psilocybin assisted therapy help to treat anxiety?{{Motivation and emotion/Multimedia/High}} {{ME-By|SammyTabrett}} # [[/Psychedelic treatment of anxiety/]] - How can psychedelics be used to help in treating anxiety? {{ME-By|SashaBrooksby}} # [[/Psychological literacy/]] - What is psychological literacy, why does it matter, and how can it be fostered?{{Motivation and emotion/Book/Low}} {{ME-By|DMaphosa}} # [[/Relief/]] - What is relief and what role does it play in our emotional lives? {{ME-By|U3216413(2023)}} # [[/Sadism and emotion/]] - What are the emotional aspects of sadism? {{ME-By|Kimberley Manyura}} # [[/Self-affirmation and psychological change/]] - How can self-affirmation be used for positive psychological change? {{ME-By|U3213910}} # [[/Self-discrepancy theory and emotion/]] - In what ways does self-discrepancy help to understand emotion? {{ME-By|U3173387}} # [[/Singing and emotion/]] - What is the relation between singing and emotion?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3213084}} # [[/Smiling and emotion/]] - What is the relationship between smiling and emotion? {{ME-By|U3230268}} # [[/Stress mindset/]] - What is stress mindset, why does it matter, and how can it be cultivated?{{Motivation and emotion/Book/Low}} {{ME-By|U3224687}} # [[/Sublimation/]] - What is sublimation and how can it be fostered? {{ME-By|U3185622}} # [[/Therapeutic recreation/]] - What is therapeutic recreation, how does it work, and what are its impacts? {{ME-By|MojoHides}} # [[/Toxic positivity/]] - What is toxic positivity, why does it occur, and how can it be managed? {{ME-By|Lauraei11}} # [[/Trauma and emotion/]] - What is the effect of trauma on emotion?{{Motivation and emotion/Book/High}} {{ME-By|Clarestuparich}} # [[/Tragic optimism/]] - What is tragic optimism, why does it matter, and how can it be fostered? {{ME-By|Alice hatcher }} # [[/Trauma-informed care/]] - What is trauma-informed care and how can it be implemented?{{Motivation and emotion/Multimedia/High}} {{ME-By|Nabila.Tursun}} # [[/Unconscious aspects of multiple selves and emotion/]] - What role do unconscious aspects play in shaping our multiple selves and their influence on emotion? {{ME-By|Natsta19}} # [[/Uncertainty and emotion/]] - How does uncertainty influence emotion? [[User:Mikaela MG|Mikaela MG]] # [[/Unemployment and depression/]] - What is the relationship between unemployment and depression? {{ME-By|U3213250}} # [[/Unemployment and mental health/]] - What is the relationship between unemployment and mental health? {{ME-By|U3211780}} # [[/Vicarious embarrassment/]] - What is vicarious embarrassment, why does it occur, and how can it be managed?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3216476}} # [[/Viewing natural scenes and emotion/]] - What is the effect of viewing natural scenes on emotion and how can this be applied? {{ME-By|Alyson131}} # [[/Yoga and emotional regulation/]] - How can yoga help with emotional regulation? {{ME-By|U3169170}} == Motivation and emotion == # [[/GABA, motivation, and emotion/]] - What is the neurological function of GABA for motivation and emotion? {{ME-By|U3227976}} # [[/Medial forebrain bundle, motivation, and emotion/]] - What are the motivational and emotional functions of the medial forebrain bundle? {{ME-By|U3226707}} # [[/Multiple selves theory, motivation, and emotion/]] - How does the concept of multiple selves impact our motivational and emotional experiences?{{Motivation and emotion/Book/Low}} {{ME-By|H.Maycock}} # [[/Overchoice, motivation, and emotion/]] - What are the motivational and emotional effects of overchoice? {{ME-By|Atu320270}} # [[/Role-playing games, motivation, and emotion/]] - What are the motivational and emotional characteristics of role-playing games?{{Motivation and emotion/Book/Low}} {{ME-By|U3224203}} # [[/Torture interrogation, motivation, and emotion/]] - What are the motivational and emotional aspects of torture interrogation and how do these impact on the effectiveness of torture interrogation? {{ME-By|U3204500}} [[Category:Motivation and emotion/Book/2023]] i3rchkc450vavrlyfwiagp3t5abfxws 2814984 2814981 2026-06-10T07:20:48Z Jtneill 10242 Move abandoned topic to 2026 2814984 wikitext text/x-wiki {{/Banner}} == Motivation == # [[/Abusive supervision/]] - What is abusive supervision, what motivates it, and what can be done about it?{{Motivation and emotion/Book/Low}} {{ME-By|Lara.Begg}} # [[/Active transport motivation/]] - What motivates use of active transport and how can people be encouraged to use it?{{Motivation and emotion/Book/Low}} {{ME-By|U3214564}} # [[/Actively open-minded thinking/]] - How can AOT be used to improve human performance? {{ME-By|Msherrybarr}} # [[/Ageing and motivation/]] - What is the relationship between ageing and motivation? {{ME-By|U3217153}} # [[/Anomie and motivation/]] - How does anomie affect motivation?{{Motivation and emotion/Book/High}} {{ME-By|U3223909}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour?{{Motivation and emotion/Book/Low}} {{ME-By|U3216706}} # [[/Bedtime procrastination/]] - What causes it, what are the consequences, and what can be done about it?{{Motivation and emotion/Book/High}} {{ME-By|U3227684}} # [[/Brief motivational interviewing as a health intervention/]] - How can brief motivational interviewing be used as a health intervention?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|JasnaM}} # [[/Childhood trauma and criminal behaviour/]] - How does childhood trauma influence subsequent criminal behaviour?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3218323}} # [[/Comprehensive action determination model|Comprehensive action determination model]] - What is the CADM and how can it be applied to understanding human motivation?{{Motivation and emotion/Book/Low}} {{ME-By|Rocxie}} # [[/Conspiracy theory motivation/]] - What motivates people to believe in conspiracy theories?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3223114}} # [[/Construal level theory/]] - What is construal level theory and how can it be applied? {{ME-By|U3217803|U3217803}} # [[/Dark tetrad and motivation/]] - What is the dark tetrad and how does it relate to motivation? {{ME-By|U3223849}} # [[/Death drive/]] - What is the death drive and how can it be negotiated? {{ME-By|U3197031}} # [[/Dopamine and extraversion-introversion/]] - How does dopamine affect extraversion-introversion? {{ME-By|SaucyTuRkLeBiRd}} # [[/Egosystem and ecosystem motivation/]] - What are egosystem and ecosystem motivations and what are their consequences? {{ME-By|U3216125}} # [[/Effective leadership/]] - What does it take to be an effective leader? {{ME-By|U103189}} # [[/Episodic future thinking and delay discounting/]] - What is the relationship between between EFT and DD?{{Motivation and emotion/Book/High}} {{ME-By|LightDragonflies}} # [[/Episodic future thinking and delay discounting 2/]] - What is the relationship between episodic future thinking and delay discounting? {{ME-By|U3224582}} # [[/Episodic memory and planning/]] - What role does episodic memory play in planning? {{ME-By|Maheenusman}} # [[/Environmental volunteering motivation/]] - What motivates environmental volunteering?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U891419}} # [[/Freedom and motivation/]] - What is the effect of freedom on motivation?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3217955}} # [[/Gamer motivation/]] - What motivates game playing?{{Motivation and emotion/Multimedia/High}} {{ME-By|Bellehines}} # [[/Grit and achievement/]] - How does grit affect achievement?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} - [[User:AnaStuart|AnaStuart]] # [[/Growth mindset and mental health/]] - What is the relationship between growth mindset and mental health? {{ME-By|Mia Pearse }} # [[/Health belief model/]] - What is the HBM and how can it be used to enhance motivation for health-promoting behaviour?{{Motivation and emotion/Book/High}} {{ME-By|U3226487}} # [[/Identity fusion and motivation/]] - What are the motivational implications of identity fusion?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3235369}} # [[/IKEA effect/]] - What is the IKEA effect and how can it be applied?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3209567}} # [[/Impact bias/]] - What is impact bias, what causes it, what are its consequences, and how can it be avoided?{{Motivation and emotion/Book/Low}} {{ME-By|De2023}} # [[/Indigenous Australian engagement in mental health services/]] - What helps Indigenous Australians to engage with mental health services? {{ME-By|CaitlinFisher01}} # [[/Intertemporal choice/]] - What are intertemporal choices and how can they be effectively negotiated?{{Motivation and emotion/Book/Low}} {{ME-By|TD3223808}} # [[/Irrational belief motivation/]] - What motivates irrational beliefs?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|Concettazicc}} # [[/Köhler effect and motivation/]] - What is the Köhler effect and how can it be applied? {{ME-By|Declan Robinson}} # [[/Learning motivation and chatbots/]] - How can student motivation to learn be enhanced by using artificial intelligence chatbots? {{ME-By|Ashhstarr}} # [[/Life history theory and human reproductive strategies/]] - How does life history theory explain human reproductive strategies? {{ME-By|U3212234}} # [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|Casey Coulton}} # [[/Mobile phone use motivation/]] - What are the motivations for mobile phone use?{{Motivation and emotion/Book/Low}} {{ME-By|Yaruki91}} # [[/Morning routine and motivation/]] - How can a morning routine be used to facilitate motivation, productivity, and well-being?{{Motivation and emotion/Book/Low}} {{ME-By|U3216585}} # [[/Motivational technologies/]] - How can motivation be fostered through technology? {{ME-By|U3196624}} # [[/Novelty-variety as a psychological need/]] - What is novelty-variety and what are its implications as a psychological need? {{ME-By|U3161239}} # [[/Nudge theory and sedentary behaviour/]] - How can nudge theory can be used reduce sedentary behaviour?{{Motivation and emotion/Book/High}} {{ME-By|U3145851}} # [[/Overchoice and motivation/]] - What is the overchoice effect, why does it occur, and who are most susceptible to it?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3230003}} # [[/Physical activity tracking and exercise motivation/]] - What role can physical activity tracking play in exercise motivation?{{Motivation and emotion/Book/Low}} {{ME-By|U3230383}} # [[/Physiological needs/]] - How do human's physiological needs affect motivation? {{ME-By|Superghostfresh}} # [[/Placebo effect and motivation/]] - What is the placebo effect and how can it affect motivation? {{ME-By|U3203797}} # [[/Protection motivation theory and COVID-19/]] - How does PMT apply to managing COVID-19?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3227759}} # [[/Sadism motivation/]] - What motivates sadism and what can be done about it? {{ME-By|BM3062384}} # [[/Signature strengths/]] - What are signature strengths and how can they be applied? {{ME-By|Rumesa.Aria}} # [[/Sport injury recovery motivation/]] - How can motivation be sustained when recovering from a sport injury? {{ME-By|U3210285}} # [[/SSRIs and motivation/]] - What are the effects of SSRIs on motivation?{{Motivation and emotion/Book/High}} {{ME-By|Josh.Cattle}} # [[/Stretch goals/]] - What are stretch goals? How do they work?{{Motivation and emotion/Book/Low}} {{ME-By|U3231448UC}} # [[/System justification theory/]] - What is SJT, how does it affect our lives, and what can be done about it? {{ME-By|Ishi}} # [[/Testosterone and dominance/]] - What is the relationship between testosterone and dominance? [[User:Giovannihbartlett|Giovannihbartlett]] # [[/Testosterone and violent crime/]] - What is the relationship between testosterone and violent crime?{{Motivation and emotion/Book/Low}} {{ME-By|U3200902}} # [[/Testosterone, winning, and losing/]] - What is the relationship between winning, losing, and testosterone? {{ME-By|Hugo DL}} # [[/Time management/]] - How can one's time be managed effectively? {{ME-By|Madison.Mayo}} # [[/To-do lists/]] - Are to-do lists a good idea? What are their pros and cons? How can they be used effectively?{{Motivation and emotion/Multimedia/High}} {{ME-By|U3162169T}} # [[/Treatment motivation in substance use disorder/]] - What is the role of treatment motivation in substance abuse disorder and how can it be enhanced?{{Motivation and emotion/Multimedia/High}} {{ME-By|KATE HERLIHY}} # [[/Violent extremism motivation/]] - What motivates people to engage in violent extremism? {{ME-By|U3216313}} # [[/Vocational identity/]] - What is vocational identity and how does it develop? {{ME-By|U3171379}} == Emotion == # [[/Artificial emotion/]] - To what extent can artificial intelligence experience emotion and how can this be applied? {{ME-By|KylieGiddings}} # [[/Attributions and emotion/]] - How do attributions affect emotion? {{ME-By|Carol Chariarse}} # [[/Autonomous sensory meridian response and emotion/]] - What emotions are involved in ASMR experiences and why do they occur? {{ME-By|Zmelmoth02}} # [[/Bewilderment/]] - What is bewilderment and how can it be dealt with? {{ME-By|Samanthagrebert}} # [[/Body functionality and body satisfaction/]] - What is the relationship between body functionality and body satisfaction? {{ME-By|Fatima2617}} # [[/Burn injury and body image/]] - What is the impact of burn injury on body image and what can be done? {{ME-By|Biancanappo}} # [[/Burnout/]] - What is burnout and how can be it be managed and prevented? {{ME-By|Grace (Tram) Chu}} # [[/Choice and emotion/]] - How does emotion affect choice? {{ME-By|Jingying Chen}} # [[/Climate change emotion/]] - What is the relationship between climate change and emotion?- [[User:Sireyes|Sireyes]] # [[/Commitment bias/]] - What motivates escalation of commitment even it does not lead to desirable outcomes? {{ME-By|U3203936}} # [[/Community resilience/]] - What is community resilience and how can it be fostered?{{Motivation and emotion/Book/High}} {{ME-By|U3213682}} # [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|Rhys Harmer}} # [[/Core emotions/]] - What are the core emotions and what is their function?{{Motivation and emotion/Book/High}} {{ME-By|U3229619}} # [[/Dark tetrad and emotion/]] - What is the dark tetrad and how does it relate to emotion? {{ME-By|U3214333}} # [[/Death and meaning in life/]] - How does confronting mortality influence the sense of meaning in life? {{ME-By|U3222680}} # [[/Default mode network and the self/]] - What is the relationship between the DMN and the self? {{ME-By|U3190506}} # [[/Deference emotion system/]] - What is the deference emotion system, how does it work, and what are its implications? {{ME-By|U3216127}} # [[/Eco-anxiety/]] - What is eco-anxiety, what are its consequences, and what can be done about it? {{ME-By|MT200107}} # [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|Emkatev}} # [[/Ecopsychology and stress/]] - How can ecopsychology help to explain and deal with stress?{{Motivation and emotion/Multimedia/High}} {{ME-By|Dnamynts}} # [[/Ego death/]] - What is ego death, what are its consequences, and how can it be facilitated?{{Motivation and emotion/Book/Low}} {{ME-By|Gaissa}} # [[/Emotion across the lifespan/]] - How does emotion develop across the lifespan? {{ME-By|Taliastefanoski}} # [[/Emotional intelligence training/]] - How can emotional intelligence be trained? {{ME-By|RolandC2023}} # [[/Empathy and psychopathy/]] - What is the relationship between empathy and psychopathy? {{ME-By|U3218292}} # [[/Empathy training/]] - How can empathy be enhanced? {{ME-By|U3224924}} # [[/Equine therapy and emotion/]] - What is the role of emotion in equine therapy? {{ME-By|SoSilverLibby}} # [[/Fear of apocalypse/]] - What is apocalyptic fear and how can it be dealt with? {{ME-By|AliciaEdwards1212}} # [[/Flourishing in the elderly/]] - How can psychological flourishing be supported in the elderly?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|JorjaFive}} # [[/Guided meditation and emotion regulation/]] - How can guided meditation be used to regulate emotion?{{Motivation and emotion/Book/Low}} {{ME-By|U3215755}} # [[/Hedonic adaptation prevention model/]] - What is the HAP model and how can it be applied? {{ME-By|Siobhanu3216316}} # [[/Hijack hypothesis of drug addiction/]] - What is the hijack hypothesis and how does it help to understand drug addiction? {{ME-By|U3189442 - K.Ryan}} # [[/Humility/]] - What is humility, what causes it, and is it desirable?{{Motivation and emotion/Book/Low}} {{ME-By|Inandonit365}} # [[/Interoception and mental health/]] - What is the role of interoception in mental health? {{ME-By|U3229576}} # [[/Indigenous Australian mindfulness/]] - How has Indigenous Australian culture traditionally conceived of, and practiced, mindfulness?{{Motivation and emotion/Book/High}} {{ME-By|U3168525}} # [[/Long-term injury in high performance field athletes and emotion/]] - How do long-term injuries impact the emotions of high performance field athletes? {{ME-By|U3216619}} # [[/Meditation and mindfulness/]] - How can meditation contribute to mindfulness? {{ME-By|APMonte}} # [[/Mental health literacy/]] - What is mental health literacy, why does it matter, and how can it be fostered? {{ME-By|U3215387}} # [[/Mental imagery and emotion/]] - How can mental imagery be used to regulate emotion? {{ME-By|MaryamNageeb}} # [[/Mental toughness in the workplace/]] - How can mental toughness be useful in the workplace and how can it be developed? {{ME-By|KubeChu}} # [[/Moral outrage/]] - What is moral outrage, what causes it, what are its consequences, and how can it be managed? {{ME-By|U3215934}} # [[/Narrative therapy and emotion/]] - What is the role of emotion in narrative therapy?{{Motivation and emotion/Book/Low}} {{ME-By|Alyssa.marin}} # [[/Placebo effect and emotion/]] - What is the placebo effect and how can it affect emotion? {{ME-By|U3213568}} # [[/Positive body image development/]] - What is a positive body image and how can it be developed? {{ME-By|U3224053}} # [[/Psilocybin assisted therapy and anxiety/]] - How can psilocybin assisted therapy help to treat anxiety?{{Motivation and emotion/Multimedia/High}} {{ME-By|SammyTabrett}} # [[/Psychedelic treatment of anxiety/]] - How can psychedelics be used to help in treating anxiety? {{ME-By|SashaBrooksby}} # [[/Psychological literacy/]] - What is psychological literacy, why does it matter, and how can it be fostered?{{Motivation and emotion/Book/Low}} {{ME-By|DMaphosa}} # [[/Relief/]] - What is relief and what role does it play in our emotional lives? {{ME-By|U3216413(2023)}} # [[/Sadism and emotion/]] - What are the emotional aspects of sadism? {{ME-By|Kimberley Manyura}} # [[/Self-affirmation and psychological change/]] - How can self-affirmation be used for positive psychological change? {{ME-By|U3213910}} # [[/Self-discrepancy theory and emotion/]] - In what ways does self-discrepancy help to understand emotion? {{ME-By|U3173387}} # [[/Singing and emotion/]] - What is the relation between singing and emotion?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3213084}} # [[/Smiling and emotion/]] - What is the relationship between smiling and emotion? {{ME-By|U3230268}} # [[/Stress mindset/]] - What is stress mindset, why does it matter, and how can it be cultivated?{{Motivation and emotion/Book/Low}} {{ME-By|U3224687}} # [[/Sublimation/]] - What is sublimation and how can it be fostered? {{ME-By|U3185622}} # [[/Therapeutic recreation/]] - What is therapeutic recreation, how does it work, and what are its impacts? {{ME-By|MojoHides}} # [[/Toxic positivity/]] - What is toxic positivity, why does it occur, and how can it be managed? {{ME-By|Lauraei11}} # [[/Trauma and emotion/]] - What is the effect of trauma on emotion?{{Motivation and emotion/Book/High}} {{ME-By|Clarestuparich}} # [[/Tragic optimism/]] - What is tragic optimism, why does it matter, and how can it be fostered? {{ME-By|Alice hatcher }} # [[/Trauma-informed care/]] - What is trauma-informed care and how can it be implemented?{{Motivation and emotion/Multimedia/High}} {{ME-By|Nabila.Tursun}} # [[/Unconscious aspects of multiple selves and emotion/]] - What role do unconscious aspects play in shaping our multiple selves and their influence on emotion? {{ME-By|Natsta19}} # [[/Uncertainty and emotion/]] - How does uncertainty influence emotion? [[User:Mikaela MG|Mikaela MG]] # [[/Unemployment and depression/]] - What is the relationship between unemployment and depression? {{ME-By|U3213250}} # [[/Unemployment and mental health/]] - What is the relationship between unemployment and mental health? {{ME-By|U3211780}} # [[/Vicarious embarrassment/]] - What is vicarious embarrassment, why does it occur, and how can it be managed?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3216476}} # [[/Viewing natural scenes and emotion/]] - What is the effect of viewing natural scenes on emotion and how can this be applied? {{ME-By|Alyson131}} # [[/Yoga and emotional regulation/]] - How can yoga help with emotional regulation? {{ME-By|U3169170}} == Motivation and emotion == # [[/GABA, motivation, and emotion/]] - What is the neurological function of GABA for motivation and emotion? {{ME-By|U3227976}} # [[/Medial forebrain bundle, motivation, and emotion/]] - What are the motivational and emotional functions of the medial forebrain bundle? {{ME-By|U3226707}} # [[/Multiple selves theory, motivation, and emotion/]] - How does the concept of multiple selves impact our motivational and emotional experiences?{{Motivation and emotion/Book/Low}} {{ME-By|H.Maycock}} # [[/Overchoice, motivation, and emotion/]] - What are the motivational and emotional effects of overchoice? {{ME-By|Atu320270}} # [[/Role-playing games, motivation, and emotion/]] - What are the motivational and emotional characteristics of role-playing games?{{Motivation and emotion/Book/Low}} {{ME-By|U3224203}} # [[/Torture interrogation, motivation, and emotion/]] - What are the motivational and emotional aspects of torture interrogation and how do these impact on the effectiveness of torture interrogation? {{ME-By|U3204500}} [[Category:Motivation and emotion/Book/2023]] kqlwnw7adk1vcnppnff8qsvfzbrlwzo 2814986 2814984 2026-06-10T07:24:15Z Jtneill 10242 Move abandoned topic to 2026 - # [[/Fear of apocalypse/]] - What is apocalyptic fear and how can it be dealt with? 2814986 wikitext text/x-wiki {{/Banner}} == Motivation == # [[/Abusive supervision/]] - What is abusive supervision, what motivates it, and what can be done about it?{{Motivation and emotion/Book/Low}} {{ME-By|Lara.Begg}} # [[/Active transport motivation/]] - What motivates use of active transport and how can people be encouraged to use it?{{Motivation and emotion/Book/Low}} {{ME-By|U3214564}} # [[/Actively open-minded thinking/]] - How can AOT be used to improve human performance? {{ME-By|Msherrybarr}} # [[/Ageing and motivation/]] - What is the relationship between ageing and motivation? {{ME-By|U3217153}} # [[/Anomie and motivation/]] - How does anomie affect motivation?{{Motivation and emotion/Book/High}} {{ME-By|U3223909}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour?{{Motivation and emotion/Book/Low}} {{ME-By|U3216706}} # [[/Bedtime procrastination/]] - What causes it, what are the consequences, and what can be done about it?{{Motivation and emotion/Book/High}} {{ME-By|U3227684}} # [[/Brief motivational interviewing as a health intervention/]] - How can brief motivational interviewing be used as a health intervention?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|JasnaM}} # [[/Childhood trauma and criminal behaviour/]] - How does childhood trauma influence subsequent criminal behaviour?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3218323}} # [[/Comprehensive action determination model|Comprehensive action determination model]] - What is the CADM and how can it be applied to understanding human motivation?{{Motivation and emotion/Book/Low}} {{ME-By|Rocxie}} # [[/Conspiracy theory motivation/]] - What motivates people to believe in conspiracy theories?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3223114}} # [[/Construal level theory/]] - What is construal level theory and how can it be applied? {{ME-By|U3217803|U3217803}} # [[/Dark tetrad and motivation/]] - What is the dark tetrad and how does it relate to motivation? {{ME-By|U3223849}} # [[/Death drive/]] - What is the death drive and how can it be negotiated? {{ME-By|U3197031}} # [[/Dopamine and extraversion-introversion/]] - How does dopamine affect extraversion-introversion? {{ME-By|SaucyTuRkLeBiRd}} # [[/Egosystem and ecosystem motivation/]] - What are egosystem and ecosystem motivations and what are their consequences? {{ME-By|U3216125}} # [[/Effective leadership/]] - What does it take to be an effective leader? {{ME-By|U103189}} # [[/Episodic future thinking and delay discounting/]] - What is the relationship between between EFT and DD?{{Motivation and emotion/Book/High}} {{ME-By|LightDragonflies}} # [[/Episodic future thinking and delay discounting 2/]] - What is the relationship between episodic future thinking and delay discounting? {{ME-By|U3224582}} # [[/Episodic memory and planning/]] - What role does episodic memory play in planning? {{ME-By|Maheenusman}} # [[/Environmental volunteering motivation/]] - What motivates environmental volunteering?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U891419}} # [[/Freedom and motivation/]] - What is the effect of freedom on motivation?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3217955}} # [[/Gamer motivation/]] - What motivates game playing?{{Motivation and emotion/Multimedia/High}} {{ME-By|Bellehines}} # [[/Grit and achievement/]] - How does grit affect achievement?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} - [[User:AnaStuart|AnaStuart]] # [[/Growth mindset and mental health/]] - What is the relationship between growth mindset and mental health? {{ME-By|Mia Pearse }} # [[/Health belief model/]] - What is the HBM and how can it be used to enhance motivation for health-promoting behaviour?{{Motivation and emotion/Book/High}} {{ME-By|U3226487}} # [[/Identity fusion and motivation/]] - What are the motivational implications of identity fusion?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3235369}} # [[/IKEA effect/]] - What is the IKEA effect and how can it be applied?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3209567}} # [[/Impact bias/]] - What is impact bias, what causes it, what are its consequences, and how can it be avoided?{{Motivation and emotion/Book/Low}} {{ME-By|De2023}} # [[/Indigenous Australian engagement in mental health services/]] - What helps Indigenous Australians to engage with mental health services? {{ME-By|CaitlinFisher01}} # [[/Intertemporal choice/]] - What are intertemporal choices and how can they be effectively negotiated?{{Motivation and emotion/Book/Low}} {{ME-By|TD3223808}} # [[/Irrational belief motivation/]] - What motivates irrational beliefs?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|Concettazicc}} # [[/Köhler effect and motivation/]] - What is the Köhler effect and how can it be applied? {{ME-By|Declan Robinson}} # [[/Learning motivation and chatbots/]] - How can student motivation to learn be enhanced by using artificial intelligence chatbots? {{ME-By|Ashhstarr}} # [[/Life history theory and human reproductive strategies/]] - How does life history theory explain human reproductive strategies? {{ME-By|U3212234}} # [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|Casey Coulton}} # [[/Mobile phone use motivation/]] - What are the motivations for mobile phone use?{{Motivation and emotion/Book/Low}} {{ME-By|Yaruki91}} # [[/Morning routine and motivation/]] - How can a morning routine be used to facilitate motivation, productivity, and well-being?{{Motivation and emotion/Book/Low}} {{ME-By|U3216585}} # [[/Motivational technologies/]] - How can motivation be fostered through technology? {{ME-By|U3196624}} # [[/Novelty-variety as a psychological need/]] - What is novelty-variety and what are its implications as a psychological need? {{ME-By|U3161239}} # [[/Nudge theory and sedentary behaviour/]] - How can nudge theory can be used reduce sedentary behaviour?{{Motivation and emotion/Book/High}} {{ME-By|U3145851}} # [[/Overchoice and motivation/]] - What is the overchoice effect, why does it occur, and who are most susceptible to it?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3230003}} # [[/Physical activity tracking and exercise motivation/]] - What role can physical activity tracking play in exercise motivation?{{Motivation and emotion/Book/Low}} {{ME-By|U3230383}} # [[/Physiological needs/]] - How do human's physiological needs affect motivation? {{ME-By|Superghostfresh}} # [[/Placebo effect and motivation/]] - What is the placebo effect and how can it affect motivation? {{ME-By|U3203797}} # [[/Protection motivation theory and COVID-19/]] - How does PMT apply to managing COVID-19?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3227759}} # [[/Sadism motivation/]] - What motivates sadism and what can be done about it? {{ME-By|BM3062384}} # [[/Signature strengths/]] - What are signature strengths and how can they be applied? {{ME-By|Rumesa.Aria}} # [[/Sport injury recovery motivation/]] - How can motivation be sustained when recovering from a sport injury? {{ME-By|U3210285}} # [[/SSRIs and motivation/]] - What are the effects of SSRIs on motivation?{{Motivation and emotion/Book/High}} {{ME-By|Josh.Cattle}} # [[/Stretch goals/]] - What are stretch goals? How do they work?{{Motivation and emotion/Book/Low}} {{ME-By|U3231448UC}} # [[/System justification theory/]] - What is SJT, how does it affect our lives, and what can be done about it? {{ME-By|Ishi}} # [[/Testosterone and dominance/]] - What is the relationship between testosterone and dominance? [[User:Giovannihbartlett|Giovannihbartlett]] # [[/Testosterone and violent crime/]] - What is the relationship between testosterone and violent crime?{{Motivation and emotion/Book/Low}} {{ME-By|U3200902}} # [[/Testosterone, winning, and losing/]] - What is the relationship between winning, losing, and testosterone? {{ME-By|Hugo DL}} # [[/Time management/]] - How can one's time be managed effectively? {{ME-By|Madison.Mayo}} # [[/To-do lists/]] - Are to-do lists a good idea? What are their pros and cons? How can they be used effectively?{{Motivation and emotion/Multimedia/High}} {{ME-By|U3162169T}} # [[/Treatment motivation in substance use disorder/]] - What is the role of treatment motivation in substance abuse disorder and how can it be enhanced?{{Motivation and emotion/Multimedia/High}} {{ME-By|KATE HERLIHY}} # [[/Violent extremism motivation/]] - What motivates people to engage in violent extremism? {{ME-By|U3216313}} # [[/Vocational identity/]] - What is vocational identity and how does it develop? {{ME-By|U3171379}} == Emotion == # [[/Artificial emotion/]] - To what extent can artificial intelligence experience emotion and how can this be applied? {{ME-By|KylieGiddings}} # [[/Attributions and emotion/]] - How do attributions affect emotion? {{ME-By|Carol Chariarse}} # [[/Autonomous sensory meridian response and emotion/]] - What emotions are involved in ASMR experiences and why do they occur? {{ME-By|Zmelmoth02}} # [[/Bewilderment/]] - What is bewilderment and how can it be dealt with? {{ME-By|Samanthagrebert}} # [[/Body functionality and body satisfaction/]] - What is the relationship between body functionality and body satisfaction? {{ME-By|Fatima2617}} # [[/Burn injury and body image/]] - What is the impact of burn injury on body image and what can be done? {{ME-By|Biancanappo}} # [[/Burnout/]] - What is burnout and how can be it be managed and prevented? {{ME-By|Grace (Tram) Chu}} # [[/Choice and emotion/]] - How does emotion affect choice? {{ME-By|Jingying Chen}} # [[/Climate change emotion/]] - What is the relationship between climate change and emotion?- [[User:Sireyes|Sireyes]] # [[/Commitment bias/]] - What motivates escalation of commitment even it does not lead to desirable outcomes? {{ME-By|U3203936}} # [[/Community resilience/]] - What is community resilience and how can it be fostered?{{Motivation and emotion/Book/High}} {{ME-By|U3213682}} # [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|Rhys Harmer}} # [[/Core emotions/]] - What are the core emotions and what is their function?{{Motivation and emotion/Book/High}} {{ME-By|U3229619}} # [[/Dark tetrad and emotion/]] - What is the dark tetrad and how does it relate to emotion? {{ME-By|U3214333}} # [[/Death and meaning in life/]] - How does confronting mortality influence the sense of meaning in life? {{ME-By|U3222680}} # [[/Default mode network and the self/]] - What is the relationship between the DMN and the self? {{ME-By|U3190506}} # [[/Deference emotion system/]] - What is the deference emotion system, how does it work, and what are its implications? {{ME-By|U3216127}} # [[/Eco-anxiety/]] - What is eco-anxiety, what are its consequences, and what can be done about it? {{ME-By|MT200107}} # [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|Emkatev}} # [[/Ecopsychology and stress/]] - How can ecopsychology help to explain and deal with stress?{{Motivation and emotion/Multimedia/High}} {{ME-By|Dnamynts}} # [[/Ego death/]] - What is ego death, what are its consequences, and how can it be facilitated?{{Motivation and emotion/Book/Low}} {{ME-By|Gaissa}} # [[/Emotion across the lifespan/]] - How does emotion develop across the lifespan? {{ME-By|Taliastefanoski}} # [[/Emotional intelligence training/]] - How can emotional intelligence be trained? {{ME-By|RolandC2023}} # [[/Empathy and psychopathy/]] - What is the relationship between empathy and psychopathy? {{ME-By|U3218292}} # [[/Empathy training/]] - How can empathy be enhanced? {{ME-By|U3224924}} # [[/Equine therapy and emotion/]] - What is the role of emotion in equine therapy? {{ME-By|SoSilverLibby}} # [[/Flourishing in the elderly/]] - How can psychological flourishing be supported in the elderly?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|JorjaFive}} # [[/Guided meditation and emotion regulation/]] - How can guided meditation be used to regulate emotion?{{Motivation and emotion/Book/Low}} {{ME-By|U3215755}} # [[/Hedonic adaptation prevention model/]] - What is the HAP model and how can it be applied? {{ME-By|Siobhanu3216316}} # [[/Hijack hypothesis of drug addiction/]] - What is the hijack hypothesis and how does it help to understand drug addiction? {{ME-By|U3189442 - K.Ryan}} # [[/Humility/]] - What is humility, what causes it, and is it desirable?{{Motivation and emotion/Book/Low}} {{ME-By|Inandonit365}} # [[/Interoception and mental health/]] - What is the role of interoception in mental health? {{ME-By|U3229576}} # [[/Indigenous Australian mindfulness/]] - How has Indigenous Australian culture traditionally conceived of, and practiced, mindfulness?{{Motivation and emotion/Book/High}} {{ME-By|U3168525}} # [[/Long-term injury in high performance field athletes and emotion/]] - How do long-term injuries impact the emotions of high performance field athletes? {{ME-By|U3216619}} # [[/Meditation and mindfulness/]] - How can meditation contribute to mindfulness? {{ME-By|APMonte}} # [[/Mental health literacy/]] - What is mental health literacy, why does it matter, and how can it be fostered? {{ME-By|U3215387}} # [[/Mental imagery and emotion/]] - How can mental imagery be used to regulate emotion? {{ME-By|MaryamNageeb}} # [[/Mental toughness in the workplace/]] - How can mental toughness be useful in the workplace and how can it be developed? {{ME-By|KubeChu}} # [[/Moral outrage/]] - What is moral outrage, what causes it, what are its consequences, and how can it be managed? {{ME-By|U3215934}} # [[/Narrative therapy and emotion/]] - What is the role of emotion in narrative therapy?{{Motivation and emotion/Book/Low}} {{ME-By|Alyssa.marin}} # [[/Placebo effect and emotion/]] - What is the placebo effect and how can it affect emotion? {{ME-By|U3213568}} # [[/Positive body image development/]] - What is a positive body image and how can it be developed? {{ME-By|U3224053}} # [[/Psilocybin assisted therapy and anxiety/]] - How can psilocybin assisted therapy help to treat anxiety?{{Motivation and emotion/Multimedia/High}} {{ME-By|SammyTabrett}} # [[/Psychedelic treatment of anxiety/]] - How can psychedelics be used to help in treating anxiety? {{ME-By|SashaBrooksby}} # [[/Psychological literacy/]] - What is psychological literacy, why does it matter, and how can it be fostered?{{Motivation and emotion/Book/Low}} {{ME-By|DMaphosa}} # [[/Relief/]] - What is relief and what role does it play in our emotional lives? {{ME-By|U3216413(2023)}} # [[/Sadism and emotion/]] - What are the emotional aspects of sadism? {{ME-By|Kimberley Manyura}} # [[/Self-affirmation and psychological change/]] - How can self-affirmation be used for positive psychological change? {{ME-By|U3213910}} # [[/Self-discrepancy theory and emotion/]] - In what ways does self-discrepancy help to understand emotion? {{ME-By|U3173387}} # [[/Singing and emotion/]] - What is the relation between singing and emotion?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3213084}} # [[/Smiling and emotion/]] - What is the relationship between smiling and emotion? {{ME-By|U3230268}} # [[/Stress mindset/]] - What is stress mindset, why does it matter, and how can it be cultivated?{{Motivation and emotion/Book/Low}} {{ME-By|U3224687}} # [[/Sublimation/]] - What is sublimation and how can it be fostered? {{ME-By|U3185622}} # [[/Therapeutic recreation/]] - What is therapeutic recreation, how does it work, and what are its impacts? {{ME-By|MojoHides}} # [[/Toxic positivity/]] - What is toxic positivity, why does it occur, and how can it be managed? {{ME-By|Lauraei11}} # [[/Trauma and emotion/]] - What is the effect of trauma on emotion?{{Motivation and emotion/Book/High}} {{ME-By|Clarestuparich}} # [[/Tragic optimism/]] - What is tragic optimism, why does it matter, and how can it be fostered? {{ME-By|Alice hatcher }} # [[/Trauma-informed care/]] - What is trauma-informed care and how can it be implemented?{{Motivation and emotion/Multimedia/High}} {{ME-By|Nabila.Tursun}} # [[/Unconscious aspects of multiple selves and emotion/]] - What role do unconscious aspects play in shaping our multiple selves and their influence on emotion? {{ME-By|Natsta19}} # [[/Uncertainty and emotion/]] - How does uncertainty influence emotion? [[User:Mikaela MG|Mikaela MG]] # [[/Unemployment and depression/]] - What is the relationship between unemployment and depression? {{ME-By|U3213250}} # [[/Unemployment and mental health/]] - What is the relationship between unemployment and mental health? {{ME-By|U3211780}} # [[/Vicarious embarrassment/]] - What is vicarious embarrassment, why does it occur, and how can it be managed?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3216476}} # [[/Viewing natural scenes and emotion/]] - What is the effect of viewing natural scenes on emotion and how can this be applied? {{ME-By|Alyson131}} # [[/Yoga and emotional regulation/]] - How can yoga help with emotional regulation? {{ME-By|U3169170}} == Motivation and emotion == # [[/GABA, motivation, and emotion/]] - What is the neurological function of GABA for motivation and emotion? {{ME-By|U3227976}} # [[/Medial forebrain bundle, motivation, and emotion/]] - What are the motivational and emotional functions of the medial forebrain bundle? {{ME-By|U3226707}} # [[/Multiple selves theory, motivation, and emotion/]] - How does the concept of multiple selves impact our motivational and emotional experiences?{{Motivation and emotion/Book/Low}} {{ME-By|H.Maycock}} # [[/Overchoice, motivation, and emotion/]] - What are the motivational and emotional effects of overchoice? {{ME-By|Atu320270}} # [[/Role-playing games, motivation, and emotion/]] - What are the motivational and emotional characteristics of role-playing games?{{Motivation and emotion/Book/Low}} {{ME-By|U3224203}} # [[/Torture interrogation, motivation, and emotion/]] - What are the motivational and emotional aspects of torture interrogation and how do these impact on the effectiveness of torture interrogation? {{ME-By|U3204500}} [[Category:Motivation and emotion/Book/2023]] p0qo256ub31hxhmjakiqlrb367hx3qc 2814994 2814986 2026-06-10T07:34:37Z Jtneill 10242 /* Emotion */ Move 1 abandoned topic to 2026 - # [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? 2814994 wikitext text/x-wiki {{/Banner}} == Motivation == # [[/Abusive supervision/]] - What is abusive supervision, what motivates it, and what can be done about it?{{Motivation and emotion/Book/Low}} {{ME-By|Lara.Begg}} # [[/Active transport motivation/]] - What motivates use of active transport and how can people be encouraged to use it?{{Motivation and emotion/Book/Low}} {{ME-By|U3214564}} # [[/Actively open-minded thinking/]] - How can AOT be used to improve human performance? {{ME-By|Msherrybarr}} # [[/Ageing and motivation/]] - What is the relationship between ageing and motivation? {{ME-By|U3217153}} # [[/Anomie and motivation/]] - How does anomie affect motivation?{{Motivation and emotion/Book/High}} {{ME-By|U3223909}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour?{{Motivation and emotion/Book/Low}} {{ME-By|U3216706}} # [[/Bedtime procrastination/]] - What causes it, what are the consequences, and what can be done about it?{{Motivation and emotion/Book/High}} {{ME-By|U3227684}} # [[/Brief motivational interviewing as a health intervention/]] - How can brief motivational interviewing be used as a health intervention?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|JasnaM}} # [[/Childhood trauma and criminal behaviour/]] - How does childhood trauma influence subsequent criminal behaviour?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3218323}} # [[/Comprehensive action determination model|Comprehensive action determination model]] - What is the CADM and how can it be applied to understanding human motivation?{{Motivation and emotion/Book/Low}} {{ME-By|Rocxie}} # [[/Conspiracy theory motivation/]] - What motivates people to believe in conspiracy theories?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3223114}} # [[/Construal level theory/]] - What is construal level theory and how can it be applied? {{ME-By|U3217803|U3217803}} # [[/Dark tetrad and motivation/]] - What is the dark tetrad and how does it relate to motivation? {{ME-By|U3223849}} # [[/Death drive/]] - What is the death drive and how can it be negotiated? {{ME-By|U3197031}} # [[/Dopamine and extraversion-introversion/]] - How does dopamine affect extraversion-introversion? {{ME-By|SaucyTuRkLeBiRd}} # [[/Egosystem and ecosystem motivation/]] - What are egosystem and ecosystem motivations and what are their consequences? {{ME-By|U3216125}} # [[/Effective leadership/]] - What does it take to be an effective leader? {{ME-By|U103189}} # [[/Episodic future thinking and delay discounting/]] - What is the relationship between between EFT and DD?{{Motivation and emotion/Book/High}} {{ME-By|LightDragonflies}} # [[/Episodic future thinking and delay discounting 2/]] - What is the relationship between episodic future thinking and delay discounting? {{ME-By|U3224582}} # [[/Episodic memory and planning/]] - What role does episodic memory play in planning? {{ME-By|Maheenusman}} # [[/Environmental volunteering motivation/]] - What motivates environmental volunteering?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U891419}} # [[/Freedom and motivation/]] - What is the effect of freedom on motivation?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3217955}} # [[/Gamer motivation/]] - What motivates game playing?{{Motivation and emotion/Multimedia/High}} {{ME-By|Bellehines}} # [[/Grit and achievement/]] - How does grit affect achievement?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} - [[User:AnaStuart|AnaStuart]] # [[/Growth mindset and mental health/]] - What is the relationship between growth mindset and mental health? {{ME-By|Mia Pearse }} # [[/Health belief model/]] - What is the HBM and how can it be used to enhance motivation for health-promoting behaviour?{{Motivation and emotion/Book/High}} {{ME-By|U3226487}} # [[/Identity fusion and motivation/]] - What are the motivational implications of identity fusion?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3235369}} # [[/IKEA effect/]] - What is the IKEA effect and how can it be applied?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3209567}} # [[/Impact bias/]] - What is impact bias, what causes it, what are its consequences, and how can it be avoided?{{Motivation and emotion/Book/Low}} {{ME-By|De2023}} # [[/Indigenous Australian engagement in mental health services/]] - What helps Indigenous Australians to engage with mental health services? {{ME-By|CaitlinFisher01}} # [[/Intertemporal choice/]] - What are intertemporal choices and how can they be effectively negotiated?{{Motivation and emotion/Book/Low}} {{ME-By|TD3223808}} # [[/Irrational belief motivation/]] - What motivates irrational beliefs?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|Concettazicc}} # [[/Köhler effect and motivation/]] - What is the Köhler effect and how can it be applied? {{ME-By|Declan Robinson}} # [[/Learning motivation and chatbots/]] - How can student motivation to learn be enhanced by using artificial intelligence chatbots? {{ME-By|Ashhstarr}} # [[/Life history theory and human reproductive strategies/]] - How does life history theory explain human reproductive strategies? {{ME-By|U3212234}} # [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|Casey Coulton}} # [[/Mobile phone use motivation/]] - What are the motivations for mobile phone use?{{Motivation and emotion/Book/Low}} {{ME-By|Yaruki91}} # [[/Morning routine and motivation/]] - How can a morning routine be used to facilitate motivation, productivity, and well-being?{{Motivation and emotion/Book/Low}} {{ME-By|U3216585}} # [[/Motivational technologies/]] - How can motivation be fostered through technology? {{ME-By|U3196624}} # [[/Novelty-variety as a psychological need/]] - What is novelty-variety and what are its implications as a psychological need? {{ME-By|U3161239}} # [[/Nudge theory and sedentary behaviour/]] - How can nudge theory can be used reduce sedentary behaviour?{{Motivation and emotion/Book/High}} {{ME-By|U3145851}} # [[/Overchoice and motivation/]] - What is the overchoice effect, why does it occur, and who are most susceptible to it?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3230003}} # [[/Physical activity tracking and exercise motivation/]] - What role can physical activity tracking play in exercise motivation?{{Motivation and emotion/Book/Low}} {{ME-By|U3230383}} # [[/Physiological needs/]] - How do human's physiological needs affect motivation? {{ME-By|Superghostfresh}} # [[/Placebo effect and motivation/]] - What is the placebo effect and how can it affect motivation? {{ME-By|U3203797}} # [[/Protection motivation theory and COVID-19/]] - How does PMT apply to managing COVID-19?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3227759}} # [[/Sadism motivation/]] - What motivates sadism and what can be done about it? {{ME-By|BM3062384}} # [[/Signature strengths/]] - What are signature strengths and how can they be applied? {{ME-By|Rumesa.Aria}} # [[/Sport injury recovery motivation/]] - How can motivation be sustained when recovering from a sport injury? {{ME-By|U3210285}} # [[/SSRIs and motivation/]] - What are the effects of SSRIs on motivation?{{Motivation and emotion/Book/High}} {{ME-By|Josh.Cattle}} # [[/Stretch goals/]] - What are stretch goals? How do they work?{{Motivation and emotion/Book/Low}} {{ME-By|U3231448UC}} # [[/System justification theory/]] - What is SJT, how does it affect our lives, and what can be done about it? {{ME-By|Ishi}} # [[/Testosterone and dominance/]] - What is the relationship between testosterone and dominance? [[User:Giovannihbartlett|Giovannihbartlett]] # [[/Testosterone and violent crime/]] - What is the relationship between testosterone and violent crime?{{Motivation and emotion/Book/Low}} {{ME-By|U3200902}} # [[/Testosterone, winning, and losing/]] - What is the relationship between winning, losing, and testosterone? {{ME-By|Hugo DL}} # [[/Time management/]] - How can one's time be managed effectively? {{ME-By|Madison.Mayo}} # [[/To-do lists/]] - Are to-do lists a good idea? What are their pros and cons? How can they be used effectively?{{Motivation and emotion/Multimedia/High}} {{ME-By|U3162169T}} # [[/Treatment motivation in substance use disorder/]] - What is the role of treatment motivation in substance abuse disorder and how can it be enhanced?{{Motivation and emotion/Multimedia/High}} {{ME-By|KATE HERLIHY}} # [[/Violent extremism motivation/]] - What motivates people to engage in violent extremism? {{ME-By|U3216313}} # [[/Vocational identity/]] - What is vocational identity and how does it develop? {{ME-By|U3171379}} == Emotion == # [[/Artificial emotion/]] - To what extent can artificial intelligence experience emotion and how can this be applied? {{ME-By|KylieGiddings}} # [[/Attributions and emotion/]] - How do attributions affect emotion? {{ME-By|Carol Chariarse}} # [[/Autonomous sensory meridian response and emotion/]] - What emotions are involved in ASMR experiences and why do they occur? {{ME-By|Zmelmoth02}} # [[/Bewilderment/]] - What is bewilderment and how can it be dealt with? {{ME-By|Samanthagrebert}} # [[/Body functionality and body satisfaction/]] - What is the relationship between body functionality and body satisfaction? {{ME-By|Fatima2617}} # [[/Burn injury and body image/]] - What is the impact of burn injury on body image and what can be done? {{ME-By|Biancanappo}} # [[/Burnout/]] - What is burnout and how can be it be managed and prevented? {{ME-By|Grace (Tram) Chu}} # [[/Choice and emotion/]] - How does emotion affect choice? {{ME-By|Jingying Chen}} # [[/Climate change emotion/]] - What is the relationship between climate change and emotion?- [[User:Sireyes|Sireyes]] # [[/Commitment bias/]] - What motivates escalation of commitment even it does not lead to desirable outcomes? {{ME-By|U3203936}} # [[/Community resilience/]] - What is community resilience and how can it be fostered?{{Motivation and emotion/Book/High}} {{ME-By|U3213682}} # [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|Rhys Harmer}} # [[/Core emotions/]] - What are the core emotions and what is their function?{{Motivation and emotion/Book/High}} {{ME-By|U3229619}} # [[/Dark tetrad and emotion/]] - What is the dark tetrad and how does it relate to emotion? {{ME-By|U3214333}} # [[/Death and meaning in life/]] - How does confronting mortality influence the sense of meaning in life? {{ME-By|U3222680}} # [[/Default mode network and the self/]] - What is the relationship between the DMN and the self? {{ME-By|U3190506}} # [[/Deference emotion system/]] - What is the deference emotion system, how does it work, and what are its implications? {{ME-By|U3216127}} # [[/Eco-anxiety/]] - What is eco-anxiety, what are its consequences, and what can be done about it? {{ME-By|MT200107}} # [[/Ecopsychology and stress/]] - How can ecopsychology help to explain and deal with stress?{{Motivation and emotion/Multimedia/High}} {{ME-By|Dnamynts}} # [[/Ego death/]] - What is ego death, what are its consequences, and how can it be facilitated?{{Motivation and emotion/Book/Low}} {{ME-By|Gaissa}} # [[/Emotion across the lifespan/]] - How does emotion develop across the lifespan? {{ME-By|Taliastefanoski}} # [[/Emotional intelligence training/]] - How can emotional intelligence be trained? {{ME-By|RolandC2023}} # [[/Empathy and psychopathy/]] - What is the relationship between empathy and psychopathy? {{ME-By|U3218292}} # [[/Empathy training/]] - How can empathy be enhanced? {{ME-By|U3224924}} # [[/Equine therapy and emotion/]] - What is the role of emotion in equine therapy? {{ME-By|SoSilverLibby}} # [[/Flourishing in the elderly/]] - How can psychological flourishing be supported in the elderly?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|JorjaFive}} # [[/Guided meditation and emotion regulation/]] - How can guided meditation be used to regulate emotion?{{Motivation and emotion/Book/Low}} {{ME-By|U3215755}} # [[/Hedonic adaptation prevention model/]] - What is the HAP model and how can it be applied? {{ME-By|Siobhanu3216316}} # [[/Hijack hypothesis of drug addiction/]] - What is the hijack hypothesis and how does it help to understand drug addiction? {{ME-By|U3189442 - K.Ryan}} # [[/Humility/]] - What is humility, what causes it, and is it desirable?{{Motivation and emotion/Book/Low}} {{ME-By|Inandonit365}} # [[/Interoception and mental health/]] - What is the role of interoception in mental health? {{ME-By|U3229576}} # [[/Indigenous Australian mindfulness/]] - How has Indigenous Australian culture traditionally conceived of, and practiced, mindfulness?{{Motivation and emotion/Book/High}} {{ME-By|U3168525}} # [[/Long-term injury in high performance field athletes and emotion/]] - How do long-term injuries impact the emotions of high performance field athletes? {{ME-By|U3216619}} # [[/Meditation and mindfulness/]] - How can meditation contribute to mindfulness? {{ME-By|APMonte}} # [[/Mental health literacy/]] - What is mental health literacy, why does it matter, and how can it be fostered? {{ME-By|U3215387}} # [[/Mental imagery and emotion/]] - How can mental imagery be used to regulate emotion? {{ME-By|MaryamNageeb}} # [[/Mental toughness in the workplace/]] - How can mental toughness be useful in the workplace and how can it be developed? {{ME-By|KubeChu}} # [[/Moral outrage/]] - What is moral outrage, what causes it, what are its consequences, and how can it be managed? {{ME-By|U3215934}} # [[/Narrative therapy and emotion/]] - What is the role of emotion in narrative therapy?{{Motivation and emotion/Book/Low}} {{ME-By|Alyssa.marin}} # [[/Placebo effect and emotion/]] - What is the placebo effect and how can it affect emotion? {{ME-By|U3213568}} # [[/Positive body image development/]] - What is a positive body image and how can it be developed? {{ME-By|U3224053}} # [[/Psilocybin assisted therapy and anxiety/]] - How can psilocybin assisted therapy help to treat anxiety?{{Motivation and emotion/Multimedia/High}} {{ME-By|SammyTabrett}} # [[/Psychedelic treatment of anxiety/]] - How can psychedelics be used to help in treating anxiety? {{ME-By|SashaBrooksby}} # [[/Psychological literacy/]] - What is psychological literacy, why does it matter, and how can it be fostered?{{Motivation and emotion/Book/Low}} {{ME-By|DMaphosa}} # [[/Relief/]] - What is relief and what role does it play in our emotional lives? {{ME-By|U3216413(2023)}} # [[/Sadism and emotion/]] - What are the emotional aspects of sadism? {{ME-By|Kimberley Manyura}} # [[/Self-affirmation and psychological change/]] - How can self-affirmation be used for positive psychological change? {{ME-By|U3213910}} # [[/Self-discrepancy theory and emotion/]] - In what ways does self-discrepancy help to understand emotion? {{ME-By|U3173387}} # [[/Singing and emotion/]] - What is the relation between singing and emotion?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3213084}} # [[/Smiling and emotion/]] - What is the relationship between smiling and emotion? {{ME-By|U3230268}} # [[/Stress mindset/]] - What is stress mindset, why does it matter, and how can it be cultivated?{{Motivation and emotion/Book/Low}} {{ME-By|U3224687}} # [[/Sublimation/]] - What is sublimation and how can it be fostered? {{ME-By|U3185622}} # [[/Therapeutic recreation/]] - What is therapeutic recreation, how does it work, and what are its impacts? {{ME-By|MojoHides}} # [[/Toxic positivity/]] - What is toxic positivity, why does it occur, and how can it be managed? {{ME-By|Lauraei11}} # [[/Trauma and emotion/]] - What is the effect of trauma on emotion?{{Motivation and emotion/Book/High}} {{ME-By|Clarestuparich}} # [[/Tragic optimism/]] - What is tragic optimism, why does it matter, and how can it be fostered? {{ME-By|Alice hatcher }} # [[/Trauma-informed care/]] - What is trauma-informed care and how can it be implemented?{{Motivation and emotion/Multimedia/High}} {{ME-By|Nabila.Tursun}} # [[/Unconscious aspects of multiple selves and emotion/]] - What role do unconscious aspects play in shaping our multiple selves and their influence on emotion? {{ME-By|Natsta19}} # [[/Uncertainty and emotion/]] - How does uncertainty influence emotion? [[User:Mikaela MG|Mikaela MG]] # [[/Unemployment and depression/]] - What is the relationship between unemployment and depression? {{ME-By|U3213250}} # [[/Unemployment and mental health/]] - What is the relationship between unemployment and mental health? {{ME-By|U3211780}} # [[/Vicarious embarrassment/]] - What is vicarious embarrassment, why does it occur, and how can it be managed?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3216476}} # [[/Viewing natural scenes and emotion/]] - What is the effect of viewing natural scenes on emotion and how can this be applied? {{ME-By|Alyson131}} # [[/Yoga and emotional regulation/]] - How can yoga help with emotional regulation? {{ME-By|U3169170}} == Motivation and emotion == # [[/GABA, motivation, and emotion/]] - What is the neurological function of GABA for motivation and emotion? {{ME-By|U3227976}} # [[/Medial forebrain bundle, motivation, and emotion/]] - What are the motivational and emotional functions of the medial forebrain bundle? {{ME-By|U3226707}} # [[/Multiple selves theory, motivation, and emotion/]] - How does the concept of multiple selves impact our motivational and emotional experiences?{{Motivation and emotion/Book/Low}} {{ME-By|H.Maycock}} # [[/Overchoice, motivation, and emotion/]] - What are the motivational and emotional effects of overchoice? {{ME-By|Atu320270}} # [[/Role-playing games, motivation, and emotion/]] - What are the motivational and emotional characteristics of role-playing games?{{Motivation and emotion/Book/Low}} {{ME-By|U3224203}} # [[/Torture interrogation, motivation, and emotion/]] - What are the motivational and emotional aspects of torture interrogation and how do these impact on the effectiveness of torture interrogation? {{ME-By|U3204500}} [[Category:Motivation and emotion/Book/2023]] 7jbkvh97nx4r6i8bnr1v25xcc7b486l 2815004 2814994 2026-06-10T07:43:03Z Jtneill 10242 /* Motivation */ Move 1 abandoned resource to 2026 - # [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? 2815004 wikitext text/x-wiki {{/Banner}} == Motivation == # [[/Abusive supervision/]] - What is abusive supervision, what motivates it, and what can be done about it?{{Motivation and emotion/Book/Low}} {{ME-By|Lara.Begg}} # [[/Active transport motivation/]] - What motivates use of active transport and how can people be encouraged to use it?{{Motivation and emotion/Book/Low}} {{ME-By|U3214564}} # [[/Actively open-minded thinking/]] - How can AOT be used to improve human performance? {{ME-By|Msherrybarr}} # [[/Ageing and motivation/]] - What is the relationship between ageing and motivation? {{ME-By|U3217153}} # [[/Anomie and motivation/]] - How does anomie affect motivation?{{Motivation and emotion/Book/High}} {{ME-By|U3223909}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour?{{Motivation and emotion/Book/Low}} {{ME-By|U3216706}} # [[/Bedtime procrastination/]] - What causes it, what are the consequences, and what can be done about it?{{Motivation and emotion/Book/High}} {{ME-By|U3227684}} # [[/Brief motivational interviewing as a health intervention/]] - How can brief motivational interviewing be used as a health intervention?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|JasnaM}} # [[/Childhood trauma and criminal behaviour/]] - How does childhood trauma influence subsequent criminal behaviour?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3218323}} # [[/Comprehensive action determination model|Comprehensive action determination model]] - What is the CADM and how can it be applied to understanding human motivation?{{Motivation and emotion/Book/Low}} {{ME-By|Rocxie}} # [[/Conspiracy theory motivation/]] - What motivates people to believe in conspiracy theories?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3223114}} # [[/Construal level theory/]] - What is construal level theory and how can it be applied? {{ME-By|U3217803|U3217803}} # [[/Dark tetrad and motivation/]] - What is the dark tetrad and how does it relate to motivation? {{ME-By|U3223849}} # [[/Death drive/]] - What is the death drive and how can it be negotiated? {{ME-By|U3197031}} # [[/Dopamine and extraversion-introversion/]] - How does dopamine affect extraversion-introversion? {{ME-By|SaucyTuRkLeBiRd}} # [[/Egosystem and ecosystem motivation/]] - What are egosystem and ecosystem motivations and what are their consequences? {{ME-By|U3216125}} # [[/Effective leadership/]] - What does it take to be an effective leader? {{ME-By|U103189}} # [[/Episodic future thinking and delay discounting/]] - What is the relationship between between EFT and DD?{{Motivation and emotion/Book/High}} {{ME-By|LightDragonflies}} # [[/Episodic future thinking and delay discounting 2/]] - What is the relationship between episodic future thinking and delay discounting? {{ME-By|U3224582}} # [[/Episodic memory and planning/]] - What role does episodic memory play in planning? {{ME-By|Maheenusman}} # [[/Environmental volunteering motivation/]] - What motivates environmental volunteering?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U891419}} # [[/Freedom and motivation/]] - What is the effect of freedom on motivation?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3217955}} # [[/Gamer motivation/]] - What motivates game playing?{{Motivation and emotion/Multimedia/High}} {{ME-By|Bellehines}} # [[/Grit and achievement/]] - How does grit affect achievement?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} - [[User:AnaStuart|AnaStuart]] # [[/Growth mindset and mental health/]] - What is the relationship between growth mindset and mental health? {{ME-By|Mia Pearse }} # [[/Health belief model/]] - What is the HBM and how can it be used to enhance motivation for health-promoting behaviour?{{Motivation and emotion/Book/High}} {{ME-By|U3226487}} # [[/Identity fusion and motivation/]] - What are the motivational implications of identity fusion?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3235369}} # [[/IKEA effect/]] - What is the IKEA effect and how can it be applied?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3209567}} # [[/Impact bias/]] - What is impact bias, what causes it, what are its consequences, and how can it be avoided?{{Motivation and emotion/Book/Low}} {{ME-By|De2023}} # [[/Indigenous Australian engagement in mental health services/]] - What helps Indigenous Australians to engage with mental health services? {{ME-By|CaitlinFisher01}} # [[/Intertemporal choice/]] - What are intertemporal choices and how can they be effectively negotiated?{{Motivation and emotion/Book/Low}} {{ME-By|TD3223808}} # [[/Irrational belief motivation/]] - What motivates irrational beliefs?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|Concettazicc}} # [[/Köhler effect and motivation/]] - What is the Köhler effect and how can it be applied? {{ME-By|Declan Robinson}} # [[/Learning motivation and chatbots/]] - How can student motivation to learn be enhanced by using artificial intelligence chatbots? {{ME-By|Ashhstarr}} # [[/Life history theory and human reproductive strategies/]] - How does life history theory explain human reproductive strategies? {{ME-By|U3212234}} # [[/Mobile phone use motivation/]] - What are the motivations for mobile phone use?{{Motivation and emotion/Book/Low}} {{ME-By|Yaruki91}} # [[/Morning routine and motivation/]] - How can a morning routine be used to facilitate motivation, productivity, and well-being?{{Motivation and emotion/Book/Low}} {{ME-By|U3216585}} # [[/Motivational technologies/]] - How can motivation be fostered through technology? {{ME-By|U3196624}} # [[/Novelty-variety as a psychological need/]] - What is novelty-variety and what are its implications as a psychological need? {{ME-By|U3161239}} # [[/Nudge theory and sedentary behaviour/]] - How can nudge theory can be used reduce sedentary behaviour?{{Motivation and emotion/Book/High}} {{ME-By|U3145851}} # [[/Overchoice and motivation/]] - What is the overchoice effect, why does it occur, and who are most susceptible to it?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3230003}} # [[/Physical activity tracking and exercise motivation/]] - What role can physical activity tracking play in exercise motivation?{{Motivation and emotion/Book/Low}} {{ME-By|U3230383}} # [[/Physiological needs/]] - How do human's physiological needs affect motivation? {{ME-By|Superghostfresh}} # [[/Placebo effect and motivation/]] - What is the placebo effect and how can it affect motivation? {{ME-By|U3203797}} # [[/Protection motivation theory and COVID-19/]] - How does PMT apply to managing COVID-19?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3227759}} # [[/Sadism motivation/]] - What motivates sadism and what can be done about it? {{ME-By|BM3062384}} # [[/Signature strengths/]] - What are signature strengths and how can they be applied? {{ME-By|Rumesa.Aria}} # [[/Sport injury recovery motivation/]] - How can motivation be sustained when recovering from a sport injury? {{ME-By|U3210285}} # [[/SSRIs and motivation/]] - What are the effects of SSRIs on motivation?{{Motivation and emotion/Book/High}} {{ME-By|Josh.Cattle}} # [[/Stretch goals/]] - What are stretch goals? How do they work?{{Motivation and emotion/Book/Low}} {{ME-By|U3231448UC}} # [[/System justification theory/]] - What is SJT, how does it affect our lives, and what can be done about it? {{ME-By|Ishi}} # [[/Testosterone and dominance/]] - What is the relationship between testosterone and dominance? [[User:Giovannihbartlett|Giovannihbartlett]] # [[/Testosterone and violent crime/]] - What is the relationship between testosterone and violent crime?{{Motivation and emotion/Book/Low}} {{ME-By|U3200902}} # [[/Testosterone, winning, and losing/]] - What is the relationship between winning, losing, and testosterone? {{ME-By|Hugo DL}} # [[/Time management/]] - How can one's time be managed effectively? {{ME-By|Madison.Mayo}} # [[/To-do lists/]] - Are to-do lists a good idea? What are their pros and cons? How can they be used effectively?{{Motivation and emotion/Multimedia/High}} {{ME-By|U3162169T}} # [[/Treatment motivation in substance use disorder/]] - What is the role of treatment motivation in substance abuse disorder and how can it be enhanced?{{Motivation and emotion/Multimedia/High}} {{ME-By|KATE HERLIHY}} # [[/Violent extremism motivation/]] - What motivates people to engage in violent extremism? {{ME-By|U3216313}} # [[/Vocational identity/]] - What is vocational identity and how does it develop? {{ME-By|U3171379}} == Emotion == # [[/Artificial emotion/]] - To what extent can artificial intelligence experience emotion and how can this be applied? {{ME-By|KylieGiddings}} # [[/Attributions and emotion/]] - How do attributions affect emotion? {{ME-By|Carol Chariarse}} # [[/Autonomous sensory meridian response and emotion/]] - What emotions are involved in ASMR experiences and why do they occur? {{ME-By|Zmelmoth02}} # [[/Bewilderment/]] - What is bewilderment and how can it be dealt with? {{ME-By|Samanthagrebert}} # [[/Body functionality and body satisfaction/]] - What is the relationship between body functionality and body satisfaction? {{ME-By|Fatima2617}} # [[/Burn injury and body image/]] - What is the impact of burn injury on body image and what can be done? {{ME-By|Biancanappo}} # [[/Burnout/]] - What is burnout and how can be it be managed and prevented? {{ME-By|Grace (Tram) Chu}} # [[/Choice and emotion/]] - How does emotion affect choice? {{ME-By|Jingying Chen}} # [[/Climate change emotion/]] - What is the relationship between climate change and emotion?- [[User:Sireyes|Sireyes]] # [[/Commitment bias/]] - What motivates escalation of commitment even it does not lead to desirable outcomes? {{ME-By|U3203936}} # [[/Community resilience/]] - What is community resilience and how can it be fostered?{{Motivation and emotion/Book/High}} {{ME-By|U3213682}} # [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|Rhys Harmer}} # [[/Core emotions/]] - What are the core emotions and what is their function?{{Motivation and emotion/Book/High}} {{ME-By|U3229619}} # [[/Dark tetrad and emotion/]] - What is the dark tetrad and how does it relate to emotion? {{ME-By|U3214333}} # [[/Death and meaning in life/]] - How does confronting mortality influence the sense of meaning in life? {{ME-By|U3222680}} # [[/Default mode network and the self/]] - What is the relationship between the DMN and the self? {{ME-By|U3190506}} # [[/Deference emotion system/]] - What is the deference emotion system, how does it work, and what are its implications? {{ME-By|U3216127}} # [[/Eco-anxiety/]] - What is eco-anxiety, what are its consequences, and what can be done about it? {{ME-By|MT200107}} # [[/Ecopsychology and stress/]] - How can ecopsychology help to explain and deal with stress?{{Motivation and emotion/Multimedia/High}} {{ME-By|Dnamynts}} # [[/Ego death/]] - What is ego death, what are its consequences, and how can it be facilitated?{{Motivation and emotion/Book/Low}} {{ME-By|Gaissa}} # [[/Emotion across the lifespan/]] - How does emotion develop across the lifespan? {{ME-By|Taliastefanoski}} # [[/Emotional intelligence training/]] - How can emotional intelligence be trained? {{ME-By|RolandC2023}} # [[/Empathy and psychopathy/]] - What is the relationship between empathy and psychopathy? {{ME-By|U3218292}} # [[/Empathy training/]] - How can empathy be enhanced? {{ME-By|U3224924}} # [[/Equine therapy and emotion/]] - What is the role of emotion in equine therapy? {{ME-By|SoSilverLibby}} # [[/Flourishing in the elderly/]] - How can psychological flourishing be supported in the elderly?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|JorjaFive}} # [[/Guided meditation and emotion regulation/]] - How can guided meditation be used to regulate emotion?{{Motivation and emotion/Book/Low}} {{ME-By|U3215755}} # [[/Hedonic adaptation prevention model/]] - What is the HAP model and how can it be applied? {{ME-By|Siobhanu3216316}} # [[/Hijack hypothesis of drug addiction/]] - What is the hijack hypothesis and how does it help to understand drug addiction? {{ME-By|U3189442 - K.Ryan}} # [[/Humility/]] - What is humility, what causes it, and is it desirable?{{Motivation and emotion/Book/Low}} {{ME-By|Inandonit365}} # [[/Interoception and mental health/]] - What is the role of interoception in mental health? {{ME-By|U3229576}} # [[/Indigenous Australian mindfulness/]] - How has Indigenous Australian culture traditionally conceived of, and practiced, mindfulness?{{Motivation and emotion/Book/High}} {{ME-By|U3168525}} # [[/Long-term injury in high performance field athletes and emotion/]] - How do long-term injuries impact the emotions of high performance field athletes? {{ME-By|U3216619}} # [[/Meditation and mindfulness/]] - How can meditation contribute to mindfulness? {{ME-By|APMonte}} # [[/Mental health literacy/]] - What is mental health literacy, why does it matter, and how can it be fostered? {{ME-By|U3215387}} # [[/Mental imagery and emotion/]] - How can mental imagery be used to regulate emotion? {{ME-By|MaryamNageeb}} # [[/Mental toughness in the workplace/]] - How can mental toughness be useful in the workplace and how can it be developed? {{ME-By|KubeChu}} # [[/Moral outrage/]] - What is moral outrage, what causes it, what are its consequences, and how can it be managed? {{ME-By|U3215934}} # [[/Narrative therapy and emotion/]] - What is the role of emotion in narrative therapy?{{Motivation and emotion/Book/Low}} {{ME-By|Alyssa.marin}} # [[/Placebo effect and emotion/]] - What is the placebo effect and how can it affect emotion? {{ME-By|U3213568}} # [[/Positive body image development/]] - What is a positive body image and how can it be developed? {{ME-By|U3224053}} # [[/Psilocybin assisted therapy and anxiety/]] - How can psilocybin assisted therapy help to treat anxiety?{{Motivation and emotion/Multimedia/High}} {{ME-By|SammyTabrett}} # [[/Psychedelic treatment of anxiety/]] - How can psychedelics be used to help in treating anxiety? {{ME-By|SashaBrooksby}} # [[/Psychological literacy/]] - What is psychological literacy, why does it matter, and how can it be fostered?{{Motivation and emotion/Book/Low}} {{ME-By|DMaphosa}} # [[/Relief/]] - What is relief and what role does it play in our emotional lives? {{ME-By|U3216413(2023)}} # [[/Sadism and emotion/]] - What are the emotional aspects of sadism? {{ME-By|Kimberley Manyura}} # [[/Self-affirmation and psychological change/]] - How can self-affirmation be used for positive psychological change? {{ME-By|U3213910}} # [[/Self-discrepancy theory and emotion/]] - In what ways does self-discrepancy help to understand emotion? {{ME-By|U3173387}} # [[/Singing and emotion/]] - What is the relation between singing and emotion?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3213084}} # [[/Smiling and emotion/]] - What is the relationship between smiling and emotion? {{ME-By|U3230268}} # [[/Stress mindset/]] - What is stress mindset, why does it matter, and how can it be cultivated?{{Motivation and emotion/Book/Low}} {{ME-By|U3224687}} # [[/Sublimation/]] - What is sublimation and how can it be fostered? {{ME-By|U3185622}} # [[/Therapeutic recreation/]] - What is therapeutic recreation, how does it work, and what are its impacts? {{ME-By|MojoHides}} # [[/Toxic positivity/]] - What is toxic positivity, why does it occur, and how can it be managed? {{ME-By|Lauraei11}} # [[/Trauma and emotion/]] - What is the effect of trauma on emotion?{{Motivation and emotion/Book/High}} {{ME-By|Clarestuparich}} # [[/Tragic optimism/]] - What is tragic optimism, why does it matter, and how can it be fostered? {{ME-By|Alice hatcher }} # [[/Trauma-informed care/]] - What is trauma-informed care and how can it be implemented?{{Motivation and emotion/Multimedia/High}} {{ME-By|Nabila.Tursun}} # [[/Unconscious aspects of multiple selves and emotion/]] - What role do unconscious aspects play in shaping our multiple selves and their influence on emotion? {{ME-By|Natsta19}} # [[/Uncertainty and emotion/]] - How does uncertainty influence emotion? [[User:Mikaela MG|Mikaela MG]] # [[/Unemployment and depression/]] - What is the relationship between unemployment and depression? {{ME-By|U3213250}} # [[/Unemployment and mental health/]] - What is the relationship between unemployment and mental health? {{ME-By|U3211780}} # [[/Vicarious embarrassment/]] - What is vicarious embarrassment, why does it occur, and how can it be managed?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3216476}} # [[/Viewing natural scenes and emotion/]] - What is the effect of viewing natural scenes on emotion and how can this be applied? {{ME-By|Alyson131}} # [[/Yoga and emotional regulation/]] - How can yoga help with emotional regulation? {{ME-By|U3169170}} == Motivation and emotion == # [[/GABA, motivation, and emotion/]] - What is the neurological function of GABA for motivation and emotion? {{ME-By|U3227976}} # [[/Medial forebrain bundle, motivation, and emotion/]] - What are the motivational and emotional functions of the medial forebrain bundle? {{ME-By|U3226707}} # [[/Multiple selves theory, motivation, and emotion/]] - How does the concept of multiple selves impact our motivational and emotional experiences?{{Motivation and emotion/Book/Low}} {{ME-By|H.Maycock}} # [[/Overchoice, motivation, and emotion/]] - What are the motivational and emotional effects of overchoice? {{ME-By|Atu320270}} # [[/Role-playing games, motivation, and emotion/]] - What are the motivational and emotional characteristics of role-playing games?{{Motivation and emotion/Book/Low}} {{ME-By|U3224203}} # [[/Torture interrogation, motivation, and emotion/]] - What are the motivational and emotional aspects of torture interrogation and how do these impact on the effectiveness of torture interrogation? {{ME-By|U3204500}} [[Category:Motivation and emotion/Book/2023]] aeoz6j9vjbmokb6ths8e1gn1wu99hg1 2815007 2815004 2026-06-10T07:45:26Z Jtneill 10242 Move 1 abandoned topic to 2026 - # [[/Consumer emotion measurement/]] - How can consumer emotion be measured? 2815007 wikitext text/x-wiki {{/Banner}} == Motivation == # [[/Abusive supervision/]] - What is abusive supervision, what motivates it, and what can be done about it?{{Motivation and emotion/Book/Low}} {{ME-By|Lara.Begg}} # [[/Active transport motivation/]] - What motivates use of active transport and how can people be encouraged to use it?{{Motivation and emotion/Book/Low}} {{ME-By|U3214564}} # [[/Actively open-minded thinking/]] - How can AOT be used to improve human performance? {{ME-By|Msherrybarr}} # [[/Ageing and motivation/]] - What is the relationship between ageing and motivation? {{ME-By|U3217153}} # [[/Anomie and motivation/]] - How does anomie affect motivation?{{Motivation and emotion/Book/High}} {{ME-By|U3223909}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour?{{Motivation and emotion/Book/Low}} {{ME-By|U3216706}} # [[/Bedtime procrastination/]] - What causes it, what are the consequences, and what can be done about it?{{Motivation and emotion/Book/High}} {{ME-By|U3227684}} # [[/Brief motivational interviewing as a health intervention/]] - How can brief motivational interviewing be used as a health intervention?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|JasnaM}} # [[/Childhood trauma and criminal behaviour/]] - How does childhood trauma influence subsequent criminal behaviour?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3218323}} # [[/Comprehensive action determination model|Comprehensive action determination model]] - What is the CADM and how can it be applied to understanding human motivation?{{Motivation and emotion/Book/Low}} {{ME-By|Rocxie}} # [[/Conspiracy theory motivation/]] - What motivates people to believe in conspiracy theories?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3223114}} # [[/Construal level theory/]] - What is construal level theory and how can it be applied? {{ME-By|U3217803|U3217803}} # [[/Dark tetrad and motivation/]] - What is the dark tetrad and how does it relate to motivation? {{ME-By|U3223849}} # [[/Death drive/]] - What is the death drive and how can it be negotiated? {{ME-By|U3197031}} # [[/Dopamine and extraversion-introversion/]] - How does dopamine affect extraversion-introversion? {{ME-By|SaucyTuRkLeBiRd}} # [[/Egosystem and ecosystem motivation/]] - What are egosystem and ecosystem motivations and what are their consequences? {{ME-By|U3216125}} # [[/Effective leadership/]] - What does it take to be an effective leader? {{ME-By|U103189}} # [[/Episodic future thinking and delay discounting/]] - What is the relationship between between EFT and DD?{{Motivation and emotion/Book/High}} {{ME-By|LightDragonflies}} # [[/Episodic future thinking and delay discounting 2/]] - What is the relationship between episodic future thinking and delay discounting? {{ME-By|U3224582}} # [[/Episodic memory and planning/]] - What role does episodic memory play in planning? {{ME-By|Maheenusman}} # [[/Environmental volunteering motivation/]] - What motivates environmental volunteering?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U891419}} # [[/Freedom and motivation/]] - What is the effect of freedom on motivation?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3217955}} # [[/Gamer motivation/]] - What motivates game playing?{{Motivation and emotion/Multimedia/High}} {{ME-By|Bellehines}} # [[/Grit and achievement/]] - How does grit affect achievement?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} - [[User:AnaStuart|AnaStuart]] # [[/Growth mindset and mental health/]] - What is the relationship between growth mindset and mental health? {{ME-By|Mia Pearse }} # [[/Health belief model/]] - What is the HBM and how can it be used to enhance motivation for health-promoting behaviour?{{Motivation and emotion/Book/High}} {{ME-By|U3226487}} # [[/Identity fusion and motivation/]] - What are the motivational implications of identity fusion?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3235369}} # [[/IKEA effect/]] - What is the IKEA effect and how can it be applied?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3209567}} # [[/Impact bias/]] - What is impact bias, what causes it, what are its consequences, and how can it be avoided?{{Motivation and emotion/Book/Low}} {{ME-By|De2023}} # [[/Indigenous Australian engagement in mental health services/]] - What helps Indigenous Australians to engage with mental health services? {{ME-By|CaitlinFisher01}} # [[/Intertemporal choice/]] - What are intertemporal choices and how can they be effectively negotiated?{{Motivation and emotion/Book/Low}} {{ME-By|TD3223808}} # [[/Irrational belief motivation/]] - What motivates irrational beliefs?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|Concettazicc}} # [[/Köhler effect and motivation/]] - What is the Köhler effect and how can it be applied? {{ME-By|Declan Robinson}} # [[/Learning motivation and chatbots/]] - How can student motivation to learn be enhanced by using artificial intelligence chatbots? {{ME-By|Ashhstarr}} # [[/Life history theory and human reproductive strategies/]] - How does life history theory explain human reproductive strategies? {{ME-By|U3212234}} # [[/Mobile phone use motivation/]] - What are the motivations for mobile phone use?{{Motivation and emotion/Book/Low}} {{ME-By|Yaruki91}} # [[/Morning routine and motivation/]] - How can a morning routine be used to facilitate motivation, productivity, and well-being?{{Motivation and emotion/Book/Low}} {{ME-By|U3216585}} # [[/Motivational technologies/]] - How can motivation be fostered through technology? {{ME-By|U3196624}} # [[/Novelty-variety as a psychological need/]] - What is novelty-variety and what are its implications as a psychological need? {{ME-By|U3161239}} # [[/Nudge theory and sedentary behaviour/]] - How can nudge theory can be used reduce sedentary behaviour?{{Motivation and emotion/Book/High}} {{ME-By|U3145851}} # [[/Overchoice and motivation/]] - What is the overchoice effect, why does it occur, and who are most susceptible to it?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3230003}} # [[/Physical activity tracking and exercise motivation/]] - What role can physical activity tracking play in exercise motivation?{{Motivation and emotion/Book/Low}} {{ME-By|U3230383}} # [[/Physiological needs/]] - How do human's physiological needs affect motivation? {{ME-By|Superghostfresh}} # [[/Placebo effect and motivation/]] - What is the placebo effect and how can it affect motivation? {{ME-By|U3203797}} # [[/Protection motivation theory and COVID-19/]] - How does PMT apply to managing COVID-19?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3227759}} # [[/Sadism motivation/]] - What motivates sadism and what can be done about it? {{ME-By|BM3062384}} # [[/Signature strengths/]] - What are signature strengths and how can they be applied? {{ME-By|Rumesa.Aria}} # [[/Sport injury recovery motivation/]] - How can motivation be sustained when recovering from a sport injury? {{ME-By|U3210285}} # [[/SSRIs and motivation/]] - What are the effects of SSRIs on motivation?{{Motivation and emotion/Book/High}} {{ME-By|Josh.Cattle}} # [[/Stretch goals/]] - What are stretch goals? How do they work?{{Motivation and emotion/Book/Low}} {{ME-By|U3231448UC}} # [[/System justification theory/]] - What is SJT, how does it affect our lives, and what can be done about it? {{ME-By|Ishi}} # [[/Testosterone and dominance/]] - What is the relationship between testosterone and dominance? [[User:Giovannihbartlett|Giovannihbartlett]] # [[/Testosterone and violent crime/]] - What is the relationship between testosterone and violent crime?{{Motivation and emotion/Book/Low}} {{ME-By|U3200902}} # [[/Testosterone, winning, and losing/]] - What is the relationship between winning, losing, and testosterone? {{ME-By|Hugo DL}} # [[/Time management/]] - How can one's time be managed effectively? {{ME-By|Madison.Mayo}} # [[/To-do lists/]] - Are to-do lists a good idea? What are their pros and cons? How can they be used effectively?{{Motivation and emotion/Multimedia/High}} {{ME-By|U3162169T}} # [[/Treatment motivation in substance use disorder/]] - What is the role of treatment motivation in substance abuse disorder and how can it be enhanced?{{Motivation and emotion/Multimedia/High}} {{ME-By|KATE HERLIHY}} # [[/Violent extremism motivation/]] - What motivates people to engage in violent extremism? {{ME-By|U3216313}} # [[/Vocational identity/]] - What is vocational identity and how does it develop? {{ME-By|U3171379}} == Emotion == # [[/Artificial emotion/]] - To what extent can artificial intelligence experience emotion and how can this be applied? {{ME-By|KylieGiddings}} # [[/Attributions and emotion/]] - How do attributions affect emotion? {{ME-By|Carol Chariarse}} # [[/Autonomous sensory meridian response and emotion/]] - What emotions are involved in ASMR experiences and why do they occur? {{ME-By|Zmelmoth02}} # [[/Bewilderment/]] - What is bewilderment and how can it be dealt with? {{ME-By|Samanthagrebert}} # [[/Body functionality and body satisfaction/]] - What is the relationship between body functionality and body satisfaction? {{ME-By|Fatima2617}} # [[/Burn injury and body image/]] - What is the impact of burn injury on body image and what can be done? {{ME-By|Biancanappo}} # [[/Burnout/]] - What is burnout and how can be it be managed and prevented? {{ME-By|Grace (Tram) Chu}} # [[/Choice and emotion/]] - How does emotion affect choice? {{ME-By|Jingying Chen}} # [[/Climate change emotion/]] - What is the relationship between climate change and emotion?- [[User:Sireyes|Sireyes]] # [[/Commitment bias/]] - What motivates escalation of commitment even it does not lead to desirable outcomes? {{ME-By|U3203936}} # [[/Community resilience/]] - What is community resilience and how can it be fostered?{{Motivation and emotion/Book/High}} {{ME-By|U3213682}} # [[/Core emotions/]] - What are the core emotions and what is their function?{{Motivation and emotion/Book/High}} {{ME-By|U3229619}} # [[/Dark tetrad and emotion/]] - What is the dark tetrad and how does it relate to emotion? {{ME-By|U3214333}} # [[/Death and meaning in life/]] - How does confronting mortality influence the sense of meaning in life? {{ME-By|U3222680}} # [[/Default mode network and the self/]] - What is the relationship between the DMN and the self? {{ME-By|U3190506}} # [[/Deference emotion system/]] - What is the deference emotion system, how does it work, and what are its implications? {{ME-By|U3216127}} # [[/Eco-anxiety/]] - What is eco-anxiety, what are its consequences, and what can be done about it? {{ME-By|MT200107}} # [[/Ecopsychology and stress/]] - How can ecopsychology help to explain and deal with stress?{{Motivation and emotion/Multimedia/High}} {{ME-By|Dnamynts}} # [[/Ego death/]] - What is ego death, what are its consequences, and how can it be facilitated?{{Motivation and emotion/Book/Low}} {{ME-By|Gaissa}} # [[/Emotion across the lifespan/]] - How does emotion develop across the lifespan? {{ME-By|Taliastefanoski}} # [[/Emotional intelligence training/]] - How can emotional intelligence be trained? {{ME-By|RolandC2023}} # [[/Empathy and psychopathy/]] - What is the relationship between empathy and psychopathy? {{ME-By|U3218292}} # [[/Empathy training/]] - How can empathy be enhanced? {{ME-By|U3224924}} # [[/Equine therapy and emotion/]] - What is the role of emotion in equine therapy? {{ME-By|SoSilverLibby}} # [[/Flourishing in the elderly/]] - How can psychological flourishing be supported in the elderly?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|JorjaFive}} # [[/Guided meditation and emotion regulation/]] - How can guided meditation be used to regulate emotion?{{Motivation and emotion/Book/Low}} {{ME-By|U3215755}} # [[/Hedonic adaptation prevention model/]] - What is the HAP model and how can it be applied? {{ME-By|Siobhanu3216316}} # [[/Hijack hypothesis of drug addiction/]] - What is the hijack hypothesis and how does it help to understand drug addiction? {{ME-By|U3189442 - K.Ryan}} # [[/Humility/]] - What is humility, what causes it, and is it desirable?{{Motivation and emotion/Book/Low}} {{ME-By|Inandonit365}} # [[/Interoception and mental health/]] - What is the role of interoception in mental health? {{ME-By|U3229576}} # [[/Indigenous Australian mindfulness/]] - How has Indigenous Australian culture traditionally conceived of, and practiced, mindfulness?{{Motivation and emotion/Book/High}} {{ME-By|U3168525}} # [[/Long-term injury in high performance field athletes and emotion/]] - How do long-term injuries impact the emotions of high performance field athletes? {{ME-By|U3216619}} # [[/Meditation and mindfulness/]] - How can meditation contribute to mindfulness? {{ME-By|APMonte}} # [[/Mental health literacy/]] - What is mental health literacy, why does it matter, and how can it be fostered? {{ME-By|U3215387}} # [[/Mental imagery and emotion/]] - How can mental imagery be used to regulate emotion? {{ME-By|MaryamNageeb}} # [[/Mental toughness in the workplace/]] - How can mental toughness be useful in the workplace and how can it be developed? {{ME-By|KubeChu}} # [[/Moral outrage/]] - What is moral outrage, what causes it, what are its consequences, and how can it be managed? {{ME-By|U3215934}} # [[/Narrative therapy and emotion/]] - What is the role of emotion in narrative therapy?{{Motivation and emotion/Book/Low}} {{ME-By|Alyssa.marin}} # [[/Placebo effect and emotion/]] - What is the placebo effect and how can it affect emotion? {{ME-By|U3213568}} # [[/Positive body image development/]] - What is a positive body image and how can it be developed? {{ME-By|U3224053}} # [[/Psilocybin assisted therapy and anxiety/]] - How can psilocybin assisted therapy help to treat anxiety?{{Motivation and emotion/Multimedia/High}} {{ME-By|SammyTabrett}} # [[/Psychedelic treatment of anxiety/]] - How can psychedelics be used to help in treating anxiety? {{ME-By|SashaBrooksby}} # [[/Psychological literacy/]] - What is psychological literacy, why does it matter, and how can it be fostered?{{Motivation and emotion/Book/Low}} {{ME-By|DMaphosa}} # [[/Relief/]] - What is relief and what role does it play in our emotional lives? {{ME-By|U3216413(2023)}} # [[/Sadism and emotion/]] - What are the emotional aspects of sadism? {{ME-By|Kimberley Manyura}} # [[/Self-affirmation and psychological change/]] - How can self-affirmation be used for positive psychological change? {{ME-By|U3213910}} # [[/Self-discrepancy theory and emotion/]] - In what ways does self-discrepancy help to understand emotion? {{ME-By|U3173387}} # [[/Singing and emotion/]] - What is the relation between singing and emotion?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3213084}} # [[/Smiling and emotion/]] - What is the relationship between smiling and emotion? {{ME-By|U3230268}} # [[/Stress mindset/]] - What is stress mindset, why does it matter, and how can it be cultivated?{{Motivation and emotion/Book/Low}} {{ME-By|U3224687}} # [[/Sublimation/]] - What is sublimation and how can it be fostered? {{ME-By|U3185622}} # [[/Therapeutic recreation/]] - What is therapeutic recreation, how does it work, and what are its impacts? {{ME-By|MojoHides}} # [[/Toxic positivity/]] - What is toxic positivity, why does it occur, and how can it be managed? {{ME-By|Lauraei11}} # [[/Trauma and emotion/]] - What is the effect of trauma on emotion?{{Motivation and emotion/Book/High}} {{ME-By|Clarestuparich}} # [[/Tragic optimism/]] - What is tragic optimism, why does it matter, and how can it be fostered? {{ME-By|Alice hatcher }} # [[/Trauma-informed care/]] - What is trauma-informed care and how can it be implemented?{{Motivation and emotion/Multimedia/High}} {{ME-By|Nabila.Tursun}} # [[/Unconscious aspects of multiple selves and emotion/]] - What role do unconscious aspects play in shaping our multiple selves and their influence on emotion? {{ME-By|Natsta19}} # [[/Uncertainty and emotion/]] - How does uncertainty influence emotion? [[User:Mikaela MG|Mikaela MG]] # [[/Unemployment and depression/]] - What is the relationship between unemployment and depression? {{ME-By|U3213250}} # [[/Unemployment and mental health/]] - What is the relationship between unemployment and mental health? {{ME-By|U3211780}} # [[/Vicarious embarrassment/]] - What is vicarious embarrassment, why does it occur, and how can it be managed?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3216476}} # [[/Viewing natural scenes and emotion/]] - What is the effect of viewing natural scenes on emotion and how can this be applied? {{ME-By|Alyson131}} # [[/Yoga and emotional regulation/]] - How can yoga help with emotional regulation? {{ME-By|U3169170}} == Motivation and emotion == # [[/GABA, motivation, and emotion/]] - What is the neurological function of GABA for motivation and emotion? {{ME-By|U3227976}} # [[/Medial forebrain bundle, motivation, and emotion/]] - What are the motivational and emotional functions of the medial forebrain bundle? {{ME-By|U3226707}} # [[/Multiple selves theory, motivation, and emotion/]] - How does the concept of multiple selves impact our motivational and emotional experiences?{{Motivation and emotion/Book/Low}} {{ME-By|H.Maycock}} # [[/Overchoice, motivation, and emotion/]] - What are the motivational and emotional effects of overchoice? {{ME-By|Atu320270}} # [[/Role-playing games, motivation, and emotion/]] - What are the motivational and emotional characteristics of role-playing games?{{Motivation and emotion/Book/Low}} {{ME-By|U3224203}} # [[/Torture interrogation, motivation, and emotion/]] - What are the motivational and emotional aspects of torture interrogation and how do these impact on the effectiveness of torture interrogation? {{ME-By|U3204500}} [[Category:Motivation and emotion/Book/2023]] 7qb22jxl47o5gfpwa4vw2dqxi86m0i7 1 298415 2815003 2544168 2026-06-10T07:41:50Z Jtneill 10242 Jtneill moved page [[Talk:Motivation and emotion/Book/2023/Machiavellian motivation]] to [[Talk:Motivation and emotion/Book/2026/Machiavellian motivation]] without leaving a redirect 2544168 wikitext text/x-wiki ==Initial suggestions== {{ping|Casey Coulton}} Thanks for tackling this topic. Some initial suggestions: * Check out other related chapters and see how you can build on, link to, and integrate with that work: **[[:Category:Motivation and emotion/Book/Dark triad]] * What psychological theories can help to understand? What is the main research in this area? Let me know if I can do anything else as you go along. Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 08:10, 6 August 2023 (UTC) Category:Motivation and emotion/Book/Dark triad hlsl7wvvvgliq1n66vayj39ph5anxne 1 298567 2815006 2545310 2026-06-10T07:44:27Z Jtneill 10242 Jtneill moved page [[Talk:Motivation and emotion/Book/2023/Consumer emotion measurement]] to [[Talk:Motivation and emotion/Book/2026/Consumer emotion measurement]] without leaving a redirect 2545310 wikitext text/x-wiki ==Initial suggestions== {{ping|Rhys Harmer}} Thanks for tackling this topic. Some initial suggestions: * Check out other related chapters and see how you can build on, link to, and integrate with that work: ** [[:Category:Motivation and emotion/Book/Consumerism]] ** [[:Category:Motivation and emotion/Book/Emotion]] ** [[:Category:Motivation and emotion/Book/Measurement]] * What psychological theories can help to understand? What is the main research in this area? Let me know if I can do anything else as you go along. Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:01, 10 August 2023 (UTC) baqwv407ujrzscc3t721b7sq60bxkoz 1 298683 2814988 2546188 2026-06-10T07:26:32Z Jtneill 10242 Jtneill moved page [[Talk:Motivation and emotion/Book/2023/Fear of apocalypse]] to [[Talk:Motivation and emotion/Book/2026/Fear of apocalypse]] without leaving a redirect: Abandoned page in 2023; moving to 2026 2546188 wikitext text/x-wiki ==Initial suggestions== {{ping|AliciaEdwards1212}} Thanks for tackling this topic. Some initial suggestions: * Check out other related chapters and see how you can build on, link to, and integrate with that work: ** [[:Category:Motivation and emotion/Book/Environment]] ** [[:Category:Motivation and emotion/Book/Fear]] * What psychological theory(ies) can help to understand? What is the main research in this area? Let me know if I can do anything else as you go along. Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 09:57, 13 August 2023 (UTC) 9i9eu324pf8qwgoyacel3tstutj60nf 1 298704 2814992 2546421 2026-06-10T07:31:33Z Jtneill 10242 Jtneill moved page [[Talk:Motivation and emotion/Book/2023/Eco-emotions]] to [[Talk:Motivation and emotion/Book/2026/Eco-emotions]] without leaving a redirect 2546421 wikitext text/x-wiki ==Initial suggestions== {{ping|Emkatev}} Thanks for tackling this topic. Some initial suggestions: * Check out other related chapters and see how you can build on, link to, and integrate with that work: ** [[:Category:Motivation and emotion/Book/Climate change]] ** [[:Category:Motivation and emotion/Book/Ecopsychology]] ** [[:Category:Motivation and emotion/Book/Emotion]] ** [[:Category:Motivation and emotion/Book/Environment]] * What psychological theory(ies) can help to understand? What is the main research in this area? Let me know if I can do anything else as you go along. Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 08:39, 14 August 2023 (UTC) hhsqxtsuu7pf0irb00w16uv49sq8yxv 0 298722 2814976 2755088 2026-06-10T06:59:37Z Jtneill 10242 Remove 2 categories 2814976 wikitext text/x-wiki {{title|Role-playing games, motivation, and emotion:<br>What are the motivational and emotional characteristics of role-playing games?}} {{MECR3|1=https://youtu.be/Ji8hfKM_PPc}} __TOC__ ==Overview== [[File:Role playing gamers.jpg|thumb|'''Figure 1'''. ''A group playing Dungeons and Dragons (D&D), a role-playing game.''|286x286px]] Gaming has grown into a widely popular pastime. Some use it as a stress reliever after a long day of work, some play games for entertainment when socialising, and some simply play games because it's fun. As gaming has slowly become more involved in society, games themselves have developed into different genres such as [[wikipedia:_Role-playing_games|role-playing games]], [[wikipedia:_Virtual_reality_games|virtual reality games]], [[wikipedia:_First-person_shooters|first-person shooters]], and even more{{Inline comment|include all the classic types of games as well, mention the age old existence of role playing games}}. The more games have developed, in both quality and genre, discussions on whether gaming was a good or bad thing came to light. Do games have a negative influence on people and their emotions? This chapter looks into that argument in more detail, focusing on what positive emotions are, what negative emotions are, and how these relate to gaming{{Inline comment|requires citation somewhere in the paragraph}}. This chapter also discusses the motivational aspects of gaming, such as why people game, why people continue to game, and how gaming can help with an individual's psychological needs. To bring everything together, a discussion on ways that gaming can be used to improve our lives via emotional and motivational avenues to try and close out our three main focus questions: what are the emotional characteristics of gaming?; what are the motivational characteristics of gaming?; and how can we apply gaming to real life? {{RoundBoxTop|theme=3}} ;Focus questions # What are the emotional characteristics of gaming? # What are the motivational characteristics of gaming? # How can we apply gaming to real life? {{RoundBoxBottom}} == Emotional characteristics == A significant aspect of gaming culture revolves around the emotions experienced both within the game and those felt by players during gameplay.{{Citation needed}}. Many of the large, successful games in society often have very emotional scenes that can draw in the audience and enhance the overall gameplay experience. Below, is an expanded look at positive and negative emotions and emotional attachment to characters. === Positive emotions === Before we delve into what [[wikipedia:Positive_affectivity|positive emotion]]<nowiki/>s we can experience from games, we need to establish what positive emotions are in the first place. Most people think of joy and happiness when they hear the words “positive emotions”, which isn't necessarily wrong, but positive emotions can also be more obscure. They can also include emotions such as surprise, interest, pride, triumph and so much more, it is arguable that any emotion that can induce [[wikipedia:Dopamine|dopamine]] or has the capacity to induce dopamine can be called a positive emotion, as stated in the Handbook of Emotions (Fredrickson & Cohn, 2008). Some positive emotions can come from simply seeing other people display a positive emotion, for instance seeing someone can influence us to smile, something that we call “[[wikipedia:Mimicry|mimicry]]” or, in some cases, the chameleon effect (Chartrand & Bargh, 1999). This effect can also come from fictional characters within a game{{Rewrite}}, meaning that through mimicry we can be influenced to feel what the characters are feeling, whether it be our own custom-made character, or a Non-Playable Character (NPC). To further expand on that, however, the positive emotions that we gain from gaming don't necessarily come from what the characters are feeling. Our personal emotions and characters' emotions are not mutually exclusive, as we can feel joy, pride, and triumph from certain achievements or actions that surround a game, such as beating that one really hard boss fight, earning a really rare trophy or rare item, to even just finishing the game itself. Sometimes, characters can also just say and do things that we find funny, encouraging us to laugh and enjoy the game further. === Negative emotions === Just as we can experience positive emotions, we can experience negative ones as well. [[wikipedia:Negative_emotion|Negative emotions]] can seemingly range from anger and rage to envy and contempt to disgust and embarrassment, but interestingly, it is difficult to find what exactly a “negative emotion” is as most people tend to allocate disorders such as [[wikipedia:Depression|depression]] and [[wikipedia:Anxiety|anxiety]] as the root causes of “negative’ moods and emotion (Kristjánsson, 2003). This is partly due to the fact that emotions themselves aren't necessarily just either negative or positive, this is followed up later. Negative emotions can also be experienced from playing games in many different ways. To begin with, mimicry plays a role in experiencing emotions, just as it does with positive emotions. Alongside mimicry, however, is our ability to feel [[wikipedia:Sympathy|sympathy]] and [[wikipedia:Empathy|empathy]], for instance, if we see a character that is upset because a close friend or family member died, then we are able to sympathise with them and feel sad. To move away from the characters and plot-related emotions, frustration and anger can be experienced from gaming as well, in the form of not being able to complete a quest, continually dying from a fight, fear and shock can also be experienced often when playing [[wikipedia:Horror_games|horror games]]. {{Inline comment|would benefit from a fictional case study to highlight the emotions being discussed}} Negative emotions tend to get a bad reputation simply because of the word ‘negative’, but are they really all that bad? Emotions are too complex to really group together as Negative and Positive, they can be either, or even both at the same time in some weird circumstances. In fact, when it comes to gaming, it has been widely reported that when a game is able to induce negative emotions from a story or plot-line based perspective, it enhances the overall enjoyment of a game (Bopp et al., 2016). Having negative emotions being portrayed in media such as games, specifically role-playing games that follow a storyline allows us to become attached to characters from a sympathetic standpoint. The ability to invoke any emotion, positive or negative is a sign of a well-done literature and/or game. [[File:Character attachment.png|thumb|Figure 2. Different kinds of character attachment]] === Emotional attachment to characters === A large reason that we can become so emotional towards games is because we can become emotionally attached to the characters or the game itself on an emotional level. [[wikipedia:Emotional_attachment|Emotional attachment]] is typically defined as “the emotional bond connecting an individual with a specific target” (Jiménez & Voss, 2014), that target, in this case, is games and their characters. Emotional attachment can range from having a bond with a character that you made and designed, this type of character being coined the term “customisable character”, or “ custom character” for short. A connection is often formed with these types of characters because, as a player, you spend time creating and shaping them to look and feel how you want. In some games, such as ''[[wikipedia:Dungeons_&_Dragons|Dungeons & Dragons]], [[wikipedia:_Baldur's_Gate_3|Baldur’s Gate 3]],'' and others, you also create and develop the past of a character as well. In other games, like ''[[wikipedia:The_Sims|The Sims]]'', for example, you create a character to your liking, and then watch them grow old, and can then even play as their future generations creating a bond with the family you have made rather than a singular character. Quite often, people will also form an emotional attachment to pre-made characters, including both the main character and the supporting cast. Sometimes, this is simply due to time playing the game and growing fond of the character, but it can also be because a player may see themselves in the character, or in other words, they feel as if they relate to this character and the hardships they have been through or the way they handle different situations.{{fact}} It is important to differentiate between an emotional attachment to a character that stems from fondness and relatability and an emotional attachment born from [[wikipedia:Obsession|obsessions]]. Attachments from fondness and relatability tend to be seen as “healthy attachments”, whereas one born from obsession is “unhealthy” and in some extreme cases and create delusions that the character is real and has feelings toward the player, platonic or otherwise{{fact}}. Figure 2 shows different kinds of attachments and emotions associated with characters and character types that are often viewed as “healthy”. == Motivational characteristics == To engage in a game, an individual must first feel motivated to start and then maintain that motivation to continue playing. Without motivation, there is no drive and want, there is no interest. An individual can have family members who play games that can influence them, and that alone might be able to help motivate them to start a game but not necessarily to continue. Below is a closer look at what can motivate an individual to play games. === Self-determination theory === One of the most renowned motivational theories known to date is Ryan and Deci’s [[wikipedia:Self-determination_theory|Self-determination theory]] (Ryan & Deci, 2000). Self-determination theory surrounds the idea that an individual's motivational drive stems from their need for [[wikipedia:Autonomy|autonomy]], [[wikipedia:Competence|competence]], and [[wikipedia:Relatedness|relatedness]]. Autonomy is typically regarded as one’s ability and perceived ability to make a choice and decision, competence is the ability to complete an action and tie in with [[wikipedia:Self-efficacy|self-efficacy]], and relatedness is the need to feel connected with others on a social level. Below will expand on these three motivational characteristics in relation to gaming. === Autonomy === Autonomy in gaming comes in the form of being able to take control of certain aspects of a character. This can be what they look like and their past if they are a customisable character, or simply just being able to control the actions that said character has. Many games let you choose which direction to take the plot with multiple different endings, they allow different dialogue options. Games such as ''[[wikipedia:_The_Witcher_3:_Wild_Hunt|The Witcher 3: Wild Hunt]], [[wikipedia:_Baldur's_Gate_3|Baldur’s Gate 3]],'' and ''[[wikipedia:_Red_Dead_Redemption_2|Red Dead Redemption 2]]'' are all games that we categorise as “open-world”, meaning that you can explore anywhere on the map of the game that you choose, raising an individual's sense of autonomy and personal agency. Autonomy is still rather associated with customisation characteristics of gaming in an overall view (Kim et al., 2015){{Vague}} === Competence === Competence in gaming contains a broader range of aspects compared to autonomy. Where autonomy focuses on personal control, competence focuses on the area of personal ability. For gaming, this can be the belief of finally getting past that one boss fight, winning that really rare achievement, and simply finishing the game. People inherently love to be challenged, and challenging sections of games, whether it be confusing puzzles, different plot choices, or tough boss fights, all feed that love of challenge which is rather consistent with the theories surrounding competence and thus enhance the overall enjoyment of a game (Boyle et al., 2011). Most gaming platforms also have an Achievement system which is known as Trophies. These achievements are rewards that are given when you complete certain tasks, sometimes these are plot-related and sometimes they are battle or item-related. Most platforms have a percentage-based aspect of these achievements, showing the percentage of players that have been able to gain or unlock the achievement, allowing for competitive motivation to complete these achievements as a show of competence. === Relatedness === There are two ways that the need for relatedness expresses itself in gaming, but they both center around the need for connection, typically socially{{Citation needed}}. There is a wide genre of games called [[wikipedia:_Massively_multiplayer_online_role-playing_game|Massive Multiplayer Online Role Playing Games]], MMORPG for short. These games allow you to follow a storyline or plot while also interacting with other players in a social aspect. MMORPGs facilitate the formation of new relationships, often through team collaborations. This format not only provides social connections but also enhances [[wikipedia:_Teamwork|teamwork]] and development. Another way that gaming can influence the need for relatedness is the above-mentioned emotional attachment. Sometimes, when a person becomes attached to a character, they feel motivated to play their game more as a way of feeling connected to someone, despite being fictional, on a more personal level. Or alternatively, if they have yet to find a character that they are fond of or are attached to on a personal level due to feelings of relatedness, they might be more motivated to play more games until they find that character for themselves{{Fact}}. == Real world application == As much as we can talk about why people play games, specifically from an emotional and motivational standpoint, there has been much debate on whether games are a negative or positive influence. To a degree, this may be true, as children can learn through modeling and observation, and many games have violent actions, which can then influence children to be violent themselves. Gaming’s influence, however, is not simple enough to simply label as positive and negative, below is a look at some of the positive ways that gaming can help improve our lifestyle. === Emotional and cognitive training === Many people often struggle to identify what they're feeling, and sometimes other people as well, especially if it is a negative emotion. Many plot-based games have emotional conflict, whether that be character death, relationship fallouts, betrayal, to simple reactions of stress. As spoken before, many people can relate to characters on an emotional level, and sometimes this identification can help people identify how they feel about a personal issue in their life. Studies have shown that gaming can actually help influence emotional stability and cognitive training depending on the genre. A review was done of recent literature in 2018 that compiled all the testing that had gone into research on emotional and cognitive training. They found that action and adventure games helped produce a higher processing speed, [[wikipedia:_Reaction_time|reaction time]], general [[wikipedia:_Memory|memory]], and multitasking. They also discovered that specifically relating to horror games, the influence of the stress-induced moments helps those with trait anxiety symptoms to, in the future, have better behavior performance in anxious-related stress tasks (Pallavicini et al., 2018). Positive character relation and the positive influence that gaming has on cognitive abilities and emotional stability is overall a very positive aspect of gaming. These aspects show that gaming can be used as an educational tool when it comes to emotion. === Gamification === [[wikipedia:_Gamification|Gamification]] is a learning tool that has been developed from a gaming structure. It is important to note that games and gamification are two very different things. Gamification is a learning system that has game-like qualities, such as “side-quests” or “objectives” rather than a simple “to-do list” (Kapp, 2012). Gamification is used for professional environments, more in organisations, whereas game-learning is less professionally structured and more aimed for children and developmental learning, so it focuses more on actual games and game qualities over gamification. === Possible therapy tool === As mentioned above, certain game genres are able to influence anxiety and stress behaviours. There have been studies that have shown that gaming can be beneficial for specific phobia types and traumatic situations, such as certain animals and insects to situations such as driving after being in a car accident (Walshe et al., 2003). Gaming can be used as a form of [[wikipedia:_exposure_therapy|in vivo exposure therapy]] and a good example of this is the ''[[wikipedia:_Hogwarts_Legacy|Hogwarts Legacy]]'' game released in 2023. In ''Hogwarts Legacy,'' there is an option to turn on settings for those who have arachnophobia, a phobia of spiders. This setting changes all spiders within the game to change appearance. This setting could allow an individual with arachnophobia to slowly ease into their fear without actually being in the presence of a spider, as the option can be turned on and off. ==Conclusion== Closing out this article let's circle back to emotional characteristics. You can experience a wide variety of emotions from gaming, both negative and positive such as sadness, anger, pride, joy, triumph, contempt, and disgust, through a game's storyline, or different gaming aspects such as achievements. Motivation for gaming very much centers around the self-determination theory and an individual’s need for autonomy, competence, and relatedness which can all be achieved via gaming. Lastly, gaming can be used to help with cognitive training and developing emotional stability as well as a form of in vivo exposure. It can also be used within businesses and organisations through forms of gamification and game-learning to help enhance learning and involvement. Generally, games can still have a negative effect on people, the death of a favourite character can cause emotional stress, and children can learn bad and violent behaviour through modeling and observation, but gaming still also has a very positive influence on our lives that many people don't consider, and more often than not, outweigh the negative influence it can have. ==See also== * [[wikipedia:Self-determination_theory|Self-determination theory]] (Wikipedia) * [[wikipedia:Gamification|Gamification]] (Wikipedia) * [[Motivation and emotion/Book/2015/Video games and emotion|Video games and emotion]] (Book chapter, 2015) ==References== {{Hanging indent|1= Bopp, J. A., Mekler, E. D., & Opwis, K. (2016, May). Negative emotion, positive experience? Emotionally moving moments in digital games. ''In Proceedings of the 2016 CHI conference on human factors in computing systems'', 2996-3006. https://10.1145/2858036.2858227 Boyle, E., Connolly, T. M., & Hainey, T. (2011). The role of psychology in understanding the impact of computer games. ''Entertainment computing, 2''(2), 69-74. https://doi.org/10.1016/j.entcom.2010.12.002 Chartrand, T. L., & Bargh, J. A. (1999). The chameleon effect: The perception–behavior link and social interaction. ''Journal of personality and social psychology, 76''(6), 893. https://doi.org/10.1037/0022-3514.76.6.893 Fredrickson, B. L., & Cohn, M. A. (2008). ''Positive emotions.'' Handbook of emotions, 3, 777-796. Jiménez, F. R., & Voss, K. E. (2014). An alternative approach to the measurement of emotional attachment. ''Psychology & Marketing, 31''(5), 360-370. https://doi.org/10.1002/mar.20700 Kapp, K. M. (2012). ''What is gamification. The gamification of learning and instruction: Game-based methods and strategies for training and education, 1-23.'' Research Gate. Kim, K., Schmierbach, M. G., Chung, M. Y., Fraustino, J. D., Dardis, F., & Ahern, L. (2015). Is it a sense of autonomy, control, or attachment? Exploring the effects of in-game customization on game enjoyment. ''Computers in Human Behavior, 48'', 695-705. https://doi.org/10.1016/j.chb.2015.02.011 Kristjánsson, K. (2003). On the very idea of “negative emotions”. ''Journal for the Theory of Social Behaviour, 33''(4), 351-364. https://doi.org/10.1046/j.1468-5914.2003.00222.x Pallavicini, F., Ferrari, A., & Mantovani, F. (2018). Video games for well-being: A systematic review on the application of computer games for cognitive and emotional training in the adult population. ''Frontiers in psychology, 9,'' 2127. https://doi.org/10.3389/fpsyg.2018.02127 Ryan, R. M., & Deci, E. L. (2000). Self-determination theory and the facilitation of intrinsic motivation, social development, and well-being. ''American psychologist, 55''(1), 68. https://doi.org/10.1037/0003-066X.55.1.68 Walshe, D. G., Lewis, E. J., Kim, S. I., O'Sullivan, K., & Wiederhold, B. K. (2003). Exploring the use of computer games and virtual reality in exposure therapy for fear of driving following a motor vehicle accident. ''CyberPsychology & Behavior, 6''(3), 329-334. https://doi.org/10.1089/109493103322011641 }} ==External links== * [https://youtu.be/WlV_PZFRP70?si=aY1ZlhGEH-jehpcM The emotional effect of video games] (YouTube - Wes Keltner) * [https://youtu.be/FktsFcooIG8?si=UKQVUahshz7q9noD Your brain on video games] (YouTube - Daphne Bavelier) * [https://www.fantasygroundsacademy.com/post/mental-health-benefits-of-role-playing-games#:~:text=RPG%20provides%20an%20active%2C%20constructive,and%20sensations%20that%20are%20occurring. Mental health benefits of role-playing games] (Website, Fantasy grounds academy) * [https://gamequitters.com/how-gaming-affects-your-mental-health/ Video games and mental health: how gaming affects your mental health] - (Website, Game Quitters) * [https://wellbeings.org/the-impact-of-gaming-on-mental-health/ The impact of gaming on mental health] - (Website, Well Beings) * [https://www.webmd.com/mental-health/mental-health-benefits-of-video-games Are there mental health benefits of video games?] - (Website, WebMD) * [https://www.barnardos.org.uk/blog/why-gaming-better-your-mental-health-you-might-think Why gaming is better for your mental health than you might think] - (Website, Barnardo's) * [https://www.youtube.com/watch?v=VxU5epY91zw Video game quotes with deep meaning] - (Youtube, Brian Diaz) * [https://www.youtube.com/watch?v=aYuU1HjftF8&t=251s Top 10 speeches in video games (Mass Effect, Gears of War, Halo, and More!)] - (Youtube, Vain'sVids) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Games/Role-playing]] [[Category:Motivation and emotion/Book/Social psychology]] mi91o6xr9t14o229zo3jzo5epvmydou 0 298896 2814989 2723659 2026-06-10T07:28:58Z Jtneill 10242 Move 1 abandoned topic to 2026 - # [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? 2814989 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Abusive supervision/]] - What motivates it and how does it affect workplace motivation? {{ME-By|TJDuus}} # [[/Adverse childhood experiences and risk-taking motivation/]] - How do ACEs affect risk-taking motivation?{{Motivation and emotion/Book/High}} {{ME-By|Krobertsonn}} # [[/Affective touch and emotion/]] - What are the emotional responses to affective touch? {{ME-By|Ratidzo Nyangu}} # [[/Analysis paralysis/]] - What causes analysis paralysis, what are its psychological mechanisms, and what are its impacts? {{ME-By|Yashvi M}} # [[/Antisocial behaviour in children/]] - What motivates antisocial behaviour in children? {{ME-By|U3191574 (PHP)}} # [[/Autism and motivation/]] - How does autism influence motivation?{{Motivation and emotion/Multimedia/High}} {{ME-By|Sophiedriscoll}} # [[/Better-than-average-effect/]] - Why do people tend to think they are better than average, what are the consequences, and what can be done about it? {{ME-By|RussellP1}} # [[/Biofeedback and motivation/]] - How does biofeedback influence motivation? {{ME-By|U3239962}} # [[/Bottom-line mentality and motivation/]] - How does a focus on the bottom line affect motivation and behaviour? {{ME-By|Biancaguina}} # [[/Breaking bad habits/]] - What strategies can be effective for breaking bad habits? {{ME-By|Pelleot}} # [[/Comprehensive action determination model|Comprehensive action determination model]] - What is the CADM and how can it be applied to understanding human motivation? {{ME-By|HassanAlsamara}} # [[/Consumer neuroscience/]] - What is consumer neuroscience and how does it contribute to understanding consumer behaviour? {{ME-By|Claudiaread}} # [[/Cultural variations in power motivation/]] - How does culture influence power motivation and its expression? {{ME-By|DFaol}} # [[/Death and motivation/]] - How does awareness of mortality influence motivation? {{ME-By|Krutipatil}} # [[/Dopamine and decision making/]] - How does dopamine influence decision making and what are the motivational implications? {{ME-By|Carter.lizzie}} # [[/Dopamine and learning/]] - What is the relationship between dopamine and learning? {{ME-By|Rhys Harmer}} # [[/Dopamine and social behaviour/]] - What role does dopamine play in social interactions and relationships?{{Motivation and emotion/Book/High}} {{ME-By|DeenMisic}} # [[/E-cigarette use motivation/]] - What motivates starting and continuing vaping of nicotine e-cigarettes?{{Motivation and emotion/Book/Low}} {{ME-By|U3242988}} # [[/Effort justification/]] - What is it and how does it influence perception and motivation? {{ME-By|Bstonehouse}} # [[/Employee recognition and work motivation/]] - What is the impact of employee recognition on work motivation?{{Motivation and emotion/Book/High}} {{ME-By|U3235875}} # [[/Environmental cues and habits/]] - How do environmental cues facilitate habit formation? {{ME-By|MoniqqueK}} # [[/ERG theory/]] - What is Alderfer's ERG theory?{{Motivation and emotion/Book/Low}} {{ME-By|U3173387}} # [[/Fandom motivation/]] - What motivates fandom? {{ME-By|TEGIANN}} # [[/Fogg behaviour model/]] - How can it be applied to understanding and changing behaviour?{{Motivation and emotion/Book/Low}} {{ME-By|HannahMooney}} # [[/Free will and neuroscience/]] - What does neuroscience reveal about the concept of free will? {{ME-By|Sienna04}} # [[/Functional approach to volunteerism/]] - How does the functional approach explain volunteer motivation and how can it be applied? {{ME-By|U32314488}} # [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|Samanthagrebert}} # [[/Gender differences in sexual motivation/]] - What are the key differences in sexual motivation between genders and what causes these differences?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3200844}} # [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes?{{Motivation and emotion/Book/Low}} {{ME-By|Rocxie}} # [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|U3194813}} # [[/Grit and conscientiousness/]] - What are the similarities and differences between grit and conscientiousness and what are the implications? {{ME-By|U3229789}} # [[/Groups and individual motivation enhancement/]] - How can group dynamics enhance individual motivation? {{ME-By|CharlotteLane55}} # [[/Groups and individual motivation reduction/]] - How can group dynamics diminish or undermine individual motivation?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3216883}} # [[/Habit stacking/]] - What is habit stacking and how can it enhance productivity?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3228753}} # [[/Inner talk in achieving high performance/]] - How does inner dialogue influence motivation for high achievement? {{ME-By|OscarDillion}} # [[/Inner voice and motivation/]] - How does the inner voice influence motivation? {{ME-By|U3248317}} # [[/Intentional pregnancy motivation/]] - What motivational factors contribute to intentional pregnancy?{{Motivation and emotion/Multimedia/High}} {{ME-By|Biancagouws}} # [[/Intertemporal choice/]] - What are intertemporal choices and how can they be effectively negotiated? {{ME-By|HDurban}} # [[/Intrinsic motivation and creativity/]] - What is the relationship between intrinsic motivation and creativity? {{ME-By|Trodrickk}} # [[/Intrinsic rewards and motivation/]] - How do intrinsic rewards affect motivation? {{ME-By|Zwyndham}} # [[/Justice motivation/]] - What is justice motivation, what causes it, and how is it pursued and satisfied? {{ME-By|Emeweme}} # [[/Lateral violence among Indigenous Australians/]] - What drives it and how can it be addressed? {{ME-By|Smritiadhikari}} # [[/Learned industriousness and motivation/]] - How does learned industriousness influence motivation and work ethic? {{ME-By|Persimmon24}} # [[/Machiavellianism and power motivation/]] - What is the relationship between Machiavellianism and power motivation?{{Motivation and emotion/Book/High}} {{ME-By|Joan-E-1405}} # [[/Money and motivation/]] - How does financial incentive influence motivation? {{ME-By|EmmanuelPaps}} # [[/Moral motivation/]] - What drives people to act according to moral principles? {{ME-By|Sophieteferi}} # [[/Morbid curiosity/]] - What is morbid curiosity, what causes it, and how does it influence behaviour?{{Motivation and emotion/Book/Low}} {{ME-By|Scarlett Neilson}} # [[/Morning routine and motivation/]] - How can a morning routine be used to facilitate motivation, productivity, and well-being?{{Motivation and emotion/Book/High}} {{ME-By|Syeda Aiman Ali}} # [[/Motivation in retirement/]] - What motivational challenges do retirees face and how can they be addressed? {{ME-By|U3243172}} # [[/Nudge theory/]] - What is nudge theory and how is it used to motivate behaviour change?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3225022}} # [[/Oxytocin and motivation/]] - How does oxytocin affect motivation? {{ME-By|U3232264}} # [[/Pain and motivation/]] - How does pain influence motivation? {{ME-By|Gabriel Geld}} # [[/Pain and placebo/]] - What is the placebo effect in pain management and how does it work? {{ME-By|U3230258}} # [[/Paradoxical interventions/]] - How can they facilitate psychological change?{{Motivation and emotion/Multimedia/High}} {{ME-By|Seth Guglielmin}} # [[/Physical activity tracking and exercise motivation/]] - What role can physical activity tracking play in exercise motivation? {{ME-By|Alyssia Myers}} # [[/Political motivation/]] - What motivates engagement in politics? {{ME-By|IvaPuskarica}} # [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness?{{Motivation and emotion/Book/Low}} {{ME-By|Zainab Zaman}} # [[/Productivity and routines/]] - How can establishing routines improve productivity?{{Motivation and emotion/Multimedia/High}} {{ME-By|U3236610 Allana}} # [[/Rivalry in the workplace/]] - What impact does workplace rivalry have on employee motivation and productivity? {{ME-By|Boomboompow4}} # [[/Self-determination theory and environmental activism/]] - How does self-determination theory explain motivation in environmental activism? {{ME-By|Zahra Karim}} # [[/Sleep and ego depletion/]] - How does sleep affect the capacity for self-control and willpower?{{Motivation and emotion/Book/Low}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3229957}} # [[/Social dominance and motivation/]] - What drives social dominance motivation, and how does it influence behaviour? {{ME-By|Cophiesollins}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation?{{Motivation and emotion/Book/Low}} {{ME-By|Dragonmoon05}} # [[/Sound and perception of food and drink/]] - How does sound influence the taste and enjoyment of food and drink?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|Ebrixon}} # [[/Stockholm syndrome motivation/]] - Why do captives bond with their captors?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3243508}} # [[/Temporal motivation theory/]] - What is temporal motivation theory and how does it influence decision-making? {{ME-By|U3236663}} # [[/Testosterone and violent crime/]] - What is the relationship between testosterone and violent crime?{{Motivation and emotion/Multimedia/High}} {{ME-By|Amarshall757}} # [[/Theoretical domains framework/]] - What is the TDF and how can it be used to guide behaviour change?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Book/High}} {{ME-By|Kiara Stone (UC)}} # [[/Treatment motivation in juvenile delinquency/]] - What is the role of treatment motivation for juvenile delinquency and how can it be enhanced?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3236338}} # [[/Truth serum drugs/]] - What are truth serum drugs, do they work, and how?{{Motivation and emotion/Multimedia/High}} {{ME-By|U3236669}} # [[/Ulysses pact and motivation/]] - What is the Ulysses pact and how does it relate to motivation? {{ME-By|L.Mercer}} # [[/Underdog versus favourite motivation/]] - How do motivation dynamics differ between underdogs and favourites? {{ME-By|Sarah Pendlebury}} # [[/Vasopressin and motivation/]] - How does vasopressin influence motivation?{{Motivation and emotion/Book/Low}} {{ME-By|U3012923}} ==Emotion== # [[/Abusive relationships and emotional dependency/]] - What psychological mechanisms drive emotional dependency and how does it perpetuate abusive relationships?{{Motivation and emotion/Book/Low}} {{ME-By|Lucywilson 546}} # [[/ADHD and emotional regulation/]] - How does ADHD affect emotional regulation and how can this be managed? {{ME-By|U3188383}} # [[/Adverse childhood experiences and emotion regulation/]] - What is the relationship between ACEs and emotion regulation? {{ME-By|U3242902}} # [[/Artificial intelligence empathy/]] - How can algorithms relate to humans? {{ME-By|U3249300}} # [[/Attribution theory and emotion/]] - How do attributions influence emotion? {{ME-By|Spring Pavlova}} # [[/Ayahuasca and the brain/]] - What are the neurological effects of ayahuasca? {{ME-By|Mkatemoore}} # [[/Bibliotherapy and emotion/]] - How can reading literature impact emotional well-being? {{ME-By|GurmeenK}} # [[/Breaking bad news/]] - How should bad news be shared to minimise emotional distress? {{ME-By|Stongu}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation?{{Motivation and emotion/Book/Low}} {{ME-By|MGibb.23}} # [[/Connection to country and well-being/]] - What is the relationship between connection to country and well-being? {{ME-By|U3237101}} # [[/Conservation of resources theory and stress/]] - How does COR theory explain stress and how can it be applied? {{ME-By|U3054914}} # [[/Digital emotion regulation/]] - How can digital tools and technologies be used to help manage emotions?{{Motivation and emotion/Book/High}} {{ME-By|U3237780}} # [[/Disgust and hygiene/]] - How does disgust influence hygiene behaviours and what are the psychological mechanisms involved?{{Motivation and emotion/Multimedia/High}} {{ME-By|U3214564}} # [[/Ecopsychological approaches to trauma/]] - How can nature-based therapies assist in emotional healing from trauma?{{Motivation and emotion/Book/High}} {{ME-By|Beabosborne}} # [[/Ego death/]] - What is ego death, what are its consequences, and how can it be facilitated?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|Danikollas}} # [[/Ego development and psychological growth/]] - How does ego development theory explain psychological growth? {{ME-By|Ryanjboulton}} # [[/Ego resilience/]] - What is ego resilience and how does it affect psychological functioning? {{ME-By|U322995}} # [[/Emotional literacy/]] - What is emotional literacy, why is it important, and how can it be developed?{{Motivation and emotion/Book/High}} {{ME-By|RBasu3243278}} # [[/Emotional self-care/]] - What is emotional self-care, why is it important, and what techniques are effective? {{ME-By|Shanaya M}} # [[/Emotion and time perception/]] - How does emotion affect perceptions of time? {{ME-By|U3243357}} # [[/Emotion preferences/]] - What do people want to feel and why? {{ME-By|U3236421}} # [[/Empathy versus sympathy/]] - What's the difference and how do they influence behaviour and relationships? {{ME-By|U3236683}} # [[/Exteroception and emotion/]] - What is the relationship between exteroception and emotional experience?{{Motivation and emotion/Book/Low}} {{ME-By|U3239399}} # [[/Fear of judgement and social media/]] - How does the fear of being judged undermine posting on social media and how can this be overcome?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|Ashdruett}} # [[/Goals and emotion/]] - What is the relationship between goals and emotion?{{Motivation and emotion/Book/High}} {{ME-By|U3236641}} # [[/Gut-brain axis and emotion/]] - What is the gut-brain axis and how does it influence emotion?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3239091}} # [[/Heart rate variability and emotion regulation/]] - How can HRV monitoring be used to regulate emotion? {{ME-By|U3239156dej}} # [[/Heart rate variability and mental health/]] - What is the relationship between HRV and mental health? {{ME-By|U3190194}} # [[/Humiliation/]] - What leads to it, what role does it play, and how can it be managed? {{ME-By|Stluciamolly}} # [[/Humility/]] - What is humility, what causes it, and is it desirable?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3237728}} # [[/Humour and social bonding/]] - How does humour facilitate social bonding? {{ME-By|Lwinter1}} # [[/Humour and stress relief/]] - How does humour help in relieving stress? {{ME-By|U3236303}} # [[/Impact bias/]] - What causes it, what are its consequences, and how can it be overcome?{{Motivation and emotion/Book/High}} {{ME-By|U3242909}} # [[/Inner voice and emotion/]] - How does the inner voice influence emotion? {{ME-By|StephenBlume}} # [[/Life satisfaction and personality/]] - What is the relationship between personality and life satisfaction?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3242288}} # [[/Moral dilemmas and emotion/]] - How can emotion aid or hinder in moral decision making? {{ME-By|U3249491}} # [[/Nature connection and emotional well-being/]] - How does connecting with nature influence emotional well-being?{{Motivation and emotion/Multimedia/High}} {{ME-By|U3229132}} # [[/Negativity bias/]] - What is the negativity bias, what are its impacts, and how can it be overcome? {{ME-By|U3167833}} # [[/Neurohormones and emotion/]] - How do neurohormones influence emotional states and regulation?{{Motivation and emotion/Multimedia/High}} {{ME-By|U3224534}} # [[/Neuroscience of unexpected positive outcomes/]] - What is the neural response to unexpected positive outcomes?{{Motivation and emotion/Book/Low}} {{ME-By|Jana2345}} # [[/News and emotion/|News and emotion]] - What role does emotion play in the presentation and consumption of news? {{ME-By|Annabel32020}} # [[/Occupational violence, emotion, and coping for educators/]] - What are the emotional impacts of occupational violence and how can educators cope? {{ME-By|U3259540}} # [[/Orthorexia and emotion/]] - What are the emotional risk factors and consequences of orthorexia? {{ME-By|U3229926}} # [[/Perimenopause and emotion regulation/]] - How does perimenopause affect emotion regulation?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Book/High}} {{ME-By|U3236447}} # [[/Positivity ratio/]] - What is the positivity ratio and what are its implications? {{ME-By|U3230301}} # [[/Post-traumatic stress disorder and emotion/]] - What is the effect of PTSD on emotion? {{ME-By|Rachel Condat}} # [[/Pregnancy loss and emotion/]] - What are the emotional consequences of pregnancy loss for parents?{{Motivation and emotion/Book/Low}} {{ME-By|Annabelle Taylor}} # [[/Psychological literacy/]] - What is psychological literacy, why does it matter, and how can it be fostered? {{ME-By|LouiseCleary}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters?{{Motivation and emotion/Book/Low}} {{ME-By|AbbeyByles}} # [[/Rejection sensitivity/]] - What is rejection sensitivity and how does it impact behaviour and relationships?{{Motivation and emotion/Book/Low}} {{ME-By|Yonis Yousufzai}} # [[/Remote work and stress/]] - How does remote work influence employee stress? {{ME-By|U3236405}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being?{{Motivation and emotion/Book/Low}} {{ME-By|ShayveSukhoo}} # [[/Seasonal affective disorder/]] - What is SAD, what are its causes, and how can it be treated? {{ME-By|U3239762}} # [[/Sense hacking/]] - How can manipulating sensory experiences enhance happiness?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3081293}} # [[/Stress mindset/]] - What is stress mindset, why does it matter, and how can it be cultivated?{{Motivation and emotion/Multimedia/High}} {{ME-By|Jacqueline Di Fronzo}} # [[/Subjective wellbeing heritability and changeability/]] - To what extent is subjective wellbeing heritable or changeable? {{ME-By|Sebastian Siakimotu}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth?{{Motivation and emotion/Book/Low}} {{ME-By|ConnorIrlam}} # [[/Therapeutic horticulture/]] - What is therapeutic horticulture, what are its effects, and how does it work? {{ME-By|Claire Keightley}} # [[/Tinnitus and emotion/]] - What are the emotional impacts of tinnitus and which emotional regulation techniques can help? {{ME-By|Kate737746}} # [[/Trauma-informed education/]] - What is trauma-informed education, and how can it benefit students? {{ME-By|U3235191}} # [[/Vagus nerve and stress//]] - What role does the vagus nerve play in the stress response? {{ME-By|Princess Brutus}} # [[/Values in action framework of strengths/]] - What is the VIA framework and how can it be applied? {{ME-By|U3185269KL}} # [[/Vitamin D and emotion regulation/]] - What is the role of vitamin D in managing emotions? {{ME-By|Chris Beaven}} # [[/Work motivation and self-determination theory/]] - How does self-determination theory explain work motivation?{{Motivation and emotion/Book/Low}} {{ME-By|U3212278}} ==Motivation and emotion== # [[/Aha! experience/]] - What are the motivational and emotional characteristics of aha! experiences? {{ME-By|Oscar Tasman}} # [[/Alcohol, dopamine, motivation, and emotion/]] - What role does dopamine play in motivational and emotional responses to alcohol consumption?{{Motivation and emotion/Book/Low}} {{ME-By|U3219927}} # [[/Dopamine fasting/]] - What is dopamine fasting and how does it affect motivation and emotion?{{Motivation and emotion/Book/Low}} {{ME-By|Emdawson02}} # [[/Lucid dream facilitation/]] - What techniques can facilitate lucid dreaming? {{ME-By|ShamusBrodie}} # [[/Multiple selves theory, motivation, and emotion/]] - How does the concept of multiple selves impact our motivational and emotional experiences?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|Tatteredwing}} # [[/Role-playing games, motivation, and emotion/]] - How do role-playing games shape player emotions, motivation, and behaviour?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|AndreaChau}} # [[/Solitary confinement/]] - What are its motivational and emotional processes and impacts?{{Motivation and emotion/Book/High}} {{ME-By|Ubaldo111}} # [[/Sleep onset optimisation/]] - What techniques assist in falling asleep promptly? {{ME-By|U3202982Isabelle}} # [[/Transcendental future time perspective, motivation, and emotion/]] - What are the motivational and emotional aspects of transcendental future time perspective? {{ME-By|StutiDoshi1}} # [[/Vulnerable dark triad, motivation, and emotion/]] - How does the vulnerable dark triad relate to motivation and emotion?{{Motivation and emotion/Multimedia/High}} {{ME-By|U3126684}} # [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom?{{Motivation and emotion/Book/Low}} {{ME-By|Abiral Shrestha}} [[Category:Motivation and emotion/Book/2024]] q6p37l0vrxt3hdbzjei0gakww3bxtw8 2814996 2814989 2026-06-10T07:36:43Z Jtneill 10242 Move 1 abandoned topic to 2026 - # [[/Getting started/]] - Why is getting started hard and how can this be overcome? 2814996 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Abusive supervision/]] - What motivates it and how does it affect workplace motivation? {{ME-By|TJDuus}} # [[/Adverse childhood experiences and risk-taking motivation/]] - How do ACEs affect risk-taking motivation?{{Motivation and emotion/Book/High}} {{ME-By|Krobertsonn}} # [[/Affective touch and emotion/]] - What are the emotional responses to affective touch? {{ME-By|Ratidzo Nyangu}} # [[/Analysis paralysis/]] - What causes analysis paralysis, what are its psychological mechanisms, and what are its impacts? {{ME-By|Yashvi M}} # [[/Antisocial behaviour in children/]] - What motivates antisocial behaviour in children? {{ME-By|U3191574 (PHP)}} # [[/Autism and motivation/]] - How does autism influence motivation?{{Motivation and emotion/Multimedia/High}} {{ME-By|Sophiedriscoll}} # [[/Better-than-average-effect/]] - Why do people tend to think they are better than average, what are the consequences, and what can be done about it? {{ME-By|RussellP1}} # [[/Biofeedback and motivation/]] - How does biofeedback influence motivation? {{ME-By|U3239962}} # [[/Bottom-line mentality and motivation/]] - How does a focus on the bottom line affect motivation and behaviour? {{ME-By|Biancaguina}} # [[/Breaking bad habits/]] - What strategies can be effective for breaking bad habits? {{ME-By|Pelleot}} # [[/Comprehensive action determination model|Comprehensive action determination model]] - What is the CADM and how can it be applied to understanding human motivation? {{ME-By|HassanAlsamara}} # [[/Consumer neuroscience/]] - What is consumer neuroscience and how does it contribute to understanding consumer behaviour? {{ME-By|Claudiaread}} # [[/Cultural variations in power motivation/]] - How does culture influence power motivation and its expression? {{ME-By|DFaol}} # [[/Death and motivation/]] - How does awareness of mortality influence motivation? {{ME-By|Krutipatil}} # [[/Dopamine and decision making/]] - How does dopamine influence decision making and what are the motivational implications? {{ME-By|Carter.lizzie}} # [[/Dopamine and learning/]] - What is the relationship between dopamine and learning? {{ME-By|Rhys Harmer}} # [[/Dopamine and social behaviour/]] - What role does dopamine play in social interactions and relationships?{{Motivation and emotion/Book/High}} {{ME-By|DeenMisic}} # [[/E-cigarette use motivation/]] - What motivates starting and continuing vaping of nicotine e-cigarettes?{{Motivation and emotion/Book/Low}} {{ME-By|U3242988}} # [[/Effort justification/]] - What is it and how does it influence perception and motivation? {{ME-By|Bstonehouse}} # [[/Employee recognition and work motivation/]] - What is the impact of employee recognition on work motivation?{{Motivation and emotion/Book/High}} {{ME-By|U3235875}} # [[/Environmental cues and habits/]] - How do environmental cues facilitate habit formation? {{ME-By|MoniqqueK}} # [[/ERG theory/]] - What is Alderfer's ERG theory?{{Motivation and emotion/Book/Low}} {{ME-By|U3173387}} # [[/Fandom motivation/]] - What motivates fandom? {{ME-By|TEGIANN}} # [[/Fogg behaviour model/]] - How can it be applied to understanding and changing behaviour?{{Motivation and emotion/Book/Low}} {{ME-By|HannahMooney}} # [[/Free will and neuroscience/]] - What does neuroscience reveal about the concept of free will? {{ME-By|Sienna04}} # [[/Functional approach to volunteerism/]] - How does the functional approach explain volunteer motivation and how can it be applied? {{ME-By|U32314488}} # [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|Samanthagrebert}} # [[/Gender differences in sexual motivation/]] - What are the key differences in sexual motivation between genders and what causes these differences?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3200844}} # [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes?{{Motivation and emotion/Book/Low}} {{ME-By|Rocxie}} # [[/Grit and conscientiousness/]] - What are the similarities and differences between grit and conscientiousness and what are the implications? {{ME-By|U3229789}} # [[/Groups and individual motivation enhancement/]] - How can group dynamics enhance individual motivation? {{ME-By|CharlotteLane55}} # [[/Groups and individual motivation reduction/]] - How can group dynamics diminish or undermine individual motivation?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3216883}} # [[/Habit stacking/]] - What is habit stacking and how can it enhance productivity?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3228753}} # [[/Inner talk in achieving high performance/]] - How does inner dialogue influence motivation for high achievement? {{ME-By|OscarDillion}} # [[/Inner voice and motivation/]] - How does the inner voice influence motivation? {{ME-By|U3248317}} # [[/Intentional pregnancy motivation/]] - What motivational factors contribute to intentional pregnancy?{{Motivation and emotion/Multimedia/High}} {{ME-By|Biancagouws}} # [[/Intertemporal choice/]] - What are intertemporal choices and how can they be effectively negotiated? {{ME-By|HDurban}} # [[/Intrinsic motivation and creativity/]] - What is the relationship between intrinsic motivation and creativity? {{ME-By|Trodrickk}} # [[/Intrinsic rewards and motivation/]] - How do intrinsic rewards affect motivation? {{ME-By|Zwyndham}} # [[/Justice motivation/]] - What is justice motivation, what causes it, and how is it pursued and satisfied? {{ME-By|Emeweme}} # [[/Lateral violence among Indigenous Australians/]] - What drives it and how can it be addressed? {{ME-By|Smritiadhikari}} # [[/Learned industriousness and motivation/]] - How does learned industriousness influence motivation and work ethic? {{ME-By|Persimmon24}} # [[/Machiavellianism and power motivation/]] - What is the relationship between Machiavellianism and power motivation?{{Motivation and emotion/Book/High}} {{ME-By|Joan-E-1405}} # [[/Money and motivation/]] - How does financial incentive influence motivation? {{ME-By|EmmanuelPaps}} # [[/Moral motivation/]] - What drives people to act according to moral principles? {{ME-By|Sophieteferi}} # [[/Morbid curiosity/]] - What is morbid curiosity, what causes it, and how does it influence behaviour?{{Motivation and emotion/Book/Low}} {{ME-By|Scarlett Neilson}} # [[/Morning routine and motivation/]] - How can a morning routine be used to facilitate motivation, productivity, and well-being?{{Motivation and emotion/Book/High}} {{ME-By|Syeda Aiman Ali}} # [[/Motivation in retirement/]] - What motivational challenges do retirees face and how can they be addressed? {{ME-By|U3243172}} # [[/Nudge theory/]] - What is nudge theory and how is it used to motivate behaviour change?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3225022}} # [[/Oxytocin and motivation/]] - How does oxytocin affect motivation? {{ME-By|U3232264}} # [[/Pain and motivation/]] - How does pain influence motivation? {{ME-By|Gabriel Geld}} # [[/Pain and placebo/]] - What is the placebo effect in pain management and how does it work? {{ME-By|U3230258}} # [[/Paradoxical interventions/]] - How can they facilitate psychological change?{{Motivation and emotion/Multimedia/High}} {{ME-By|Seth Guglielmin}} # [[/Physical activity tracking and exercise motivation/]] - What role can physical activity tracking play in exercise motivation? {{ME-By|Alyssia Myers}} # [[/Political motivation/]] - What motivates engagement in politics? {{ME-By|IvaPuskarica}} # [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness?{{Motivation and emotion/Book/Low}} {{ME-By|Zainab Zaman}} # [[/Productivity and routines/]] - How can establishing routines improve productivity?{{Motivation and emotion/Multimedia/High}} {{ME-By|U3236610 Allana}} # [[/Rivalry in the workplace/]] - What impact does workplace rivalry have on employee motivation and productivity? {{ME-By|Boomboompow4}} # [[/Self-determination theory and environmental activism/]] - How does self-determination theory explain motivation in environmental activism? {{ME-By|Zahra Karim}} # [[/Sleep and ego depletion/]] - How does sleep affect the capacity for self-control and willpower?{{Motivation and emotion/Book/Low}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3229957}} # [[/Social dominance and motivation/]] - What drives social dominance motivation, and how does it influence behaviour? {{ME-By|Cophiesollins}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation?{{Motivation and emotion/Book/Low}} {{ME-By|Dragonmoon05}} # [[/Sound and perception of food and drink/]] - How does sound influence the taste and enjoyment of food and drink?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|Ebrixon}} # [[/Stockholm syndrome motivation/]] - Why do captives bond with their captors?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3243508}} # [[/Temporal motivation theory/]] - What is temporal motivation theory and how does it influence decision-making? {{ME-By|U3236663}} # [[/Testosterone and violent crime/]] - What is the relationship between testosterone and violent crime?{{Motivation and emotion/Multimedia/High}} {{ME-By|Amarshall757}} # [[/Theoretical domains framework/]] - What is the TDF and how can it be used to guide behaviour change?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Book/High}} {{ME-By|Kiara Stone (UC)}} # [[/Treatment motivation in juvenile delinquency/]] - What is the role of treatment motivation for juvenile delinquency and how can it be enhanced?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3236338}} # [[/Truth serum drugs/]] - What are truth serum drugs, do they work, and how?{{Motivation and emotion/Multimedia/High}} {{ME-By|U3236669}} # [[/Ulysses pact and motivation/]] - What is the Ulysses pact and how does it relate to motivation? {{ME-By|L.Mercer}} # [[/Underdog versus favourite motivation/]] - How do motivation dynamics differ between underdogs and favourites? {{ME-By|Sarah Pendlebury}} # [[/Vasopressin and motivation/]] - How does vasopressin influence motivation?{{Motivation and emotion/Book/Low}} {{ME-By|U3012923}} ==Emotion== # [[/Abusive relationships and emotional dependency/]] - What psychological mechanisms drive emotional dependency and how does it perpetuate abusive relationships?{{Motivation and emotion/Book/Low}} {{ME-By|Lucywilson 546}} # [[/ADHD and emotional regulation/]] - How does ADHD affect emotional regulation and how can this be managed? {{ME-By|U3188383}} # [[/Adverse childhood experiences and emotion regulation/]] - What is the relationship between ACEs and emotion regulation? {{ME-By|U3242902}} # [[/Artificial intelligence empathy/]] - How can algorithms relate to humans? {{ME-By|U3249300}} # [[/Attribution theory and emotion/]] - How do attributions influence emotion? {{ME-By|Spring Pavlova}} # [[/Ayahuasca and the brain/]] - What are the neurological effects of ayahuasca? {{ME-By|Mkatemoore}} # [[/Bibliotherapy and emotion/]] - How can reading literature impact emotional well-being? {{ME-By|GurmeenK}} # [[/Breaking bad news/]] - How should bad news be shared to minimise emotional distress? {{ME-By|Stongu}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation?{{Motivation and emotion/Book/Low}} {{ME-By|MGibb.23}} # [[/Connection to country and well-being/]] - What is the relationship between connection to country and well-being? {{ME-By|U3237101}} # [[/Conservation of resources theory and stress/]] - How does COR theory explain stress and how can it be applied? {{ME-By|U3054914}} # [[/Digital emotion regulation/]] - How can digital tools and technologies be used to help manage emotions?{{Motivation and emotion/Book/High}} {{ME-By|U3237780}} # [[/Disgust and hygiene/]] - How does disgust influence hygiene behaviours and what are the psychological mechanisms involved?{{Motivation and emotion/Multimedia/High}} {{ME-By|U3214564}} # [[/Ecopsychological approaches to trauma/]] - How can nature-based therapies assist in emotional healing from trauma?{{Motivation and emotion/Book/High}} {{ME-By|Beabosborne}} # [[/Ego death/]] - What is ego death, what are its consequences, and how can it be facilitated?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|Danikollas}} # [[/Ego development and psychological growth/]] - How does ego development theory explain psychological growth? {{ME-By|Ryanjboulton}} # [[/Ego resilience/]] - What is ego resilience and how does it affect psychological functioning? {{ME-By|U322995}} # [[/Emotional literacy/]] - What is emotional literacy, why is it important, and how can it be developed?{{Motivation and emotion/Book/High}} {{ME-By|RBasu3243278}} # [[/Emotional self-care/]] - What is emotional self-care, why is it important, and what techniques are effective? {{ME-By|Shanaya M}} # [[/Emotion and time perception/]] - How does emotion affect perceptions of time? {{ME-By|U3243357}} # [[/Emotion preferences/]] - What do people want to feel and why? {{ME-By|U3236421}} # [[/Empathy versus sympathy/]] - What's the difference and how do they influence behaviour and relationships? {{ME-By|U3236683}} # [[/Exteroception and emotion/]] - What is the relationship between exteroception and emotional experience?{{Motivation and emotion/Book/Low}} {{ME-By|U3239399}} # [[/Fear of judgement and social media/]] - How does the fear of being judged undermine posting on social media and how can this be overcome?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|Ashdruett}} # [[/Goals and emotion/]] - What is the relationship between goals and emotion?{{Motivation and emotion/Book/High}} {{ME-By|U3236641}} # [[/Gut-brain axis and emotion/]] - What is the gut-brain axis and how does it influence emotion?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3239091}} # [[/Heart rate variability and emotion regulation/]] - How can HRV monitoring be used to regulate emotion? {{ME-By|U3239156dej}} # [[/Heart rate variability and mental health/]] - What is the relationship between HRV and mental health? {{ME-By|U3190194}} # [[/Humiliation/]] - What leads to it, what role does it play, and how can it be managed? {{ME-By|Stluciamolly}} # [[/Humility/]] - What is humility, what causes it, and is it desirable?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3237728}} # [[/Humour and social bonding/]] - How does humour facilitate social bonding? {{ME-By|Lwinter1}} # [[/Humour and stress relief/]] - How does humour help in relieving stress? {{ME-By|U3236303}} # [[/Impact bias/]] - What causes it, what are its consequences, and how can it be overcome?{{Motivation and emotion/Book/High}} {{ME-By|U3242909}} # [[/Inner voice and emotion/]] - How does the inner voice influence emotion? {{ME-By|StephenBlume}} # [[/Life satisfaction and personality/]] - What is the relationship between personality and life satisfaction?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3242288}} # [[/Moral dilemmas and emotion/]] - How can emotion aid or hinder in moral decision making? {{ME-By|U3249491}} # [[/Nature connection and emotional well-being/]] - How does connecting with nature influence emotional well-being?{{Motivation and emotion/Multimedia/High}} {{ME-By|U3229132}} # [[/Negativity bias/]] - What is the negativity bias, what are its impacts, and how can it be overcome? {{ME-By|U3167833}} # [[/Neurohormones and emotion/]] - How do neurohormones influence emotional states and regulation?{{Motivation and emotion/Multimedia/High}} {{ME-By|U3224534}} # [[/Neuroscience of unexpected positive outcomes/]] - What is the neural response to unexpected positive outcomes?{{Motivation and emotion/Book/Low}} {{ME-By|Jana2345}} # [[/News and emotion/|News and emotion]] - What role does emotion play in the presentation and consumption of news? {{ME-By|Annabel32020}} # [[/Occupational violence, emotion, and coping for educators/]] - What are the emotional impacts of occupational violence and how can educators cope? {{ME-By|U3259540}} # [[/Orthorexia and emotion/]] - What are the emotional risk factors and consequences of orthorexia? {{ME-By|U3229926}} # [[/Perimenopause and emotion regulation/]] - How does perimenopause affect emotion regulation?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Book/High}} {{ME-By|U3236447}} # [[/Positivity ratio/]] - What is the positivity ratio and what are its implications? {{ME-By|U3230301}} # [[/Post-traumatic stress disorder and emotion/]] - What is the effect of PTSD on emotion? {{ME-By|Rachel Condat}} # [[/Pregnancy loss and emotion/]] - What are the emotional consequences of pregnancy loss for parents?{{Motivation and emotion/Book/Low}} {{ME-By|Annabelle Taylor}} # [[/Psychological literacy/]] - What is psychological literacy, why does it matter, and how can it be fostered? {{ME-By|LouiseCleary}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters?{{Motivation and emotion/Book/Low}} {{ME-By|AbbeyByles}} # [[/Rejection sensitivity/]] - What is rejection sensitivity and how does it impact behaviour and relationships?{{Motivation and emotion/Book/Low}} {{ME-By|Yonis Yousufzai}} # [[/Remote work and stress/]] - How does remote work influence employee stress? {{ME-By|U3236405}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being?{{Motivation and emotion/Book/Low}} {{ME-By|ShayveSukhoo}} # [[/Seasonal affective disorder/]] - What is SAD, what are its causes, and how can it be treated? {{ME-By|U3239762}} # [[/Sense hacking/]] - How can manipulating sensory experiences enhance happiness?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3081293}} # [[/Stress mindset/]] - What is stress mindset, why does it matter, and how can it be cultivated?{{Motivation and emotion/Multimedia/High}} {{ME-By|Jacqueline Di Fronzo}} # [[/Subjective wellbeing heritability and changeability/]] - To what extent is subjective wellbeing heritable or changeable? {{ME-By|Sebastian Siakimotu}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth?{{Motivation and emotion/Book/Low}} {{ME-By|ConnorIrlam}} # [[/Therapeutic horticulture/]] - What is therapeutic horticulture, what are its effects, and how does it work? {{ME-By|Claire Keightley}} # [[/Tinnitus and emotion/]] - What are the emotional impacts of tinnitus and which emotional regulation techniques can help? {{ME-By|Kate737746}} # [[/Trauma-informed education/]] - What is trauma-informed education, and how can it benefit students? {{ME-By|U3235191}} # [[/Vagus nerve and stress//]] - What role does the vagus nerve play in the stress response? {{ME-By|Princess Brutus}} # [[/Values in action framework of strengths/]] - What is the VIA framework and how can it be applied? {{ME-By|U3185269KL}} # [[/Vitamin D and emotion regulation/]] - What is the role of vitamin D in managing emotions? {{ME-By|Chris Beaven}} # [[/Work motivation and self-determination theory/]] - How does self-determination theory explain work motivation?{{Motivation and emotion/Book/Low}} {{ME-By|U3212278}} ==Motivation and emotion== # [[/Aha! experience/]] - What are the motivational and emotional characteristics of aha! experiences? {{ME-By|Oscar Tasman}} # [[/Alcohol, dopamine, motivation, and emotion/]] - What role does dopamine play in motivational and emotional responses to alcohol consumption?{{Motivation and emotion/Book/Low}} {{ME-By|U3219927}} # [[/Dopamine fasting/]] - What is dopamine fasting and how does it affect motivation and emotion?{{Motivation and emotion/Book/Low}} {{ME-By|Emdawson02}} # [[/Lucid dream facilitation/]] - What techniques can facilitate lucid dreaming? {{ME-By|ShamusBrodie}} # [[/Multiple selves theory, motivation, and emotion/]] - How does the concept of multiple selves impact our motivational and emotional experiences?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|Tatteredwing}} # [[/Role-playing games, motivation, and emotion/]] - How do role-playing games shape player emotions, motivation, and behaviour?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|AndreaChau}} # [[/Solitary confinement/]] - What are its motivational and emotional processes and impacts?{{Motivation and emotion/Book/High}} {{ME-By|Ubaldo111}} # [[/Sleep onset optimisation/]] - What techniques assist in falling asleep promptly? {{ME-By|U3202982Isabelle}} # [[/Transcendental future time perspective, motivation, and emotion/]] - What are the motivational and emotional aspects of transcendental future time perspective? {{ME-By|StutiDoshi1}} # [[/Vulnerable dark triad, motivation, and emotion/]] - How does the vulnerable dark triad relate to motivation and emotion?{{Motivation and emotion/Multimedia/High}} {{ME-By|U3126684}} # [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom?{{Motivation and emotion/Book/Low}} {{ME-By|Abiral Shrestha}} [[Category:Motivation and emotion/Book/2024]] g22i3sobzv6zw6nuflvvubj5xy13ufz 2815000 2814996 2026-06-10T07:39:45Z Jtneill 10242 Move 1 abandoned topic to 2026 - # [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? 2815000 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Abusive supervision/]] - What motivates it and how does it affect workplace motivation? {{ME-By|TJDuus}} # [[/Adverse childhood experiences and risk-taking motivation/]] - How do ACEs affect risk-taking motivation?{{Motivation and emotion/Book/High}} {{ME-By|Krobertsonn}} # [[/Affective touch and emotion/]] - What are the emotional responses to affective touch? {{ME-By|Ratidzo Nyangu}} # [[/Analysis paralysis/]] - What causes analysis paralysis, what are its psychological mechanisms, and what are its impacts? {{ME-By|Yashvi M}} # [[/Antisocial behaviour in children/]] - What motivates antisocial behaviour in children? {{ME-By|U3191574 (PHP)}} # [[/Autism and motivation/]] - How does autism influence motivation?{{Motivation and emotion/Multimedia/High}} {{ME-By|Sophiedriscoll}} # [[/Better-than-average-effect/]] - Why do people tend to think they are better than average, what are the consequences, and what can be done about it? {{ME-By|RussellP1}} # [[/Biofeedback and motivation/]] - How does biofeedback influence motivation? {{ME-By|U3239962}} # [[/Bottom-line mentality and motivation/]] - How does a focus on the bottom line affect motivation and behaviour? {{ME-By|Biancaguina}} # [[/Breaking bad habits/]] - What strategies can be effective for breaking bad habits? {{ME-By|Pelleot}} # [[/Comprehensive action determination model|Comprehensive action determination model]] - What is the CADM and how can it be applied to understanding human motivation? {{ME-By|HassanAlsamara}} # [[/Consumer neuroscience/]] - What is consumer neuroscience and how does it contribute to understanding consumer behaviour? {{ME-By|Claudiaread}} # [[/Cultural variations in power motivation/]] - How does culture influence power motivation and its expression? {{ME-By|DFaol}} # [[/Death and motivation/]] - How does awareness of mortality influence motivation? {{ME-By|Krutipatil}} # [[/Dopamine and decision making/]] - How does dopamine influence decision making and what are the motivational implications? {{ME-By|Carter.lizzie}} # [[/Dopamine and learning/]] - What is the relationship between dopamine and learning? {{ME-By|Rhys Harmer}} # [[/Dopamine and social behaviour/]] - What role does dopamine play in social interactions and relationships?{{Motivation and emotion/Book/High}} {{ME-By|DeenMisic}} # [[/E-cigarette use motivation/]] - What motivates starting and continuing vaping of nicotine e-cigarettes?{{Motivation and emotion/Book/Low}} {{ME-By|U3242988}} # [[/Effort justification/]] - What is it and how does it influence perception and motivation? {{ME-By|Bstonehouse}} # [[/Employee recognition and work motivation/]] - What is the impact of employee recognition on work motivation?{{Motivation and emotion/Book/High}} {{ME-By|U3235875}} # [[/Environmental cues and habits/]] - How do environmental cues facilitate habit formation? {{ME-By|MoniqqueK}} # [[/ERG theory/]] - What is Alderfer's ERG theory?{{Motivation and emotion/Book/Low}} {{ME-By|U3173387}} # [[/Fandom motivation/]] - What motivates fandom? {{ME-By|TEGIANN}} # [[/Fogg behaviour model/]] - How can it be applied to understanding and changing behaviour?{{Motivation and emotion/Book/Low}} {{ME-By|HannahMooney}} # [[/Free will and neuroscience/]] - What does neuroscience reveal about the concept of free will? {{ME-By|Sienna04}} # [[/Functional approach to volunteerism/]] - How does the functional approach explain volunteer motivation and how can it be applied? {{ME-By|U32314488}} # [[/Gender differences in sexual motivation/]] - What are the key differences in sexual motivation between genders and what causes these differences?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3200844}} # [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes?{{Motivation and emotion/Book/Low}} {{ME-By|Rocxie}} # [[/Grit and conscientiousness/]] - What are the similarities and differences between grit and conscientiousness and what are the implications? {{ME-By|U3229789}} # [[/Groups and individual motivation enhancement/]] - How can group dynamics enhance individual motivation? {{ME-By|CharlotteLane55}} # [[/Groups and individual motivation reduction/]] - How can group dynamics diminish or undermine individual motivation?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3216883}} # [[/Habit stacking/]] - What is habit stacking and how can it enhance productivity?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3228753}} # [[/Inner talk in achieving high performance/]] - How does inner dialogue influence motivation for high achievement? {{ME-By|OscarDillion}} # [[/Inner voice and motivation/]] - How does the inner voice influence motivation? {{ME-By|U3248317}} # [[/Intentional pregnancy motivation/]] - What motivational factors contribute to intentional pregnancy?{{Motivation and emotion/Multimedia/High}} {{ME-By|Biancagouws}} # [[/Intertemporal choice/]] - What are intertemporal choices and how can they be effectively negotiated? {{ME-By|HDurban}} # [[/Intrinsic motivation and creativity/]] - What is the relationship between intrinsic motivation and creativity? {{ME-By|Trodrickk}} # [[/Intrinsic rewards and motivation/]] - How do intrinsic rewards affect motivation? {{ME-By|Zwyndham}} # [[/Justice motivation/]] - What is justice motivation, what causes it, and how is it pursued and satisfied? {{ME-By|Emeweme}} # [[/Lateral violence among Indigenous Australians/]] - What drives it and how can it be addressed? {{ME-By|Smritiadhikari}} # [[/Learned industriousness and motivation/]] - How does learned industriousness influence motivation and work ethic? {{ME-By|Persimmon24}} # [[/Machiavellianism and power motivation/]] - What is the relationship between Machiavellianism and power motivation?{{Motivation and emotion/Book/High}} {{ME-By|Joan-E-1405}} # [[/Money and motivation/]] - How does financial incentive influence motivation? {{ME-By|EmmanuelPaps}} # [[/Moral motivation/]] - What drives people to act according to moral principles? {{ME-By|Sophieteferi}} # [[/Morbid curiosity/]] - What is morbid curiosity, what causes it, and how does it influence behaviour?{{Motivation and emotion/Book/Low}} {{ME-By|Scarlett Neilson}} # [[/Morning routine and motivation/]] - How can a morning routine be used to facilitate motivation, productivity, and well-being?{{Motivation and emotion/Book/High}} {{ME-By|Syeda Aiman Ali}} # [[/Motivation in retirement/]] - What motivational challenges do retirees face and how can they be addressed? {{ME-By|U3243172}} # [[/Nudge theory/]] - What is nudge theory and how is it used to motivate behaviour change?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3225022}} # [[/Oxytocin and motivation/]] - How does oxytocin affect motivation? {{ME-By|U3232264}} # [[/Pain and motivation/]] - How does pain influence motivation? {{ME-By|Gabriel Geld}} # [[/Pain and placebo/]] - What is the placebo effect in pain management and how does it work? {{ME-By|U3230258}} # [[/Paradoxical interventions/]] - How can they facilitate psychological change?{{Motivation and emotion/Multimedia/High}} {{ME-By|Seth Guglielmin}} # [[/Physical activity tracking and exercise motivation/]] - What role can physical activity tracking play in exercise motivation? {{ME-By|Alyssia Myers}} # [[/Political motivation/]] - What motivates engagement in politics? {{ME-By|IvaPuskarica}} # [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness?{{Motivation and emotion/Book/Low}} {{ME-By|Zainab Zaman}} # [[/Productivity and routines/]] - How can establishing routines improve productivity?{{Motivation and emotion/Multimedia/High}} {{ME-By|U3236610 Allana}} # [[/Rivalry in the workplace/]] - What impact does workplace rivalry have on employee motivation and productivity? {{ME-By|Boomboompow4}} # [[/Self-determination theory and environmental activism/]] - How does self-determination theory explain motivation in environmental activism? {{ME-By|Zahra Karim}} # [[/Sleep and ego depletion/]] - How does sleep affect the capacity for self-control and willpower?{{Motivation and emotion/Book/Low}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3229957}} # [[/Social dominance and motivation/]] - What drives social dominance motivation, and how does it influence behaviour? {{ME-By|Cophiesollins}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation?{{Motivation and emotion/Book/Low}} {{ME-By|Dragonmoon05}} # [[/Sound and perception of food and drink/]] - How does sound influence the taste and enjoyment of food and drink?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|Ebrixon}} # [[/Stockholm syndrome motivation/]] - Why do captives bond with their captors?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3243508}} # [[/Temporal motivation theory/]] - What is temporal motivation theory and how does it influence decision-making? {{ME-By|U3236663}} # [[/Testosterone and violent crime/]] - What is the relationship between testosterone and violent crime?{{Motivation and emotion/Multimedia/High}} {{ME-By|Amarshall757}} # [[/Theoretical domains framework/]] - What is the TDF and how can it be used to guide behaviour change?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Book/High}} {{ME-By|Kiara Stone (UC)}} # [[/Treatment motivation in juvenile delinquency/]] - What is the role of treatment motivation for juvenile delinquency and how can it be enhanced?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3236338}} # [[/Truth serum drugs/]] - What are truth serum drugs, do they work, and how?{{Motivation and emotion/Multimedia/High}} {{ME-By|U3236669}} # [[/Ulysses pact and motivation/]] - What is the Ulysses pact and how does it relate to motivation? {{ME-By|L.Mercer}} # [[/Underdog versus favourite motivation/]] - How do motivation dynamics differ between underdogs and favourites? {{ME-By|Sarah Pendlebury}} # [[/Vasopressin and motivation/]] - How does vasopressin influence motivation?{{Motivation and emotion/Book/Low}} {{ME-By|U3012923}} ==Emotion== # [[/Abusive relationships and emotional dependency/]] - What psychological mechanisms drive emotional dependency and how does it perpetuate abusive relationships?{{Motivation and emotion/Book/Low}} {{ME-By|Lucywilson 546}} # [[/ADHD and emotional regulation/]] - How does ADHD affect emotional regulation and how can this be managed? {{ME-By|U3188383}} # [[/Adverse childhood experiences and emotion regulation/]] - What is the relationship between ACEs and emotion regulation? {{ME-By|U3242902}} # [[/Artificial intelligence empathy/]] - How can algorithms relate to humans? {{ME-By|U3249300}} # [[/Attribution theory and emotion/]] - How do attributions influence emotion? {{ME-By|Spring Pavlova}} # [[/Ayahuasca and the brain/]] - What are the neurological effects of ayahuasca? {{ME-By|Mkatemoore}} # [[/Bibliotherapy and emotion/]] - How can reading literature impact emotional well-being? {{ME-By|GurmeenK}} # [[/Breaking bad news/]] - How should bad news be shared to minimise emotional distress? {{ME-By|Stongu}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation?{{Motivation and emotion/Book/Low}} {{ME-By|MGibb.23}} # [[/Connection to country and well-being/]] - What is the relationship between connection to country and well-being? {{ME-By|U3237101}} # [[/Conservation of resources theory and stress/]] - How does COR theory explain stress and how can it be applied? {{ME-By|U3054914}} # [[/Digital emotion regulation/]] - How can digital tools and technologies be used to help manage emotions?{{Motivation and emotion/Book/High}} {{ME-By|U3237780}} # [[/Disgust and hygiene/]] - How does disgust influence hygiene behaviours and what are the psychological mechanisms involved?{{Motivation and emotion/Multimedia/High}} {{ME-By|U3214564}} # [[/Ecopsychological approaches to trauma/]] - How can nature-based therapies assist in emotional healing from trauma?{{Motivation and emotion/Book/High}} {{ME-By|Beabosborne}} # [[/Ego death/]] - What is ego death, what are its consequences, and how can it be facilitated?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|Danikollas}} # [[/Ego development and psychological growth/]] - How does ego development theory explain psychological growth? {{ME-By|Ryanjboulton}} # [[/Ego resilience/]] - What is ego resilience and how does it affect psychological functioning? {{ME-By|U322995}} # [[/Emotional literacy/]] - What is emotional literacy, why is it important, and how can it be developed?{{Motivation and emotion/Book/High}} {{ME-By|RBasu3243278}} # [[/Emotional self-care/]] - What is emotional self-care, why is it important, and what techniques are effective? {{ME-By|Shanaya M}} # [[/Emotion and time perception/]] - How does emotion affect perceptions of time? {{ME-By|U3243357}} # [[/Emotion preferences/]] - What do people want to feel and why? {{ME-By|U3236421}} # [[/Empathy versus sympathy/]] - What's the difference and how do they influence behaviour and relationships? {{ME-By|U3236683}} # [[/Exteroception and emotion/]] - What is the relationship between exteroception and emotional experience?{{Motivation and emotion/Book/Low}} {{ME-By|U3239399}} # [[/Fear of judgement and social media/]] - How does the fear of being judged undermine posting on social media and how can this be overcome?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|Ashdruett}} # [[/Goals and emotion/]] - What is the relationship between goals and emotion?{{Motivation and emotion/Book/High}} {{ME-By|U3236641}} # [[/Gut-brain axis and emotion/]] - What is the gut-brain axis and how does it influence emotion?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3239091}} # [[/Heart rate variability and emotion regulation/]] - How can HRV monitoring be used to regulate emotion? {{ME-By|U3239156dej}} # [[/Heart rate variability and mental health/]] - What is the relationship between HRV and mental health? {{ME-By|U3190194}} # [[/Humiliation/]] - What leads to it, what role does it play, and how can it be managed? {{ME-By|Stluciamolly}} # [[/Humility/]] - What is humility, what causes it, and is it desirable?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3237728}} # [[/Humour and social bonding/]] - How does humour facilitate social bonding? {{ME-By|Lwinter1}} # [[/Humour and stress relief/]] - How does humour help in relieving stress? {{ME-By|U3236303}} # [[/Impact bias/]] - What causes it, what are its consequences, and how can it be overcome?{{Motivation and emotion/Book/High}} {{ME-By|U3242909}} # [[/Inner voice and emotion/]] - How does the inner voice influence emotion? {{ME-By|StephenBlume}} # [[/Life satisfaction and personality/]] - What is the relationship between personality and life satisfaction?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3242288}} # [[/Moral dilemmas and emotion/]] - How can emotion aid or hinder in moral decision making? {{ME-By|U3249491}} # [[/Nature connection and emotional well-being/]] - How does connecting with nature influence emotional well-being?{{Motivation and emotion/Multimedia/High}} {{ME-By|U3229132}} # [[/Negativity bias/]] - What is the negativity bias, what are its impacts, and how can it be overcome? {{ME-By|U3167833}} # [[/Neurohormones and emotion/]] - How do neurohormones influence emotional states and regulation?{{Motivation and emotion/Multimedia/High}} {{ME-By|U3224534}} # [[/Neuroscience of unexpected positive outcomes/]] - What is the neural response to unexpected positive outcomes?{{Motivation and emotion/Book/Low}} {{ME-By|Jana2345}} # [[/News and emotion/|News and emotion]] - What role does emotion play in the presentation and consumption of news? {{ME-By|Annabel32020}} # [[/Occupational violence, emotion, and coping for educators/]] - What are the emotional impacts of occupational violence and how can educators cope? {{ME-By|U3259540}} # [[/Orthorexia and emotion/]] - What are the emotional risk factors and consequences of orthorexia? {{ME-By|U3229926}} # [[/Perimenopause and emotion regulation/]] - How does perimenopause affect emotion regulation?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Book/High}} {{ME-By|U3236447}} # [[/Positivity ratio/]] - What is the positivity ratio and what are its implications? {{ME-By|U3230301}} # [[/Post-traumatic stress disorder and emotion/]] - What is the effect of PTSD on emotion? {{ME-By|Rachel Condat}} # [[/Pregnancy loss and emotion/]] - What are the emotional consequences of pregnancy loss for parents?{{Motivation and emotion/Book/Low}} {{ME-By|Annabelle Taylor}} # [[/Psychological literacy/]] - What is psychological literacy, why does it matter, and how can it be fostered? {{ME-By|LouiseCleary}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters?{{Motivation and emotion/Book/Low}} {{ME-By|AbbeyByles}} # [[/Rejection sensitivity/]] - What is rejection sensitivity and how does it impact behaviour and relationships?{{Motivation and emotion/Book/Low}} {{ME-By|Yonis Yousufzai}} # [[/Remote work and stress/]] - How does remote work influence employee stress? {{ME-By|U3236405}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being?{{Motivation and emotion/Book/Low}} {{ME-By|ShayveSukhoo}} # [[/Seasonal affective disorder/]] - What is SAD, what are its causes, and how can it be treated? {{ME-By|U3239762}} # [[/Sense hacking/]] - How can manipulating sensory experiences enhance happiness?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|U3081293}} # [[/Stress mindset/]] - What is stress mindset, why does it matter, and how can it be cultivated?{{Motivation and emotion/Multimedia/High}} {{ME-By|Jacqueline Di Fronzo}} # [[/Subjective wellbeing heritability and changeability/]] - To what extent is subjective wellbeing heritable or changeable? {{ME-By|Sebastian Siakimotu}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth?{{Motivation and emotion/Book/Low}} {{ME-By|ConnorIrlam}} # [[/Therapeutic horticulture/]] - What is therapeutic horticulture, what are its effects, and how does it work? {{ME-By|Claire Keightley}} # [[/Tinnitus and emotion/]] - What are the emotional impacts of tinnitus and which emotional regulation techniques can help? {{ME-By|Kate737746}} # [[/Trauma-informed education/]] - What is trauma-informed education, and how can it benefit students? {{ME-By|U3235191}} # [[/Vagus nerve and stress//]] - What role does the vagus nerve play in the stress response? {{ME-By|Princess Brutus}} # [[/Values in action framework of strengths/]] - What is the VIA framework and how can it be applied? {{ME-By|U3185269KL}} # [[/Vitamin D and emotion regulation/]] - What is the role of vitamin D in managing emotions? {{ME-By|Chris Beaven}} # [[/Work motivation and self-determination theory/]] - How does self-determination theory explain work motivation?{{Motivation and emotion/Book/Low}} {{ME-By|U3212278}} ==Motivation and emotion== # [[/Aha! experience/]] - What are the motivational and emotional characteristics of aha! experiences? {{ME-By|Oscar Tasman}} # [[/Alcohol, dopamine, motivation, and emotion/]] - What role does dopamine play in motivational and emotional responses to alcohol consumption?{{Motivation and emotion/Book/Low}} {{ME-By|U3219927}} # [[/Dopamine fasting/]] - What is dopamine fasting and how does it affect motivation and emotion?{{Motivation and emotion/Book/Low}} {{ME-By|Emdawson02}} # [[/Lucid dream facilitation/]] - What techniques can facilitate lucid dreaming? {{ME-By|ShamusBrodie}} # [[/Multiple selves theory, motivation, and emotion/]] - How does the concept of multiple selves impact our motivational and emotional experiences?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|Tatteredwing}} # [[/Role-playing games, motivation, and emotion/]] - How do role-playing games shape player emotions, motivation, and behaviour?{{Motivation and emotion/Book/High}}{{Motivation and emotion/Multimedia/High}} {{ME-By|AndreaChau}} # [[/Solitary confinement/]] - What are its motivational and emotional processes and impacts?{{Motivation and emotion/Book/High}} {{ME-By|Ubaldo111}} # [[/Sleep onset optimisation/]] - What techniques assist in falling asleep promptly? {{ME-By|U3202982Isabelle}} # [[/Transcendental future time perspective, motivation, and emotion/]] - What are the motivational and emotional aspects of transcendental future time perspective? {{ME-By|StutiDoshi1}} # [[/Vulnerable dark triad, motivation, and emotion/]] - How does the vulnerable dark triad relate to motivation and emotion?{{Motivation and emotion/Multimedia/High}} {{ME-By|U3126684}} # [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom?{{Motivation and emotion/Book/Low}} {{ME-By|Abiral Shrestha}} [[Category:Motivation and emotion/Book/2024]] gar9n133cqs3nf8v3le269oouag9mj6 0 299218 2814991 2553893 2026-06-10T07:31:32Z Jtneill 10242 Jtneill moved page [[Motivation and emotion/Book/2023/Eco-emotions]] to [[Motivation and emotion/Book/2026/Eco-emotions]] without leaving a redirect 2553893 wikitext text/x-wiki = Eco-Emotions: The Advocate For Change = {{MECR3|1=https://yourlinkgoeshere.com}} <div align=center>Replace this link once the multimedia presentation has been published.<br> This template provides tips for the [[Motivation and emotion/Assessment/Topic|topic development]] exercise.<br>Gradually remove these suggestions as the chapter develops.<br>Also consult the [[Motivation and emotion/Assessment/Chapter|book chapter guidelines]].</div> __TOC__ ==Overview== [[File:A picture is worth a thousand words.jpg|right|thumb|150px|'''Figure 1'''. Explore the topic, then develop a structure for the book chapter.]] Ecological Emotions - an emotional sense reactive to the interconnected changing of our natural and social environment. There will be a focus on the eco-emotional response to the changing social climate and the influence of social media the promote change targeting the human ecological emotive response. The social issue of Gender Inequality will be used as an ongoing example with reference to both current and historical feminist movements. ''\Imagine'' ... a [[#Case studies|scenario or case study]] which illustrates the problem and engages reader interest. Consider including an image (see Figure 1). The scenario could be presented in a [[#Feature box|feature box]]. The Overview is typically 180 to 330 words. Eco-emotions; the advocate for change {{RoundBoxTop|theme=3}} '''Focus questions:''' {{RoundBoxBottom}} ==Main headings== * Why should we care? Indirect issues on a personal basis. * * '#MeToo' * Factors limiting Action: (Online Caution, Cultural Differences) * Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]] * Sub-headings can also be used, but avoid having sections with only one sub-heading ==Key points== * Why should we care? Indirect issues on a personal basis: * what are ecological emotions? * how to deal with ecological emotions? * Example: Suffragettes * * * Revolution: Sexual revolution, why protest? * What are the consequences of acting. * social / cultural context. * * #MeToo: * Influence of Social Media on choosing to act/speak. - diffusion of responsibility and social influence * What influenced people to come forward? * Act global not local? - Why not to the attention of local authorities? - Anonymous * Gender Inequality issues historically: Birth Control, Sexual Revolution etc., and how these eco-emotions were more/less apparent, yet, always present referring to psychological theory. I will use a historical movement as a case study. *Provide at least 3 bullet-points per heading, including for the Overview and Conclusion * Include key citations ==Learning features== * Interactive learning features bring online book chapters to life and can be embedded throughout the chapter. * Collective Action Theory - belief that through collective action ones group could make change. ===Case studies=== * Bystander Effect on Climate Change as example. * Suffragettes * Case studies describe real-world examples of concepts in action. * Case studies can be real or fictional. * A case study could be split into multiple sections throughout a chapter to illustrate different theories or stages. * It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Feature boxes=== * Important content can be highlighted in a feature box. But don't overuse feature boxes, otherwise they lose their effect. There are several ways of creating boxes. Recommended: [[Help:Pretty boxes|Pretty boxes]]). * Consider using feature boxes for: ** Focus questions ** [[#Case studies|Case studies]] or examples ** Quiz questions ** Take-home messages A very simple box can be created by using a space at the start of the line {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border * Change the theme number for different colours {{RoundBoxBottom}} ===Figures=== [[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]] * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, etcetera * Figures can be embedded throughout the chapter, including the Overview section * Figures should be captioned (using '''Figure #.''' and a description). Captions explain the relevance of the image to the text/ * [[commons:|Wikimedia Commons]] provides a library of embeddable images * Images can also be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Refer to each figure at least once in the main text (e.g., see Figure 2) ===Links=== * When key words are introduced, use [[Help:Links|interwiki links]] * These links can go to: ** Wikipedia (e.g., [[w:Sigmund Freud|Sigmund Freud]] wrote about (e.g., [[w:Dreams|dreams]]) or ** Related book chapters (e.g., if your are struggling, you might be interested to read the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]) ===Tables=== * Use tables to organise and summarise information * As with [[#Figures|figures]], tables should be captioned (e.g., see Table 1) * Refer to each table at least once in the main text (e.g., see Table 1) * [[Motivation and emotion/Wikiversity/Tables|Example 3 x 3 tables]] which could be adapted '''Table 1.''' Descriptive Caption Which Explains The Table and its Relevant to the Text - Johari Window Model {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ===Quizzes=== * Using one or two review questions per major section is usually better than a long quiz at the end * Quiz conceptual understanding, rather than trivia * Don't make quizzes too hard * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion is arguably the most important section * The Conclusion is typically 150 to 330 words * What are the take-home messages likely to be? * It should be possible for someone to only read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the problem and what is known based on psychological science {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide up to 6 [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2021/Cognitive dissonance and motivation|Cognitive dissonance and motivation]] (Book chapter, 2021) * [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity) * [[w:David McClelland|David McClelland]] (Wikipedia) * [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021) * [[w:Self determination theory|Self determination theory]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order * Use [[w:Letter case#Sentence casing|sentence casing]] * Include the source in parentheses }} ==References== List cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: Kałwak, W., & Weihgold, V. (2022). The Relationality of Ecological Emotions: An Interdisciplinary Critique of Individual Resilience as Psychology's Response to the Climate Crisis. ''Frontiers in psychology'', ''13'', 823620. <nowiki>https://doi.org/10.3389/fpsyg.2022.823620</nowiki> ==External links== Provide up to 6 [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]]. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[File:A picture is worth a thousand words.jpg|right|thumb|150px|'''Figure 1'''. Explore the topic, then develop a structure for the book chapter.]] ''Imagine'' ... a [[#Case studies|scenario or case study]] which illustrates the problem and engages reader interest. Consider including an image (see Figure 1). The scenario could be presented in a [[#Feature box|feature box]]. The Overview is typically 180 to 330 words. {{tip|Suggestions for this section: * Engage the reader with a scenario, example, or case study, and an accompanying image * Explain the problem and why it is important * Outline how psychological science can help * Present focus questions }} {{RoundBoxTop|theme=3}} '''Focus questions:''' Break the problem (see the sub-title) down into three to five focus questions. Focus questions could also be used as top-level headings. * What is the first focus question? * What is the second focus question? * What is the third focus question? Ask ''[[w:Open-ended question|open-ended]]'' focus questions. For example: * ''Is there'' a relationship between motivation and success? (closed-ended) {{sad}} * ''What is the'' relationship between motivation and success? (open-ended) {{smile}} {{RoundBoxBottom}} I am using the issue of Gender Inequality, particularly in Western Society, as an ongoing example of how eco emotions are used, and acknowledged to motivate change. The exact chapter I’m writing will discuss this in direct reference to modern western society, and the prevalence of eco-emotions to act, using the recent ‘MeToo’ movement, to discuss the influence and use of social media to promote world issues, igniting an ecological emotional response, motivating others to act (what made them come forward), however small, even from a phone. Following, the rest of the book chapter headings would discuss why people respond to in-direct issues on a personal basis using psychological theory. The other sub headings would refer to briefly the Gender Inequality issues historically, such as the Suffragette Movement, Birth Control, Sexual Revolution etc, and how these eco-emotions were more/less apparent, yet, always present referring to psychological theory. I will use a historical movement as a case study. Further subheadings would cover the influence and prominence of eco-emotions and if eco-emotions on Feminism differ in cultures. Prior conclusion, I will discuss the limitations of eco-emotions to motivate change, based on cultural and political limitations to act on ecological-emotions (eg government / political restrictions / resources / repercussions) Attribution Theory ==Main headings== * Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]] * Sub-headings can also be used, but avoid having sections with only one sub-heading ==Key points== * Provide at least 3 bullet-points per heading, including for the Overview and Conclusion * Include key citations ==Learning features== * Interactive learning features bring online book chapters to life and can be embedded throughout the chapter. ===Case studies=== * Case studies describe real-world examples of concepts in action. * Case studies can be real or fictional. * A case study could be split into multiple sections throughout a chapter to illustrate different theories or stages. * It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Feature boxes=== * Important content can be highlighted in a feature box. But don't overuse feature boxes, otherwise they lose their effect. There are several ways of creating boxes. Recommended: [[Help:Pretty boxes|Pretty boxes]]). * Consider using feature boxes for: ** Focus questions ** [[#Case studies|Case studies]] or examples ** Quiz questions ** Take-home messages A very simple box can be created by using a space at the start of the line {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border * Change the theme number for different colours {{RoundBoxBottom}} ===Figures=== [[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]] * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, etcetera * Figures can be embedded throughout the chapter, including the Overview section * Figures should be captioned (using '''Figure #.''' and a description). Captions explain the relevance of the image to the text/ * [[commons:|Wikimedia Commons]] provides a library of embeddable images * Images can also be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Refer to each figure at least once in the main text (e.g., see Figure 2) ===Links=== * When key words are introduced, use [[Help:Links|interwiki links]] * These links can go to: ** Wikipedia (e.g., [[w:Sigmund Freud|Sigmund Freud]] wrote about (e.g., [[w:Dreams|dreams]]) or ** Related book chapters (e.g., if your are struggling, you might be interested to read the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]) ===Tables=== * Use tables to organise and summarise information * As with [[#Figures|figures]], tables should be captioned (e.g., see Table 1) * Refer to each table at least once in the main text (e.g., see Table 1) * [[Motivation and emotion/Wikiversity/Tables|Example 3 x 3 tables]] which could be adapted '''Table 1.''' Descriptive Caption Which Explains The Table and its Relevant to the Text - Johari Window Model {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ===Quizzes=== * Using one or two review questions per major section is usually better than a long quiz at the end * Quiz conceptual understanding, rather than trivia * Don't make quizzes too hard * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion is arguably the most important section * The Conclusion is typically 150 to 330 words * What are the take-home messages likely to be? * It should be possible for someone to only read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the problem and what is known based on psychological science {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide up to 6 [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2021/Cognitive dissonance and motivation|Cognitive dissonance and motivation]] (Book chapter, 2021) * [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity) * [[w:David McClelland|David McClelland]] (Wikipedia) * [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021) * [[w:Self determination theory|Self determination theory]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order * Use [[w:Letter case#Sentence casing|sentence casing]] * Include the source in parentheses }} ==References== List cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Mercer-Mapstone, L., Dvorakova, S. L., Matthews, K. E., Abbot, S., Cheng, B., Felten, P., Knorr, K., Marquis, E., Shammas, R., & Swaim, K. (2017). A systematic literature review of students as partners in higher education. ''International Journal for Students as Partners'', ''1''(1). https://doi.org/10.15173/ijsap.v1i1.3119 Rosenberg, B. D., & Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091 Sears, C. R., Boyce, M. A., Boon, S. D., Goghari, V. M., Irwin, K., & Boyes, M. (2017). Predictors of student satisfaction in a large psychology undergraduate program. ''Canadian Psychology/Psychologie Canadienne'', ''58''(2), 148–160. https://doi.org/10.1037/cap0000082 }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]. Use "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** Provide the full doi as a URL and working hyperlink * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** Citing sources that weren't read or consulted }} ==External links== Provide up to 6 [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]]. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] {{MECR3|1=https://yourlinkgoeshere.com}} <div align=center>Replace this link once the multimedia presentation has been published.<br> This template provides tips for the [[Motivation and emotion/Assessment/Topic|topic development]] exercise.<br>Gradually remove these suggestions as the chapter develops.<br>Also consult the [[Motivation and emotion/Assessment/Chapter|book chapter guidelines]].</div> __TOC__ ==Overview== [[File:A picture is worth a thousand words.jpg|right|thumb|150px|'''Figure 1'''. Explore the topic, then develop a structure for the book chapter.]] ''Imagine'' ... a [[#Case studies|scenario or case study]] which illustrates the problem and engages reader interest. Consider including an image (see Figure 1). The scenario could be presented in a [[#Feature box|feature box]]. The Overview is typically 180 to 330 words. {{tip|Suggestions for this section: * Engage the reader with a scenario, example, or case study, and an accompanying image * Explain the problem and why it is important * Outline how psychological science can help * Present focus questions }} {{RoundBoxTop|theme=3}} '''Focus questions:''' {{RoundBoxBottom}} ==Main headings== * Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]] * Sub-headings can also be used, but avoid having sections with only one sub-heading ==Key points== * Provide at least 3 bullet-points per heading, including for the Overview and Conclusion * Include key citations ==Learning features== * Interactive learning features bring online book chapters to life and can be embedded throughout the chapter. ===Case studies=== * Case studies describe real-world examples of concepts in action. * Case studies can be real or fictional. * A case study could be split into multiple sections throughout a chapter to illustrate different theories or stages. * It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Feature boxes=== * Important content can be highlighted in a feature box. But don't overuse feature boxes, otherwise they lose their effect. There are several ways of creating boxes. Recommended: [[Help:Pretty boxes|Pretty boxes]]). * Consider using feature boxes for: ** Focus questions ** [[#Case studies|Case studies]] or examples ** Quiz questions ** Take-home messages A very simple box can be created by using a space at the start of the line {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border * Change the theme number for different colours {{RoundBoxBottom}} ===Figures=== [[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]] * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, etcetera * Figures can be embedded throughout the chapter, including the Overview section * Figures should be captioned (using '''Figure #.''' and a description). Captions explain the relevance of the image to the text/ * [[commons:|Wikimedia Commons]] provides a library of embeddable images * Images can also be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Refer to each figure at least once in the main text (e.g., see Figure 2) ===Links=== * When key words are introduced, use [[Help:Links|interwiki links]] * These links can go to: ** Wikipedia (e.g., [[w:Sigmund Freud|Sigmund Freud]] wrote about (e.g., [[w:Dreams|dreams]]) or ** Related book chapters (e.g., if your are struggling, you might be interested to read the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]) ===Tables=== * Use tables to organise and summarise information * As with [[#Figures|figures]], tables should be captioned (e.g., see Table 1) * Refer to each table at least once in the main text (e.g., see Table 1) * [[Motivation and emotion/Wikiversity/Tables|Example 3 x 3 tables]] which could be adapted '''Table 1.''' Descriptive Caption Which Explains The Table and its Relevant to the Text - Johari Window Model {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ===Quizzes=== * Using one or two review questions per major section is usually better than a long quiz at the end * Quiz conceptual understanding, rather than trivia * Don't make quizzes too hard * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion is arguably the most important section * The Conclusion is typically 150 to 330 words * What are the take-home messages likely to be? * It should be possible for someone to only read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the problem and what is known based on psychological science {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide up to 6 [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2021/Cognitive dissonance and motivation|Cognitive dissonance and motivation]] (Book chapter, 2021) * [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity) * [[w:David McClelland|David McClelland]] (Wikipedia) * [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021) * [[w:Self determination theory|Self determination theory]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order * Use [[w:Letter case#Sentence casing|sentence casing]] * Include the source in parentheses }} ==References== List cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Mercer-Mapstone, L., Dvorakova, S. L., Matthews, K. E., Abbot, S., Cheng, B., Felten, P., Knorr, K., Marquis, E., Shammas, R., & Swaim, K. (2017). A systematic literature review of students as partners in higher education. ''International Journal for Students as Partners'', ''1''(1). https://doi.org/10.15173/ijsap.v1i1.3119 Rosenberg, B. D., & Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091 Sears, C. R., Boyce, M. A., Boon, S. D., Goghari, V. M., Irwin, K., & Boyes, M. (2017). Predictors of student satisfaction in a large psychology undergraduate program. ''Canadian Psychology/Psychologie Canadienne'', ''58''(2), 148–160. https://doi.org/10.1037/cap0000082 }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]. Use "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** Provide the full doi as a URL and working hyperlink * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** Citing sources that weren't read or consulted }} ==External links== Provide up to 6 [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]]. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[File:A picture is worth a thousand words.jpg|right|thumb|150px|'''Figure 1'''. Explore the topic, then develop a structure for the book chapter.]] ''Imagine'' ... a [[#Case studies|scenario or case study]] which illustrates the problem and engages reader interest. Consider including an image (see Figure 1). The scenario could be presented in a [[#Feature box|feature box]]. The Overview is typically 180 to 330 words. {{tip|Suggestions for this section: * Engage the reader with a scenario, example, or case study, and an accompanying image * Explain the problem and why it is important * Outline how psychological science can help * Present focus questions }} {{RoundBoxTop|theme=3}} '''Focus questions:''' Break the problem (see the sub-title) down into three to five focus questions. Focus questions could also be used as top-level headings. * What is the first focus question? * What is the second focus question? * What is the third focus question? Ask ''[[w:Open-ended question|open-ended]]'' focus questions. For example: * ''Is there'' a relationship between motivation and success? (closed-ended) {{sad}} * ''What is the'' relationship between motivation and success? (open-ended) {{smile}} {{RoundBoxBottom}} ==Main headings== * Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]] * Sub-headings can also be used, but avoid having sections with only one sub-heading ==Key points== * Provide at least 3 bullet-points per heading, including for the Overview and Conclusion * Include key citations ==Learning features== * Interactive learning features bring online book chapters to life and can be embedded throughout the chapter. ===Case studies=== * Case studies describe real-world examples of concepts in action. * Case studies can be real or fictional. * A case study could be split into multiple sections throughout a chapter to illustrate different theories or stages. * It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Feature boxes=== * Important content can be highlighted in a feature box. But don't overuse feature boxes, otherwise they lose their effect. There are several ways of creating boxes. Recommended: [[Help:Pretty boxes|Pretty boxes]]). * Consider using feature boxes for: ** Focus questions ** [[#Case studies|Case studies]] or examples ** Quiz questions ** Take-home messages A very simple box can be created by using a space at the start of the line {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border * Change the theme number for different colours {{RoundBoxBottom}} ===Figures=== [[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]] * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, etcetera * Figures can be embedded throughout the chapter, including the Overview section * Figures should be captioned (using '''Figure #.''' and a description). Captions explain the relevance of the image to the text/ * [[commons:|Wikimedia Commons]] provides a library of embeddable images * Images can also be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Refer to each figure at least once in the main text (e.g., see Figure 2) ===Links=== * When key words are introduced, use [[Help:Links|interwiki links]] * These links can go to: ** Wikipedia (e.g., [[w:Sigmund Freud|Sigmund Freud]] wrote about (e.g., [[w:Dreams|dreams]]) or ** Related book chapters (e.g., if your are struggling, you might be interested to read the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]) ===Tables=== * Use tables to organise and summarise information * As with [[#Figures|figures]], tables should be captioned (e.g., see Table 1) * Refer to each table at least once in the main text (e.g., see Table 1) * [[Motivation and emotion/Wikiversity/Tables|Example 3 x 3 tables]] which could be adapted '''Table 1.''' Descriptive Caption Which Explains The Table and its Relevant to the Text - Johari Window Model {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ===Quizzes=== * Using one or two review questions per major section is usually better than a long quiz at the end * Quiz conceptual understanding, rather than trivia * Don't make quizzes too hard * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion is arguably the most important section * The Conclusion is typically 150 to 330 words * What are the take-home messages likely to be? * It should be possible for someone to only read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the problem and what is known based on psychological science {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide up to 6 [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2021/Cognitive dissonance and motivation|Cognitive dissonance and motivation]] (Book chapter, 2021) * [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity) * [[w:David McClelland|David McClelland]] (Wikipedia) * [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021) * [[w:Self determination theory|Self determination theory]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order * Use [[w:Letter case#Sentence casing|sentence casing]] * Include the source in parentheses }} ==References== List cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Mercer-Mapstone, L., Dvorakova, S. L., Matthews, K. E., Abbot, S., Cheng, B., Felten, P., Knorr, K., Marquis, E., Shammas, R., & Swaim, K. (2017). A systematic literature review of students as partners in higher education. ''International Journal for Students as Partners'', ''1''(1). https://doi.org/10.15173/ijsap.v1i1.3119 Rosenberg, B. D., & Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091 Sears, C. R., Boyce, M. A., Boon, S. D., Goghari, V. M., Irwin, K., & Boyes, M. (2017). Predictors of student satisfaction in a large psychology undergraduate program. ''Canadian Psychology/Psychologie Canadienne'', ''58''(2), 148–160. https://doi.org/10.1037/cap0000082 }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]. Use "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** Provide the full doi as a URL and working hyperlink * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** Citing sources that weren't read or consulted }} ==External links== Provide up to 6 [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]]. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] rn46xka78jhhnp13ygpt9dv6e6gimk8 2814993 2814991 2026-06-10T07:33:09Z Jtneill 10242 Adjust title layout 2814993 wikitext text/x-wiki {{METE}} {{title|Eco-emotions:<br>Subtitle goes here}} This template provides tips for the [[Motivation and emotion/Assessment/Topic|topic development]] exercise.<br>Gradually remove these suggestions as the chapter develops.<br>Also consult the [[Motivation and emotion/Assessment/Chapter|book chapter guidelines]].</div> __TOC__ ==Overview== [[File:A picture is worth a thousand words.jpg|right|thumb|150px|'''Figure 1'''. Explore the topic, then develop a structure for the book chapter.]] Ecological Emotions - an emotional sense reactive to the interconnected changing of our natural and social environment. There will be a focus on the eco-emotional response to the changing social climate and the influence of social media the promote change targeting the human ecological emotive response. The social issue of Gender Inequality will be used as an ongoing example with reference to both current and historical feminist movements. ''\Imagine'' ... a [[#Case studies|scenario or case study]] which illustrates the problem and engages reader interest. Consider including an image (see Figure 1). The scenario could be presented in a [[#Feature box|feature box]]. The Overview is typically 180 to 330 words. Eco-emotions; the advocate for change {{RoundBoxTop|theme=3}} '''Focus questions:''' {{RoundBoxBottom}} ==Main headings== * Why should we care? Indirect issues on a personal basis. * * '#MeToo' * Factors limiting Action: (Online Caution, Cultural Differences) * Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]] * Sub-headings can also be used, but avoid having sections with only one sub-heading ==Key points== * Why should we care? Indirect issues on a personal basis: * what are ecological emotions? * how to deal with ecological emotions? * Example: Suffragettes * * * Revolution: Sexual revolution, why protest? * What are the consequences of acting. * social / cultural context. * * #MeToo: * Influence of Social Media on choosing to act/speak. - diffusion of responsibility and social influence * What influenced people to come forward? * Act global not local? - Why not to the attention of local authorities? - Anonymous * Gender Inequality issues historically: Birth Control, Sexual Revolution etc., and how these eco-emotions were more/less apparent, yet, always present referring to psychological theory. I will use a historical movement as a case study. *Provide at least 3 bullet-points per heading, including for the Overview and Conclusion * Include key citations ==Learning features== * Interactive learning features bring online book chapters to life and can be embedded throughout the chapter. * Collective Action Theory - belief that through collective action ones group could make change. ===Case studies=== * Bystander Effect on Climate Change as example. * Suffragettes * Case studies describe real-world examples of concepts in action. * Case studies can be real or fictional. * A case study could be split into multiple sections throughout a chapter to illustrate different theories or stages. * It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Feature boxes=== * Important content can be highlighted in a feature box. But don't overuse feature boxes, otherwise they lose their effect. There are several ways of creating boxes. Recommended: [[Help:Pretty boxes|Pretty boxes]]). * Consider using feature boxes for: ** Focus questions ** [[#Case studies|Case studies]] or examples ** Quiz questions ** Take-home messages A very simple box can be created by using a space at the start of the line {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border * Change the theme number for different colours {{RoundBoxBottom}} ===Figures=== [[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]] * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, etcetera * Figures can be embedded throughout the chapter, including the Overview section * Figures should be captioned (using '''Figure #.''' and a description). Captions explain the relevance of the image to the text/ * [[commons:|Wikimedia Commons]] provides a library of embeddable images * Images can also be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Refer to each figure at least once in the main text (e.g., see Figure 2) ===Links=== * When key words are introduced, use [[Help:Links|interwiki links]] * These links can go to: ** Wikipedia (e.g., [[w:Sigmund Freud|Sigmund Freud]] wrote about (e.g., [[w:Dreams|dreams]]) or ** Related book chapters (e.g., if your are struggling, you might be interested to read the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]) ===Tables=== * Use tables to organise and summarise information * As with [[#Figures|figures]], tables should be captioned (e.g., see Table 1) * Refer to each table at least once in the main text (e.g., see Table 1) * [[Motivation and emotion/Wikiversity/Tables|Example 3 x 3 tables]] which could be adapted '''Table 1.''' Descriptive Caption Which Explains The Table and its Relevant to the Text - Johari Window Model {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ===Quizzes=== * Using one or two review questions per major section is usually better than a long quiz at the end * Quiz conceptual understanding, rather than trivia * Don't make quizzes too hard * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion is arguably the most important section * The Conclusion is typically 150 to 330 words * What are the take-home messages likely to be? * It should be possible for someone to only read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the problem and what is known based on psychological science {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide up to 6 [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2021/Cognitive dissonance and motivation|Cognitive dissonance and motivation]] (Book chapter, 2021) * [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity) * [[w:David McClelland|David McClelland]] (Wikipedia) * [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021) * [[w:Self determination theory|Self determination theory]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order * Use [[w:Letter case#Sentence casing|sentence casing]] * Include the source in parentheses }} ==References== List cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: Kałwak, W., & Weihgold, V. (2022). The Relationality of Ecological Emotions: An Interdisciplinary Critique of Individual Resilience as Psychology's Response to the Climate Crisis. ''Frontiers in psychology'', ''13'', 823620. <nowiki>https://doi.org/10.3389/fpsyg.2022.823620</nowiki> ==External links== Provide up to 6 [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]]. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[File:A picture is worth a thousand words.jpg|right|thumb|150px|'''Figure 1'''. Explore the topic, then develop a structure for the book chapter.]] ''Imagine'' ... a [[#Case studies|scenario or case study]] which illustrates the problem and engages reader interest. Consider including an image (see Figure 1). The scenario could be presented in a [[#Feature box|feature box]]. The Overview is typically 180 to 330 words. {{tip|Suggestions for this section: * Engage the reader with a scenario, example, or case study, and an accompanying image * Explain the problem and why it is important * Outline how psychological science can help * Present focus questions }} {{RoundBoxTop|theme=3}} '''Focus questions:''' Break the problem (see the sub-title) down into three to five focus questions. Focus questions could also be used as top-level headings. * What is the first focus question? * What is the second focus question? * What is the third focus question? Ask ''[[w:Open-ended question|open-ended]]'' focus questions. For example: * ''Is there'' a relationship between motivation and success? (closed-ended) {{sad}} * ''What is the'' relationship between motivation and success? (open-ended) {{smile}} {{RoundBoxBottom}} I am using the issue of Gender Inequality, particularly in Western Society, as an ongoing example of how eco emotions are used, and acknowledged to motivate change. The exact chapter I’m writing will discuss this in direct reference to modern western society, and the prevalence of eco-emotions to act, using the recent ‘MeToo’ movement, to discuss the influence and use of social media to promote world issues, igniting an ecological emotional response, motivating others to act (what made them come forward), however small, even from a phone. Following, the rest of the book chapter headings would discuss why people respond to in-direct issues on a personal basis using psychological theory. The other sub headings would refer to briefly the Gender Inequality issues historically, such as the Suffragette Movement, Birth Control, Sexual Revolution etc, and how these eco-emotions were more/less apparent, yet, always present referring to psychological theory. I will use a historical movement as a case study. Further subheadings would cover the influence and prominence of eco-emotions and if eco-emotions on Feminism differ in cultures. Prior conclusion, I will discuss the limitations of eco-emotions to motivate change, based on cultural and political limitations to act on ecological-emotions (eg government / political restrictions / resources / repercussions) Attribution Theory ==Main headings== * Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]] * Sub-headings can also be used, but avoid having sections with only one sub-heading ==Key points== * Provide at least 3 bullet-points per heading, including for the Overview and Conclusion * Include key citations ==Learning features== * Interactive learning features bring online book chapters to life and can be embedded throughout the chapter. ===Case studies=== * Case studies describe real-world examples of concepts in action. * Case studies can be real or fictional. * A case study could be split into multiple sections throughout a chapter to illustrate different theories or stages. * It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Feature boxes=== * Important content can be highlighted in a feature box. But don't overuse feature boxes, otherwise they lose their effect. There are several ways of creating boxes. Recommended: [[Help:Pretty boxes|Pretty boxes]]). * Consider using feature boxes for: ** Focus questions ** [[#Case studies|Case studies]] or examples ** Quiz questions ** Take-home messages A very simple box can be created by using a space at the start of the line {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border * Change the theme number for different colours {{RoundBoxBottom}} ===Figures=== [[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]] * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, etcetera * Figures can be embedded throughout the chapter, including the Overview section * Figures should be captioned (using '''Figure #.''' and a description). Captions explain the relevance of the image to the text/ * [[commons:|Wikimedia Commons]] provides a library of embeddable images * Images can also be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Refer to each figure at least once in the main text (e.g., see Figure 2) ===Links=== * When key words are introduced, use [[Help:Links|interwiki links]] * These links can go to: ** Wikipedia (e.g., [[w:Sigmund Freud|Sigmund Freud]] wrote about (e.g., [[w:Dreams|dreams]]) or ** Related book chapters (e.g., if your are struggling, you might be interested to read the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]) ===Tables=== * Use tables to organise and summarise information * As with [[#Figures|figures]], tables should be captioned (e.g., see Table 1) * Refer to each table at least once in the main text (e.g., see Table 1) * [[Motivation and emotion/Wikiversity/Tables|Example 3 x 3 tables]] which could be adapted '''Table 1.''' Descriptive Caption Which Explains The Table and its Relevant to the Text - Johari Window Model {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ===Quizzes=== * Using one or two review questions per major section is usually better than a long quiz at the end * Quiz conceptual understanding, rather than trivia * Don't make quizzes too hard * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion is arguably the most important section * The Conclusion is typically 150 to 330 words * What are the take-home messages likely to be? * It should be possible for someone to only read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the problem and what is known based on psychological science {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide up to 6 [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2021/Cognitive dissonance and motivation|Cognitive dissonance and motivation]] (Book chapter, 2021) * [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity) * [[w:David McClelland|David McClelland]] (Wikipedia) * [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021) * [[w:Self determination theory|Self determination theory]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order * Use [[w:Letter case#Sentence casing|sentence casing]] * Include the source in parentheses }} ==References== List cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Mercer-Mapstone, L., Dvorakova, S. L., Matthews, K. E., Abbot, S., Cheng, B., Felten, P., Knorr, K., Marquis, E., Shammas, R., & Swaim, K. (2017). A systematic literature review of students as partners in higher education. ''International Journal for Students as Partners'', ''1''(1). https://doi.org/10.15173/ijsap.v1i1.3119 Rosenberg, B. D., & Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091 Sears, C. R., Boyce, M. A., Boon, S. D., Goghari, V. M., Irwin, K., & Boyes, M. (2017). Predictors of student satisfaction in a large psychology undergraduate program. ''Canadian Psychology/Psychologie Canadienne'', ''58''(2), 148–160. https://doi.org/10.1037/cap0000082 }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]. Use "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** Provide the full doi as a URL and working hyperlink * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** Citing sources that weren't read or consulted }} ==External links== Provide up to 6 [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]]. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] {{MECR3|1=https://yourlinkgoeshere.com}} <div align=center>Replace this link once the multimedia presentation has been published.<br> This template provides tips for the [[Motivation and emotion/Assessment/Topic|topic development]] exercise.<br>Gradually remove these suggestions as the chapter develops.<br>Also consult the [[Motivation and emotion/Assessment/Chapter|book chapter guidelines]].</div> __TOC__ ==Overview== [[File:A picture is worth a thousand words.jpg|right|thumb|150px|'''Figure 1'''. Explore the topic, then develop a structure for the book chapter.]] ''Imagine'' ... a [[#Case studies|scenario or case study]] which illustrates the problem and engages reader interest. Consider including an image (see Figure 1). The scenario could be presented in a [[#Feature box|feature box]]. The Overview is typically 180 to 330 words. {{tip|Suggestions for this section: * Engage the reader with a scenario, example, or case study, and an accompanying image * Explain the problem and why it is important * Outline how psychological science can help * Present focus questions }} {{RoundBoxTop|theme=3}} '''Focus questions:''' {{RoundBoxBottom}} ==Main headings== * Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]] * Sub-headings can also be used, but avoid having sections with only one sub-heading ==Key points== * Provide at least 3 bullet-points per heading, including for the Overview and Conclusion * Include key citations ==Learning features== * Interactive learning features bring online book chapters to life and can be embedded throughout the chapter. ===Case studies=== * Case studies describe real-world examples of concepts in action. * Case studies can be real or fictional. * A case study could be split into multiple sections throughout a chapter to illustrate different theories or stages. * It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Feature boxes=== * Important content can be highlighted in a feature box. But don't overuse feature boxes, otherwise they lose their effect. There are several ways of creating boxes. Recommended: [[Help:Pretty boxes|Pretty boxes]]). * Consider using feature boxes for: ** Focus questions ** [[#Case studies|Case studies]] or examples ** Quiz questions ** Take-home messages A very simple box can be created by using a space at the start of the line {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border * Change the theme number for different colours {{RoundBoxBottom}} ===Figures=== [[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]] * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, etcetera * Figures can be embedded throughout the chapter, including the Overview section * Figures should be captioned (using '''Figure #.''' and a description). Captions explain the relevance of the image to the text/ * [[commons:|Wikimedia Commons]] provides a library of embeddable images * Images can also be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Refer to each figure at least once in the main text (e.g., see Figure 2) ===Links=== * When key words are introduced, use [[Help:Links|interwiki links]] * These links can go to: ** Wikipedia (e.g., [[w:Sigmund Freud|Sigmund Freud]] wrote about (e.g., [[w:Dreams|dreams]]) or ** Related book chapters (e.g., if your are struggling, you might be interested to read the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]) ===Tables=== * Use tables to organise and summarise information * As with [[#Figures|figures]], tables should be captioned (e.g., see Table 1) * Refer to each table at least once in the main text (e.g., see Table 1) * [[Motivation and emotion/Wikiversity/Tables|Example 3 x 3 tables]] which could be adapted '''Table 1.''' Descriptive Caption Which Explains The Table and its Relevant to the Text - Johari Window Model {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ===Quizzes=== * Using one or two review questions per major section is usually better than a long quiz at the end * Quiz conceptual understanding, rather than trivia * Don't make quizzes too hard * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion is arguably the most important section * The Conclusion is typically 150 to 330 words * What are the take-home messages likely to be? * It should be possible for someone to only read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the problem and what is known based on psychological science {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide up to 6 [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2021/Cognitive dissonance and motivation|Cognitive dissonance and motivation]] (Book chapter, 2021) * [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity) * [[w:David McClelland|David McClelland]] (Wikipedia) * [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021) * [[w:Self determination theory|Self determination theory]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order * Use [[w:Letter case#Sentence casing|sentence casing]] * Include the source in parentheses }} ==References== List cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Mercer-Mapstone, L., Dvorakova, S. L., Matthews, K. E., Abbot, S., Cheng, B., Felten, P., Knorr, K., Marquis, E., Shammas, R., & Swaim, K. (2017). A systematic literature review of students as partners in higher education. ''International Journal for Students as Partners'', ''1''(1). https://doi.org/10.15173/ijsap.v1i1.3119 Rosenberg, B. D., & Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091 Sears, C. R., Boyce, M. A., Boon, S. D., Goghari, V. M., Irwin, K., & Boyes, M. (2017). Predictors of student satisfaction in a large psychology undergraduate program. ''Canadian Psychology/Psychologie Canadienne'', ''58''(2), 148–160. https://doi.org/10.1037/cap0000082 }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]. Use "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** Provide the full doi as a URL and working hyperlink * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** Citing sources that weren't read or consulted }} ==External links== Provide up to 6 [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]]. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[File:A picture is worth a thousand words.jpg|right|thumb|150px|'''Figure 1'''. Explore the topic, then develop a structure for the book chapter.]] ''Imagine'' ... a [[#Case studies|scenario or case study]] which illustrates the problem and engages reader interest. Consider including an image (see Figure 1). The scenario could be presented in a [[#Feature box|feature box]]. The Overview is typically 180 to 330 words. {{tip|Suggestions for this section: * Engage the reader with a scenario, example, or case study, and an accompanying image * Explain the problem and why it is important * Outline how psychological science can help * Present focus questions }} {{RoundBoxTop|theme=3}} '''Focus questions:''' Break the problem (see the sub-title) down into three to five focus questions. Focus questions could also be used as top-level headings. * What is the first focus question? * What is the second focus question? * What is the third focus question? Ask ''[[w:Open-ended question|open-ended]]'' focus questions. For example: * ''Is there'' a relationship between motivation and success? (closed-ended) {{sad}} * ''What is the'' relationship between motivation and success? (open-ended) {{smile}} {{RoundBoxBottom}} ==Main headings== * Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]] * Sub-headings can also be used, but avoid having sections with only one sub-heading ==Key points== * Provide at least 3 bullet-points per heading, including for the Overview and Conclusion * Include key citations ==Learning features== * Interactive learning features bring online book chapters to life and can be embedded throughout the chapter. ===Case studies=== * Case studies describe real-world examples of concepts in action. * Case studies can be real or fictional. * A case study could be split into multiple sections throughout a chapter to illustrate different theories or stages. * It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Feature boxes=== * Important content can be highlighted in a feature box. But don't overuse feature boxes, otherwise they lose their effect. There are several ways of creating boxes. Recommended: [[Help:Pretty boxes|Pretty boxes]]). * Consider using feature boxes for: ** Focus questions ** [[#Case studies|Case studies]] or examples ** Quiz questions ** Take-home messages A very simple box can be created by using a space at the start of the line {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border * Change the theme number for different colours {{RoundBoxBottom}} ===Figures=== [[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]] * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, etcetera * Figures can be embedded throughout the chapter, including the Overview section * Figures should be captioned (using '''Figure #.''' and a description). Captions explain the relevance of the image to the text/ * [[commons:|Wikimedia Commons]] provides a library of embeddable images * Images can also be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Refer to each figure at least once in the main text (e.g., see Figure 2) ===Links=== * When key words are introduced, use [[Help:Links|interwiki links]] * These links can go to: ** Wikipedia (e.g., [[w:Sigmund Freud|Sigmund Freud]] wrote about (e.g., [[w:Dreams|dreams]]) or ** Related book chapters (e.g., if your are struggling, you might be interested to read the chapter about [[Motivation and emotion/Book/2020/Writer's block|writer's block]]) ===Tables=== * Use tables to organise and summarise information * As with [[#Figures|figures]], tables should be captioned (e.g., see Table 1) * Refer to each table at least once in the main text (e.g., see Table 1) * [[Motivation and emotion/Wikiversity/Tables|Example 3 x 3 tables]] which could be adapted '''Table 1.''' Descriptive Caption Which Explains The Table and its Relevant to the Text - Johari Window Model {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ===Quizzes=== * Using one or two review questions per major section is usually better than a long quiz at the end * Quiz conceptual understanding, rather than trivia * Don't make quizzes too hard * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion is arguably the most important section * The Conclusion is typically 150 to 330 words * What are the take-home messages likely to be? * It should be possible for someone to only read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the problem and what is known based on psychological science {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide up to 6 [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2021/Cognitive dissonance and motivation|Cognitive dissonance and motivation]] (Book chapter, 2021) * [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity) * [[w:David McClelland|David McClelland]] (Wikipedia) * [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021) * [[w:Self determination theory|Self determination theory]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order * Use [[w:Letter case#Sentence casing|sentence casing]] * Include the source in parentheses }} ==References== List cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Mercer-Mapstone, L., Dvorakova, S. L., Matthews, K. E., Abbot, S., Cheng, B., Felten, P., Knorr, K., Marquis, E., Shammas, R., & Swaim, K. (2017). A systematic literature review of students as partners in higher education. ''International Journal for Students as Partners'', ''1''(1). https://doi.org/10.15173/ijsap.v1i1.3119 Rosenberg, B. D., & Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091 Sears, C. R., Boyce, M. A., Boon, S. D., Goghari, V. M., Irwin, K., & Boyes, M. (2017). Predictors of student satisfaction in a large psychology undergraduate program. ''Canadian Psychology/Psychologie Canadienne'', ''58''(2), 148–160. https://doi.org/10.1037/cap0000082 }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]. Use "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** Provide the full doi as a URL and working hyperlink * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** Citing sources that weren't read or consulted }} ==External links== Provide up to 6 [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]]. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] aphlus6lszcrk5kdc9o59rbpkd6nv4d 10 299642 2814941 2556764 2026-06-10T00:12:42Z Atcovi 276019 2814941 wikitext text/x-wiki {{Userbox | border-c = rgb(160,175,183) | id = [[Image:MediaWiki-2020-icon.svg|38px]] | id-c = #fff | info = This user is an '''[[:wikt:Wiktionary:Administrators|administrator]]''' on [[MediaWiki]]. {{#if:{{{1|}}} |<span style="font-size:0.9em;" class="plainlinks">([{{fullurl:mw:{{{lang_code|en}}}:Special:ListUsers|limit=1&username={{urlencode:{{{username|{{BASEPAGENAME}}}}}}}}} <span style="color:#5871C6;">verify{{#if:{{{lang_code|}}}|&nbsp;{{{lang_code}}}|}}</span>])</span>|<span style="font-size:0.9em;" class="plainlinks">([{{fullurl:mw:Special:ListUsers|limit=1&username={{urlencode:{{{username|{{BASEPAGENAME}}}}}}}}} <span style="color:#5871C6;">verify</span>])</span>}} }}<noinclude> {{Clear}} [[Category:Userboxes]] [[Category:Interwiki templates]] </noinclude> fnqqh3mqwuryxr2eopt6583pwem11dd 0 307088 2814998 2644331 2026-06-10T07:39:09Z Jtneill 10242 Jtneill moved page [[Motivation and emotion/Book/2024/Functional motives theory and environmental activism]] to [[Motivation and emotion/Book/2026/Functional motives theory and environmental activism]] without leaving a redirect 2644331 wikitext text/x-wiki {{title|Functional motives theory:<br>How does this theory link to environmental activism?}} {{MECR3|1=https://yourlinkgoeshere.com}} <div align=center>Replace the link in the box above once the multimedia presentation has been published.</div> __TOC__ ==Overview== Environmental activists are passionate individuals, who are strongly committed to positively influencing a change for the environment. Although environmental activists are highly committed to their work, the pace of the change they want to achieve can become painful and frustrating. There are many uncertainties that arise when following a path of environmental activism, such as 'will this benefit the environment in 20 years time?' or 'how do we know this will implement a positive change?'. In knowing this, we ask the question, how do environmental activists stay motivated and committed to their work with the uncertainty and frustration that what they are doing may not influence beneficial change?{{RoundBoxTop|theme=5}}[[Scenario]] [[File:A picture is worth a thousand words.jpg|right|thumb|170px|'''Figure 1'''. Explore the topic, then brainstorm a structure.]] An environmental activist is adamant on reducing environmental pollution such as pollution in the air and water. The activist has suggested peaceful protests as well as flyers around the city she lives in to engage more individuals in the cause. She has decided to leave a link to a website she created in hopes others will sign up and join her to help make a change. After one week, she noticed there had only been three people sign up, which is not enough to start a protest. The activist feels hopeless and uncertain that her initiatives are not enough to contribute to a sustainable environment. So how can we assure the activist remains motivated? {{RoundBoxBottom}} This template provides tips for the [[Motivation and emotion/Assessment/Topic|topic development]] exercise. Gradually remove these suggestions as the chapter develops. It is OK to retain some of this template content for the topic development exercise. Also consult the [[Motivation and emotion/Assessment/Chapter|book chapter guidelines]]. The Overview is typically consists of one to four paragraphs inbetween the scenario and focus questions. Suggested word count aim for the Overview: 180 to 330 words. {{tip|Suggestions for this section: * Engage the reader with a scenario, example, or case study, and an accompanying image * Explain the problem and why it is important * Outline how psychological science can help * Present focus questions }} {{RoundBoxTop|theme=3}} '''Focus questions:''' Break the problem (i.e., the sub-title) down into three to five focus questions. Focus questions can also be used as top-level headings. * What is the first focus question? * What is the second focus question? * What is the third focus question? Ask [[w:Open-ended question|open-ended]] focus questions. For example: * Is there a relationship between motivation and success? (closed-ended) {{sad}} * What is the relationship between motivation and success? (open-ended) {{smile}} {{RoundBoxBottom}} ==Headings== * Aim for three to six main headings inbetween the [[#Overview|Overview]] and [[#Conclusion|Conclusion]] * Sub-headings can also be used, but ** avoid having sections with only one sub-heading ** provide an introductory paragraph before breaking into sub-sections ==Key points== * Provide at least three bullet-points per headingʔ and sub-heading, including for the Overview and Conclusion * Include key citations ==Figures== [[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]] * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etcetera * Embed figures throughout the chapter, including the Overview section * Figures should be captioned (using '''Figure #.''' and a caption). Use captions to explain the relevance of the image to the text/ * [[commons:|Wikimedia Commons]] provides a library of embeddable images * Images can also be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Refer to each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring online book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios or case studies describe applied/real-world examples of concepts in action * Case studies can be real or fictional * A case study could be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. {{anchor|Feature box}} ;Feature boxes * Important content can be highlighted in a [[Motivation and emotion/Wikiversity/Feature box|feature box]]. But don't overuse feature boxes, otherwise they lose their effect. * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia (e.g., [[w:Sigmund Freud|Sigmund Freud]] wrote about (e.g., [[w:Dreams|dreams]]) or ** Related book chapters (e.g., [[Motivation and emotion/Book/2020/Writer's block|writer's block]]) ;Tables * Use to organise and summarise information * As with [[#Figures|figures]], tables should be captioned * Refer to each table at least once in the main text (e.g., see Table 1) * [[Motivation and emotion/Wikiversity/Tables|Example 3 x 3 tables]] which could be adapted '''Table 1.''' Descriptive Caption Which Explains The Table and its Relevant to the Text - Johari Window Model {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two review questions per major section is usually better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia * Don't make quizzes too hard * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion is arguably the most important section * Suggested word count: 150 to 330 words * It should be possible for someone to only read the [[#Overview|Overview]] and the Conclusion and still get a pretty good idea of the problem and what is known based on psychological science {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? (Even for the topic development, have a go at the likely take-home message) }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use these formats: * [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021) * [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity) * [[w:Self determination theory|Self determination theory]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order * Use [[w:Letter case#Sentence casing|sentence casing]] * Include the source in parentheses }} ==References== List cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Rosenberg, B. D., & Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091 }} {{tip|Suggestions for this section: * Important aspects of APA style for references include: ** Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]. Use "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** Provide the full doi as a URL and working hyperlink * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** Citing sources that weren't read or consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as presentations, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]]. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] a9f4fb4zg16cdjvrfcisda2abhccyk3 2815002 2814998 2026-06-10T07:41:03Z Jtneill 10242 + {{METE}} 2815002 wikitext text/x-wiki {{METE}} {{title|Functional motives theory:<br>How does this theory link to environmental activism?}} __TOC__ ==Overview== Environmental activists are passionate individuals, who are strongly committed to positively influencing a change for the environment. Although environmental activists are highly committed to their work, the pace of the change they want to achieve can become painful and frustrating. There are many uncertainties that arise when following a path of environmental activism, such as 'will this benefit the environment in 20 years time?' or 'how do we know this will implement a positive change?'. In knowing this, we ask the question, how do environmental activists stay motivated and committed to their work with the uncertainty and frustration that what they are doing may not influence beneficial change?{{RoundBoxTop|theme=5}}[[Scenario]] [[File:A picture is worth a thousand words.jpg|right|thumb|170px|'''Figure 1'''. Explore the topic, then brainstorm a structure.]] An environmental activist is adamant on reducing environmental pollution such as pollution in the air and water. The activist has suggested peaceful protests as well as flyers around the city she lives in to engage more individuals in the cause. She has decided to leave a link to a website she created in hopes others will sign up and join her to help make a change. After one week, she noticed there had only been three people sign up, which is not enough to start a protest. The activist feels hopeless and uncertain that her initiatives are not enough to contribute to a sustainable environment. So how can we assure the activist remains motivated? {{RoundBoxBottom}} This template provides tips for the [[Motivation and emotion/Assessment/Topic|topic development]] exercise. Gradually remove these suggestions as the chapter develops. It is OK to retain some of this template content for the topic development exercise. Also consult the [[Motivation and emotion/Assessment/Chapter|book chapter guidelines]]. The Overview is typically consists of one to four paragraphs inbetween the scenario and focus questions. Suggested word count aim for the Overview: 180 to 330 words. {{tip|Suggestions for this section: * Engage the reader with a scenario, example, or case study, and an accompanying image * Explain the problem and why it is important * Outline how psychological science can help * Present focus questions }} {{RoundBoxTop|theme=3}} '''Focus questions:''' Break the problem (i.e., the sub-title) down into three to five focus questions. Focus questions can also be used as top-level headings. * What is the first focus question? * What is the second focus question? * What is the third focus question? Ask [[w:Open-ended question|open-ended]] focus questions. For example: * Is there a relationship between motivation and success? (closed-ended) {{sad}} * What is the relationship between motivation and success? (open-ended) {{smile}} {{RoundBoxBottom}} ==Headings== * Aim for three to six main headings inbetween the [[#Overview|Overview]] and [[#Conclusion|Conclusion]] * Sub-headings can also be used, but ** avoid having sections with only one sub-heading ** provide an introductory paragraph before breaking into sub-sections ==Key points== * Provide at least three bullet-points per headingʔ and sub-heading, including for the Overview and Conclusion * Include key citations ==Figures== [[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]] * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etcetera * Embed figures throughout the chapter, including the Overview section * Figures should be captioned (using '''Figure #.''' and a caption). Use captions to explain the relevance of the image to the text/ * [[commons:|Wikimedia Commons]] provides a library of embeddable images * Images can also be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Refer to each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring online book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios or case studies describe applied/real-world examples of concepts in action * Case studies can be real or fictional * A case study could be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. {{anchor|Feature box}} ;Feature boxes * Important content can be highlighted in a [[Motivation and emotion/Wikiversity/Feature box|feature box]]. But don't overuse feature boxes, otherwise they lose their effect. * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia (e.g., [[w:Sigmund Freud|Sigmund Freud]] wrote about (e.g., [[w:Dreams|dreams]]) or ** Related book chapters (e.g., [[Motivation and emotion/Book/2020/Writer's block|writer's block]]) ;Tables * Use to organise and summarise information * As with [[#Figures|figures]], tables should be captioned * Refer to each table at least once in the main text (e.g., see Table 1) * [[Motivation and emotion/Wikiversity/Tables|Example 3 x 3 tables]] which could be adapted '''Table 1.''' Descriptive Caption Which Explains The Table and its Relevant to the Text - Johari Window Model {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two review questions per major section is usually better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia * Don't make quizzes too hard * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion is arguably the most important section * Suggested word count: 150 to 330 words * It should be possible for someone to only read the [[#Overview|Overview]] and the Conclusion and still get a pretty good idea of the problem and what is known based on psychological science {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? (Even for the topic development, have a go at the likely take-home message) }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use these formats: * [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021) * [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity) * [[w:Self determination theory|Self determination theory]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order * Use [[w:Letter case#Sentence casing|sentence casing]] * Include the source in parentheses }} ==References== List cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Rosenberg, B. D., & Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091 }} {{tip|Suggestions for this section: * Important aspects of APA style for references include: ** Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]. Use "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** Provide the full doi as a URL and working hyperlink * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** Citing sources that weren't read or consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as presentations, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]]. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] n05f1dvvuapkf1an8diwgoy4qutufdr 0 307192 2815010 2673893 2026-06-10T07:47:57Z Jtneill 10242 Jtneill moved page [[Motivation and emotion/Book/2024/Hygiene motivation]] to [[Motivation and emotion/Book/2026/Hygiene motivation]] without leaving a redirect: Abandoned page in 2024; moving to 2026 2673893 wikitext text/x-wiki {{title|Hygiene motivation:<br>What motivates maintenance of personal hygiene?}} <div align=center>Edit the placeholder chapter title and sub-title above.<br>Ensure exact match of wording and casing (capitalisation) with the [[Motivation and emotion/Book/2024|2024 list of topics]].<br>All sub-titles end with a question mark.<br>Seek approval for any changes to wording and punctuation.<br>Do not list author name. Authorship is as per the page's edit history.</div> {{MECR3|1=https://yourlinkgoeshere.com}} <div align=center>Replace the link in the box above once the multimedia presentation has been published.</div> __TOC__ ==Overview== {{RoundBoxTop|theme=13}}{{Title|Case Study}}<div style="{{Robelbox/pad}}"> John, is a 30 yr old office worker who works from home. Initially John kept a regular hygiene routine such as daily showers, brushing his teeth twice a day and keeping a clean apartment, but due to increased stresses in his life his hygiene standards started to drop as he spent his energy dealing with immediate problems and letting his personal hygiene standards start to slip, slowly with brushing his teeth every second day and showering every third. As new stressors came and old stresses lingered his standards grew worse. {{expand}} (Consider adding more details about how this impacted John's well-being or relationships.) </div> {{RoundBoxBottom}} This template provides tips for the [[Motivation and emotion/Assessment/Topic|topic development]] exercise. Gradually remove these suggestions as the chapter develops. It is OK to retain some of this template content for the topic development exercise. Also consult the [[Motivation and emotion/Assessment/Chapter|book chapter guidelines]]. The Overview is typically consists of one to four paragraphs inbetween the scenario and focus questions. Suggested word count aim for the Overview: 180 to 330 words. {{tip| <quiz display="simple"> 1. What does hygiene relate to? |type="()"} - Showering and brushing your teeth - Washing sheets and towels - Cleaning the kitchen + All of the above</div> </quiz>}} {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is Hygiene and Hygiene standards? * What are the consequences of bad hygiene standards? * How is motivation involved in personal hygiene? * What are motivational methods of improving hygiene standards? * What are motivational methods of maintaining hygiene standards?{{RoundBoxBottom}} <quiz display="simple"> What are forms of hygiene? * Showering and brushing teeth * Cleaning the kitchen * Washing sheets and towels </quiz> [[File:Washing Hands.jpg|thumb|(I have a suggestion for "Washing Hands photo" you might like to use feel free to remove it)]]'''Focus on Psychology:''' Since the chapter deals with motivation, be sure to connect psychological theories of motivation (e.g., Maslow’s Hierarchy of Needs, Self-Determination Theory) with personal hygiene. This will ground the chapter in psychological science and provide a more structured foundation. ==Introduction to Hygiene standards:== === What is hygiene? === [[File:Hand washing man.jpg|thumb|Figure 1. Example of washing hands]] [[Healthy lifestyle/Hygiene|Hygiene]] is the act of taking care of one's body through washing hands (see figure 1.) or showering to remove dirt and germs from the body in an attempt to keep it clean or brushing your teeth to keep them healthy. Other forms of hygiene include keeping a clean living environment, such as throwing rubbish in bins and regularly washing clothes, sheets and towels. {{expand}} (Consider expanding the definition of hygiene by including mental and social aspects of cleanliness, not just physical.) === What are hygiene standards? === {{comment|Could include a clear definition of hygiene standards and possibly examples of different cultural or professional standards.}} === What causes bad hygiene standards? === {{comment|Could explore factors such as stress, mental health, socioeconomic conditions, etc.}} == Consequences of not maintaining hygiene == === Short term consequences === {{comment|Could add short-term effects like body odor, visible uncleanliness, or immediate discomfort from poor hygiene.}} === Long term consequences === * Aim for three to six main headings inbetween the [[#Overview|Overview]] and [[#Conclusion|Conclusion]] * Sub-headings can also be used, but ** avoid having sections with only one sub-heading ** provide an introductory paragraph before breaking into sub-sections {{comment|Expand on long-term effects such as increased risk of illness, dental problems, and social isolation due to prolonged poor hygiene.}} == Motivations role in personal hygiene == === How motivation applies to personal hygiene === ==== Disgust ==== Disgust plays a motivational role in personal hygiene as shown in the Book Chapter in [[Motivation and emotion/Book/2020/Disgust and disease avoidance motivation#Disgust Motivates Hygiene Behaviours|Disgust and Disease avoidance (4.3).]] ==== Anxiety ==== {{comment|You could can elaborate on how anxiety about social judgments or health risks can motivate hygiene behavior.}} ==== Comfort ==== {{comment|Could discuss how the desire for comfort, both physically and mentally, influences hygiene practices.}} === Identifying signs of bad personal hygiene maintenance === {{comment|This section could cover visible signs of poor hygiene and behavioral signs, such as avoidance of social situations.}} == Ways to increase motivation for personal hygiene == === Motivational methods to improve personal hygiene standards === === Motivation in maintaining personal hygiene standards === {{comment|Consider breaking down intrinsic and extrinsic motivators in this section, e.g., internal satisfaction from cleanliness vs. external rewards like social acceptance.}} ==Key points== * Provide at least three bullet-points per headingʔ and sub-heading, including for the Overview and Conclusion * Include key citations ==Figures== [[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]] * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etcetera * Embed figures throughout the chapter, including the Overview section * Figures should be captioned (using '''Figure #.''' and a caption). Use captions to explain the relevance of the image to the text/ * [[commons:|Wikimedia Commons]] provides a library of embeddable images * Images can also be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Refer to each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring online book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios or case studies describe applied/real-world examples of concepts in action * Case studies can be real or fictional * A case study could be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. {{anchor|Feature box}} ;Feature boxes * Important content can be highlighted in a [[Motivation and emotion/Wikiversity/Feature box|feature box]]. But don't overuse feature boxes, otherwise they lose their effect. * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia (e.g., [[w:Sigmund Freud|Sigmund Freud]] wrote about (e.g., [[w:Dreams|dreams]]) or ** Related book chapters (e.g., [[Motivation and emotion/Book/2020/Writer's block|writer's block]]) ;Tables * Use to organise and summarise information * As with [[#Figures|figures]], tables should be captioned * Refer to each table at least once in the main text (e.g., see Table 1) * [[Motivation and emotion/Wikiversity/Tables|Example 3 x 3 tables]] which could be adapted '''Table 1.''' Descriptive Caption Which Explains The Table and its Relevant to the Text - Johari Window Model {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two review questions per major section is usually better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia * Don't make quizzes too hard * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion is arguably the most important section * Suggested word count: 150 to 330 words * It should be possible for someone to only read the [[#Overview|Overview]] and the Conclusion and still get a pretty good idea of the problem and what is known based on psychological science {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? (Even for the topic development, have a go at the likely take-home message) }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use these formats: * [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021) * [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity) * [[w:Self determination theory|Self determination theory]] (Wikipedia) * [[Motivation and emotion/Book/2024/Disgust and hygiene|Disgust and hygiene]] (Book chapter, 2024) {{tip|Suggestions for this section: * Present in alphabetical order * Use [[w:Letter case#Sentence casing|sentence casing]] * Include the source in parentheses }} ==References== Healthy lifestyle/Hygiene. (2024, January 15). ''Wikiversity. &nbsp;from [https://en.wikiversity.org/w/index.php?title=Healthy_lifestyle/Hygiene&oldid=2598812. <nowiki>https://en.wikiversity.org/w/index.php?title=Healthy_lifestyle/Hygiene&oldid=2598812</nowiki>.]'' ''Personal hygiene''. (2004). [Publisher not identified],. https://online.clickview.com.au/libraries/videos/3714437/personal-hygiene {{Hanging indent|1= Rosenberg, B. D., & Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091 }} {{tip|Suggestions for this section: * Important aspects of APA style for references include: ** Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]. Use "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** Provide the full doi as a URL and working hyperlink * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** Citing sources that weren't read or consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as presentations, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]]. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Anxiety]] [[Category:Motivation and emotion/Book/Disgust]] [[Category:Motivation and emotion/Book/Motivation]] 0u6r266luwlawsuyoouh2t8x5dko8gr 2815011 2815010 2026-06-10T07:49:19Z Jtneill 10242 + {{METE}} 2815011 wikitext text/x-wiki {{METE}} {{title|Hygiene motivation:<br>What motivates maintenance of personal hygiene?}} __TOC__ ==Overview== {{RoundBoxTop|theme=13}}{{Title|Case Study}}<div style="{{Robelbox/pad}}"> John, is a 30 yr old office worker who works from home. Initially John kept a regular hygiene routine such as daily showers, brushing his teeth twice a day and keeping a clean apartment, but due to increased stresses in his life his hygiene standards started to drop as he spent his energy dealing with immediate problems and letting his personal hygiene standards start to slip, slowly with brushing his teeth every second day and showering every third. As new stressors came and old stresses lingered his standards grew worse. {{expand}} (Consider adding more details about how this impacted John's well-being or relationships.) </div> {{RoundBoxBottom}} This template provides tips for the [[Motivation and emotion/Assessment/Topic|topic development]] exercise. Gradually remove these suggestions as the chapter develops. It is OK to retain some of this template content for the topic development exercise. Also consult the [[Motivation and emotion/Assessment/Chapter|book chapter guidelines]]. The Overview is typically consists of one to four paragraphs inbetween the scenario and focus questions. Suggested word count aim for the Overview: 180 to 330 words. {{tip| <quiz display="simple"> 1. What does hygiene relate to? |type="()"} - Showering and brushing your teeth - Washing sheets and towels - Cleaning the kitchen + All of the above</div> </quiz>}} {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is Hygiene and Hygiene standards? * What are the consequences of bad hygiene standards? * How is motivation involved in personal hygiene? * What are motivational methods of improving hygiene standards? * What are motivational methods of maintaining hygiene standards?{{RoundBoxBottom}} <quiz display="simple"> What are forms of hygiene? * Showering and brushing teeth * Cleaning the kitchen * Washing sheets and towels </quiz> [[File:Washing Hands.jpg|thumb|(I have a suggestion for "Washing Hands photo" you might like to use feel free to remove it)]]'''Focus on Psychology:''' Since the chapter deals with motivation, be sure to connect psychological theories of motivation (e.g., Maslow’s Hierarchy of Needs, Self-Determination Theory) with personal hygiene. This will ground the chapter in psychological science and provide a more structured foundation. ==Introduction to Hygiene standards:== === What is hygiene? === [[File:Hand washing man.jpg|thumb|Figure 1. Example of washing hands]] [[Healthy lifestyle/Hygiene|Hygiene]] is the act of taking care of one's body through washing hands (see figure 1.) or showering to remove dirt and germs from the body in an attempt to keep it clean or brushing your teeth to keep them healthy. Other forms of hygiene include keeping a clean living environment, such as throwing rubbish in bins and regularly washing clothes, sheets and towels. {{expand}} (Consider expanding the definition of hygiene by including mental and social aspects of cleanliness, not just physical.) === What are hygiene standards? === {{comment|Could include a clear definition of hygiene standards and possibly examples of different cultural or professional standards.}} === What causes bad hygiene standards? === {{comment|Could explore factors such as stress, mental health, socioeconomic conditions, etc.}} == Consequences of not maintaining hygiene == === Short term consequences === {{comment|Could add short-term effects like body odor, visible uncleanliness, or immediate discomfort from poor hygiene.}} === Long term consequences === * Aim for three to six main headings inbetween the [[#Overview|Overview]] and [[#Conclusion|Conclusion]] * Sub-headings can also be used, but ** avoid having sections with only one sub-heading ** provide an introductory paragraph before breaking into sub-sections {{comment|Expand on long-term effects such as increased risk of illness, dental problems, and social isolation due to prolonged poor hygiene.}} == Motivations role in personal hygiene == === How motivation applies to personal hygiene === ==== Disgust ==== Disgust plays a motivational role in personal hygiene as shown in the Book Chapter in [[Motivation and emotion/Book/2020/Disgust and disease avoidance motivation#Disgust Motivates Hygiene Behaviours|Disgust and Disease avoidance (4.3).]] ==== Anxiety ==== {{comment|You could can elaborate on how anxiety about social judgments or health risks can motivate hygiene behavior.}} ==== Comfort ==== {{comment|Could discuss how the desire for comfort, both physically and mentally, influences hygiene practices.}} === Identifying signs of bad personal hygiene maintenance === {{comment|This section could cover visible signs of poor hygiene and behavioral signs, such as avoidance of social situations.}} == Ways to increase motivation for personal hygiene == === Motivational methods to improve personal hygiene standards === === Motivation in maintaining personal hygiene standards === {{comment|Consider breaking down intrinsic and extrinsic motivators in this section, e.g., internal satisfaction from cleanliness vs. external rewards like social acceptance.}} ==Key points== * Provide at least three bullet-points per headingʔ and sub-heading, including for the Overview and Conclusion * Include key citations ==Figures== [[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]] * Use figures to illustrate concepts, add interest, and to serve as examples * Figures can show photos, diagrams, graphs, video, audio, etcetera * Embed figures throughout the chapter, including the Overview section * Figures should be captioned (using '''Figure #.''' and a caption). Use captions to explain the relevance of the image to the text/ * [[commons:|Wikimedia Commons]] provides a library of embeddable images * Images can also be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed * Refer to each figure at least once in the main text (e.g., see Figure 2) ==Learning features== Interactive learning features help to bring online book chapters to life and can be embedded throughout the chapter. {{anchor|Scenarios}} ;Scenarios * Scenarios or case studies describe applied/real-world examples of concepts in action * Case studies can be real or fictional * A case study could be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages) * It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. {{anchor|Feature box}} ;Feature boxes * Important content can be highlighted in a [[Motivation and emotion/Wikiversity/Feature box|feature box]]. But don't overuse feature boxes, otherwise they lose their effect. * Consider using feature boxes for: ** [[#Scenarios|Scenarios]], case studies, or examples ** Focus questions ** Tips ** Quiz questions ** Take-home messages ;Links * When key words are introduced, use [[Help:Links|interwiki links]] to: ** Wikipedia (e.g., [[w:Sigmund Freud|Sigmund Freud]] wrote about (e.g., [[w:Dreams|dreams]]) or ** Related book chapters (e.g., [[Motivation and emotion/Book/2020/Writer's block|writer's block]]) ;Tables * Use to organise and summarise information * As with [[#Figures|figures]], tables should be captioned * Refer to each table at least once in the main text (e.g., see Table 1) * [[Motivation and emotion/Wikiversity/Tables|Example 3 x 3 tables]] which could be adapted '''Table 1.''' Descriptive Caption Which Explains The Table and its Relevant to the Text - Johari Window Model {| class="wikitable" style="margin: auto; |- ! !! Known to self !! Not known to self |- | '''Known to others''' || Open area || Blind spot |- | '''Not known to others''' || Hidden area || Unknown |} ;Quizzes * Using one or two review questions per major section is usually better than a long quiz at the end * Quiz ''conceptual'' understanding, rather than trivia * Don't make quizzes too hard * Different types of quiz questions are possible; see [[Help:Quiz|Quiz]] Example simple quiz questions. Choose your answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> ==Conclusion== * The Conclusion is arguably the most important section * Suggested word count: 150 to 330 words * It should be possible for someone to only read the [[#Overview|Overview]] and the Conclusion and still get a pretty good idea of the problem and what is known based on psychological science {{tip|Suggestions for this section: * What is the answer to the sub-title question based on psychological theory and research? * What are the answers to the focus questions? * What are the practical, take-home messages? (Even for the topic development, have a go at the likely take-home message) }} ==See also== Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use these formats: * [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021) * [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity) * [[w:Self determination theory|Self determination theory]] (Wikipedia) * [[Motivation and emotion/Book/2024/Disgust and hygiene|Disgust and hygiene]] (Book chapter, 2024) {{tip|Suggestions for this section: * Present in alphabetical order * Use [[w:Letter case#Sentence casing|sentence casing]] * Include the source in parentheses }} ==References== Healthy lifestyle/Hygiene. (2024, January 15). ''Wikiversity. &nbsp;from [https://en.wikiversity.org/w/index.php?title=Healthy_lifestyle/Hygiene&oldid=2598812. <nowiki>https://en.wikiversity.org/w/index.php?title=Healthy_lifestyle/Hygiene&oldid=2598812</nowiki>.]'' ''Personal hygiene''. (2004). [Publisher not identified],. https://online.clickview.com.au/libraries/videos/3714437/personal-hygiene {{Hanging indent|1= Rosenberg, B. D., & Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091 }} {{tip|Suggestions for this section: * Important aspects of APA style for references include: ** Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]. Use "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then the author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop ** Provide the full doi as a URL and working hyperlink * The most common mistakes include: ** Incorrect capitalisation ** Incorrect italicisation ** Citing sources that weren't read or consulted }} ==External links== Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as presentations, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]]. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Anxiety]] [[Category:Motivation and emotion/Book/Disgust]] [[Category:Motivation and emotion/Book/Motivation]] 9w3oae8ioadipwn8sxpw5rfkyg2mj39 1 307245 2815009 2665468 2026-06-10T07:47:09Z Jtneill 10242 Jtneill moved page [[Talk:Motivation and emotion/Book/2024/Hygiene motivation]] to [[Talk:Motivation and emotion/Book/2026/Hygiene motivation]] without leaving a redirect 2665468 wikitext text/x-wiki == Helpful peer reviewed APA 7 article for Hygiene Motivation if you wish to add or remove!! == The concept of hygiene motivation is examined in the study by Curtis, V., & Cairncross, S. (2003), which emphasises that people's hygiene behaviours are primarily driven by the desire to avoid disgust rather than by health education alone. The authors contend that disgust is a potent motivator that can be used in public health campaigns to promote better hygiene practices, particularly in prevention of disease transmission. The study underscores the significance of comprehending emotional drivers like disgust in designing more successful health interventions. Curtis, V., & Cairncross, S. (2003). Effect of washing hands with soap on diarrhoea risk in the community: a systematic review. ''The Lancet Infectious Diseases'', ''3''(5), 275–281. <nowiki>https://doi.org/10.1016/S1473-3099(03)00606-6</nowiki> [[User:U3214564|U3214564]] ([[User talk:U3214564|discuss]] • [[Special:Contributions/U3214564|contribs]]) 08:19, 15 August 2024 (UTC) == Feedback and Suggestions == Hey Jack, I really like your book chapter so far! I found this article from Miller in 2020 which covers motivation to enact hygienic practices in the early stages of the COVID‐19 outbreak. I think this would be an interesting topic to explore and touch on within your book chapter. This article focusses primarily on the role of human behaviour in the COVID-19 outbreak and the speed of the spread across the world, and how human hygiene motivation was associated with the progression. Here is the reference to the article: Gibson Miller, J. (2020). Capability, opportunity, and motivation to enact hygienic practices in the early stages of the COVID‐19 outbreak in the United Kingdom. ''British Journal of Health Psychology'', ''25''(4), 856–864. <nowiki>https://doi.org/10.1111/bjhp.12426</nowiki> Good luck! :) - Alyssia [[User:Alyssia Myers|Alyssia Myers]] ([[User talk:Alyssia Myers|discuss]] • [[Special:Contributions/Alyssia Myers|contribs]]) 11:15, 15 August 2024 (UTC) Hello Jack, it looks like you are hard at work on your page, I've adding another figure to your page, hopefully it'll help you flesh out your article !![https://commons.wikimedia.org/wiki/File:OCD_handwash.jpg this is the image], you could use it to talk about personal hygiene, such as washing one's hands !!! Good luck [[User:Gabriel Geld|Gabriel Geld]] ([[User talk:Gabriel Geld|discuss]] • [[Special:Contributions/Gabriel Geld|contribs]]) 08:43, 16 August 2024 (UTC) <!-- Official topic development feedback --> {{METF/2024 |1= <!-- Title --> # The title and sub-title are correctly worded; casing has been corrected # I've abbreviated the sub-title (same meaning) |2= <!-- Headings --> # Well developed 2-level heading structure. Meaningful headings clearly relate directly to the core topic. # Consider abbreviating the two introductory sections (2 and 3) into a single section; it is OK to retain but keep these sections relatively brief, with embedded links to further information so that the chapter concentrates on psychological science # Remove trailing colons <!-- Alignment with focus questions --> # Good alignment between focus questions and heading structure |3= <!-- Overview--> <!-- Scenario --> # A scenario or case study is presented in a feature box at the start of this section # Add an image to the scenario or case study to help attract reader interest <!-- Description --> # Add a brief, evocative description of the problem/topic <!-- Focus questions --> # Focus questions are aligned with sub-title and top-level headings # Use sentence casing (Hygiene -> hygiene) # Move quiz into a subsequent section |4= <!-- Key points--> # Partial development of key points for some sections, with limited use of relevant citations <!-- Theory and research --> # Strive for an integrated balance of theory and research, with practical examples # It is unclear whether the best available psychological theory and research has been consulted in the preparation of this plan <!-- Conclusion --> # Conclusion (the most important section): ## Hasn't been developed ## What might the take-home, practical messages be? (What are the answer(s) to the question(s) in the sub-title and/or focus questions?) |5= <!-- Figure --> # A relevant figure is presented, captioned, and cited # The figure caption(s) provide(s) a clear, appropriately detailed description that is meaningfully connected with the main text # Caption citations should refer to Figure rather than figure (APA style) |6= <!-- Learning feature --> <!-- Interwiki links ---> # One use of in-text [[m:Help:Interwiki linking|interwiki links]] for the first mention of key terms to relevant Wikipedia articles and/or to other relevant book chapters <!-- Examples/case studies --> # Promising use of example(s)/case study(ies) <!-- Other --> # Consider including more examples/case studies, quiz question(s), table(s) etc. |7= <!-- References --> <!-- Overall --> # Insufficient <!-- Suggestions --> # Move non-academic / non-peer reviewed sources to External links # Are there any systematic reviews about this topic? |8= <!-- Resources --> # See also ## Not developed # External links ## Not developed |9= <!-- User page --> # Used effectively <!-- Description about self --> # Brief description about self provided – consider expanding <!-- Links to profile(s) --> # Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks. <!-- Link to book chapter --> # Link provided to book chapter |10= <!-- Social contribution --> # At least three different types of contributions with direct link(s) to evidence # The links to evidence can be improved (i.e., be more direct/accurate). For more info, see [[Motivation and emotion/Assessment/Chapter#Making and summarising social contributions|Making and summarising social contributions]] }} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:53, 18 August 2024 (UTC) == social contribution == hey, I like the case study you have used. linking it to how hygiene decline directly impacts him emotionally and mentally. You could expand on the motivation factors leading to this decline. you could link it to psychological theories and discuss motivational theories like Maslow’s hierarchy of needs or self-determination theory in relation to hygiene maintenance to add to the depth of your chapter. you could also expand on methods to increase hygiene motivation with actionable steps. Great work so far and good luck :) [[User:U3236683|U3236683]] ([[User talk:U3236683|discuss]] • [[Special:Contributions/U3236683|contribs]]) 07:00, 3 October 2024 (UTC) == General tips == Hello, Here are some suggestions to enhance your chapter: First, consider simplifying John’s scenario to make it more relatable. Clearly outline how his hygiene standards declined due to increased stress. This will help readers connect with his situation better. Make sure your figure captions are clear and directly related to the content, like explaining the relevance of washing hands in Figure 1.r. Make sure your figure captions are clear and directly related to the content, like explaining the relevance of washing hands in Figure 1. [[User:U3219927|U3219927]] ([[User talk:U3219927|discuss]] • [[Special:Contributions/U3219927|contribs]]) 11:35, 6 October 2024 (UTC) p41c9ygkkiq47h6c5xtzjymb21vm02t 1 308386 2814999 2661115 2026-06-10T07:39:09Z Jtneill 10242 Jtneill moved page [[Talk:Motivation and emotion/Book/2024/Functional motives theory and environmental activism]] to [[Talk:Motivation and emotion/Book/2026/Functional motives theory and environmental activism]] without leaving a redirect 2661115 wikitext text/x-wiki Hi! Please see the attached article linked [https://onlinelibrary.wiley.com/doi/epdf/10.1002/nvsm.431 here], which looks at young peoples motivations when it comes to environmental volunteering. Good luck! Iva [[User:IvaPuskarica|IvaPuskarica]] ([[User talk:IvaPuskarica|discuss]] • [[Special:Contributions/IvaPuskarica|contribs]]) = [[User:IvaPuskarica|IvaPuskarica]] ([[User talk:IvaPuskarica|discuss]] • [[Special:Contributions/IvaPuskarica|contribs]]) 04:57, 26 September 2024 (UTC) == social contribution == hey, you have a really interesting topic. it would be good to add in a case study- have you looked into the Greta Thunberg climate movement I think it ties environmental activism and Functional motives theory together effectively. you could reference some of the research which explores the work she has done on activism and how this has inspired collective activism. google scholar has great sources on this. best of luck :) [[User:U3236683|U3236683]] ([[User talk:U3236683|discuss]] • [[Special:Contributions/U3236683|contribs]]) 20:35, 3 October 2024 (UTC) qdcvry131znrn819cudg5hgz1fy3522 2 320737 2815019 2810757 2026-06-10T09:50:07Z Tommy Kronkvist 31941 User statistics. 2815019 wikitext text/x-wiki <div style="margin: 0 0 1em 0;">{{userpage}}</div> {{Userboxtop|toptext=Babel:}} {{#babel:sv|en-4|de-2|la-1}} {{Userboxbottom}} [[File:Sorbus torminalis Trunk and canopy.jpg|thumb|310px|The intracanopy of a Wild Service Tree, i.e. <small>''Torminalis glaberrima'' (Gand.) Sennikov & Kurtto, ''Memoranda Soc. Fauna Fl. Fenn.'' 93: 32 (2017).</small>]]<br /> Most of my wiki contributions are made to [[:species:Main Page|Wikispecies]] where I'm an administrator, bureaucrat and interface admin,<small>^{[https://species.wikimedia.org/w/index.php?title=Special:ListUsers&limit=1&username=Tommy_Kronkvist (verify)]}</small> to the Swedish Wikimedia Chapter [[WMSE:|Wikimedia Sverige]] (WMSE) where I'm an administrator,<small>^{(<span class="plainlinks">[https://se.wikimedia.org/w/index.php?title=Special:Användare&limit=1&username=Tommy_Kronkvist verify]</span>)}</small> and as administrator and interface administrator at the Swedish version of [[wikivoyage:sv:Huvudsida|Wikivoyage]].<small>^{(<span class="plainlinks">[https://sv.wikivoyage.org/w/index.php?title=Special:ListUsers&limit=1&username=Tommy_Kronkvist verify]</span>)}</small> So far (June 10, 2026), I've made just over 393,100 edits to 153 of the Wikimedia sister projects&nbsp;– the majority of them to Wikispecies and Wikidata. My global account information for all of Wikimedia can be found [[meta:Special:CentralAuth/Tommy Kronkvist|here]]. Swedish is my mother tongue&nbsp;– even though I was born in Finland&nbsp;– but I feel comfortable speaking and writing English and to some extent in German as well. Odd as it may seem, unfortunately I can't speak any Finnish even though I went to school there for a few years prior to moving to Sweden (see [[w:Swedish-speaking population of Finland|Swedish-speaking population of Finland]] in Wikipedia). I've lived all over Sweden but nowadays reside in Uppsala, the fourth biggest city and former capital of Sweden. I'm only the fourth generation named "Kronkvist". My family name consists of two parts: ''kron'' – a short form of the Swedish word ''krona'' meaning 'crown', as in coronation crown or tree crown – and ''kvist'', meaning 'bough' or 'twig'. Hence the name ''Kronkvist'' refers to a twig in the canopy of a forest. I'm the fourth generation of Kronkvist's. Prior to that our family name was ''Mattus'': an oeconym meaning "Matthew's Farm", dating back to at least 1637. {{Clear}} {{User committed identity|a6edd6d2fdbf82621f0cda4e5525c71f8da9b5dfd308242c3c63365e998c32c5406b75448380903265a5403edffd1a0435b61ac943f3c65870db9250f8b884a9|SHA-512|background=#e0e8ff|border=e0e8ff}} 94rw4d2bzdurhhgd4y20u4tqmgs8etg 14 322030 2814977 2718240 2026-06-10T06:59:56Z Jtneill 10242 added [[Category:Role-playing games]] using [[Help:Gadget-HotCat|HotCat]] 2814977 wikitext text/x-wiki [[Category:Motivation and emotion/Book/Games]] [[Category:Role-playing games]] gu2ahskxxtpu2zy9cr7nz168afxze9c 2 322600 2814894 2805056 2026-06-09T18:47:41Z IanVG 2918363 2814894 wikitext text/x-wiki {{user pt-BR-2}} {{Progress|4}} [[Image:00%.svg]] [[File:25%25.svg]] [[File:50%25.svg]] [[File:75%25.svg]] [[Image:100%.svg]] Hello! I am Ian Van Giesen, a former mechanical engineer (specifically in building services engineering) by trade and education. I am currently a graduate student in a solar energy and building energy efficiency program in France. I have a fairly wide set of professional and hobby-level interest that have brought me over many diverse corners of the internet! I hold a special place in my heart for open-source <span style="text-decoration:line-through;">software, projects, learning material, media</span> anything! Some of my specific interests now are: * [[openstreetmap:User:IanVG|OpenStreetMap]] (like wikipedia but for maps) * OpenStudio (open source GUI for building energy modeling), [https://unmethours.com/users/48879/ianvg/ Unmethours User page]. * Wikiversity Find my other wiki user pages here: <span style="padding-left:30px;">[[File:Commons-logo.svg|35px|link=commons:User:IanVG|Wikimedia Commons]]</span> <span style="padding-left:30px;">[[File:Wikipedia-logo-v2.svg|35px|link=w:User:IanVG|Wikipedia]]</span> <span style="padding-left:30px;">[[File:Wikidata-logo.svg|42px|link=d:User:IanVG|Wikidata]]</span> <span style="padding-left:30px;">[[File:Wikisource-logo.svg|42px|link=s:User:IanVG|Wikisource]]</span> I also have a website I've working on-again-off-again where I link to my different internet accounts and resources. www.vangiesendesigns.com Some wikiversity topics I am interested in: * [[Building services engineering]] * [[Rainwater harvesting]] * [[Thermodynamics]] * [[Engineering thermodynamics]] * [[Solar energy]] * [[Building performance simulation]] * [[Differential equations]] I am a strong believer in the ability of open-source education to reduce the barriers to students with access to the internet. I also strongly believe in the ability of quizzes and exams to be good diagnostic tools to determine understanding and acquisition of the material. == Courses I am working on == === [[Thermodynamics]] === * eventually this should be moved to: [[Introduction to thermodynamics]] * Added some quiz problems here: [[Engineering thermodynamics/Quizzes/Turbine and compressor problems]] ** Eventually add many more problems here I think. Could be a convenient location to mass store the problems. === [[Solar energy]] === * eventually I should create a course within this topic: [[Introduction to solar energy]] === [[Rainwater harvesting]] === === [[Building performance simulation]] === [[Introduction to building performance simulation]] This course will be based on the energiebatiment website found here, and will mostly consist of a translation from French to English. * Building energy performance * Based on this course: https://energiebatiment.com/ === Tensor Analysis === == Personal resources == === Making SVG's === [[User:IanVG/Drawio]] === Tables === * [[Table of physical constants]] * [[Physics Formulae/Thermodynamics Formulae]] === WikiTable Generators === * [https://www.tablesgenerator.com/mediawiki_tables Tables Generator] I like this the most because it's easy to import a wikitable directly into it. === Course I'd like to learn from === '''[[Instructional Design]]''' '''[[Electric Circuit Analysis]]''' * This course has a really great organizational layout. Also I found the evolution of the course over (as of this writing) around 17 years to be compelling. === Sample quizzes I can copy from === [[User:IanVG/Sandbox/Quizzes]] === Personal essays === [[User:IanVG/Essays]] === Wiki-specific things I found that are useful === [[User:IanVG/Wiki-specific]] == Specific wiki principles I like == === [[Syllabus of Economics/Learning bricks|Brick in the wall concept in the school of economics]] === Just adding some nuggets of information when you get the opportunity as it may prove useful to others down the road. === [[Syllabus of Economics|Economics syllabus]] === What I liked from this page is that modules using different educational resources hosted on wikiversity can help users tailor their educational learning experience based on their objectives. === [[Other Free Learning Resources]] === The idea that wikiversity can serve as a node to other free and active or free and archived resources is interesting to me. == External Resources == === Thermodynamics Opensource and free resources === * [https://pressbooks.bccampus.ca/thermo1 British Colombia/Yukon Pressbooks] * [https://srd.nist.gov/jpcrdreprint/1.1285884.pdf NIST PDF Document from 2000] with lots of thermodynamic properties for air * https://thermodynamique.fr/ ** Olivier Cleynen's thermodynamics textbook looks very interesting! ** https://thermodynamicsbook.com/thermodynamics-free-textbook.pdf *** This link is a download of the thermodynamics textbook in English. === Thermodynamics Non-opensources free resources === https://arroweng.wordpress.com/resources-dump/ == Sandbox == [[User:IanVG/Sandbox]] == Meta-Wikiversity resources == [[Wikiversity:Colloquium]] seems to be the only place like other wiki village pumps and so forth. [[Wikiversity:Community Portal|Wikiversity:Community portal]] is a good place to find links to community conversation and platforms. * irc://irc.libera.chat/wikiversity-en * https://web.libera.chat/?channel=#wikiversity [[Wikiversity:Chat]] is also another page with links to outside platforms. Adminstration categories: [[:Category:Candidates for speedy deletion]] === Personal statistics === https://xtools.wmcloud.org/ec-generalstats/en.wikiversity.org/IanVG d5t7giad7c71unezmymcjjpunv7jx2r 2814917 2814894 2026-06-09T20:04:28Z IanVG 2918363 2814917 wikitext text/x-wiki {{user pt-BR-2}} {{Progress|4}} [[Image:00%.svg]] [[File:25%25.svg]] [[File:50%25.svg]] [[File:75%25.svg]] [[Image:100%.svg]] Hello! I am Ian Van Giesen, a former mechanical engineer (specifically in building services engineering) by trade and education. I am currently a graduate student in a solar energy and building energy efficiency program in France. I have a fairly wide set of professional and hobby-level interest that have brought me over many diverse corners of the internet! I hold a special place in my heart for open-source <span style="text-decoration:line-through;">software, projects, learning material, media</span> anything! Some of my specific interests now are: * [[openstreetmap:User:IanVG|OpenStreetMap]] (like wikipedia but for maps) * OpenStudio (open source GUI for building energy modeling), [https://unmethours.com/users/48879/ianvg/ Unmethours User page]. * Wikiversity Find my other wiki user pages here: <span style="padding-left:30px;">[[File:Commons-logo.svg|35px|link=commons:User:IanVG|Wikimedia Commons]]</span> <span style="padding-left:30px;">[[File:Wikipedia-logo-v2.svg|35px|link=w:User:IanVG|Wikipedia]]</span> <span style="padding-left:30px;">[[File:Wikidata-logo.svg|42px|link=d:User:IanVG|Wikidata]]</span> <span style="padding-left:30px;">[[File:Wikisource-logo.svg|42px|link=s:User:IanVG|Wikisource]]</span> I also have a website I've working on-again-off-again where I link to my different internet accounts and resources. www.vangiesendesigns.com Some wikiversity topics I am interested in: * [[Building services engineering]] * [[Rainwater harvesting]] * [[Thermodynamics]] * [[Engineering thermodynamics]] * [[Solar energy]] * [[Building performance simulation]] * [[Differential equations]] I am a strong believer in the ability of open-source education to reduce the barriers to students with access to the internet. I also strongly believe in the ability of quizzes and exams to be good diagnostic tools to determine understanding and acquisition of the material. == Courses I am working on == === [[Thermodynamics]] === * eventually this should be moved to: [[Introduction to thermodynamics]] * Added some quiz problems here: [[Engineering thermodynamics/Quizzes/Turbine and compressor problems]] ** Eventually add many more problems here I think. Could be a convenient location to mass store the problems. === [[Solar energy]] === * eventually I should create a course within this topic: [[Introduction to solar energy]] === [[Rainwater harvesting]] === === [[Building performance simulation]] === [[Introduction to building performance simulation]] This course will be based on the energiebatiment website found here, and will mostly consist of a translation from French to English. * Building energy performance * Based on this course: https://energiebatiment.com/ === Tensor Analysis === == Community related resources == [[User:IanVG/Welcome user]] == Personal resources == === Making SVG's === [[User:IanVG/Drawio]] === Tables === * [[Table of physical constants]] * [[Physics Formulae/Thermodynamics Formulae]] === WikiTable Generators === * [https://www.tablesgenerator.com/mediawiki_tables Tables Generator] I like this the most because it's easy to import a wikitable directly into it. === Course I'd like to learn from === '''[[Instructional Design]]''' '''[[Electric Circuit Analysis]]''' * This course has a really great organizational layout. Also I found the evolution of the course over (as of this writing) around 17 years to be compelling. === Sample quizzes I can copy from === [[User:IanVG/Sandbox/Quizzes]] === Personal essays === [[User:IanVG/Essays]] === Wiki-specific things I found that are useful === [[User:IanVG/Wiki-specific]] == Specific wiki principles I like == === [[Syllabus of Economics/Learning bricks|Brick in the wall concept in the school of economics]] === Just adding some nuggets of information when you get the opportunity as it may prove useful to others down the road. === [[Syllabus of Economics|Economics syllabus]] === What I liked from this page is that modules using different educational resources hosted on wikiversity can help users tailor their educational learning experience based on their objectives. === [[Other Free Learning Resources]] === The idea that wikiversity can serve as a node to other free and active or free and archived resources is interesting to me. == External Resources == === Thermodynamics Opensource and free resources === * [https://pressbooks.bccampus.ca/thermo1 British Colombia/Yukon Pressbooks] * [https://srd.nist.gov/jpcrdreprint/1.1285884.pdf NIST PDF Document from 2000] with lots of thermodynamic properties for air * https://thermodynamique.fr/ ** Olivier Cleynen's thermodynamics textbook looks very interesting! ** https://thermodynamicsbook.com/thermodynamics-free-textbook.pdf *** This link is a download of the thermodynamics textbook in English. === Thermodynamics Non-opensources free resources === https://arroweng.wordpress.com/resources-dump/ == Sandbox == [[User:IanVG/Sandbox]] == Meta-Wikiversity resources == [[Wikiversity:Colloquium]] seems to be the only place like other wiki village pumps and so forth. [[Wikiversity:Community Portal|Wikiversity:Community portal]] is a good place to find links to community conversation and platforms. * irc://irc.libera.chat/wikiversity-en * https://web.libera.chat/?channel=#wikiversity [[Wikiversity:Chat]] is also another page with links to outside platforms. Adminstration categories: [[:Category:Candidates for speedy deletion]] === Personal statistics === https://xtools.wmcloud.org/ec-generalstats/en.wikiversity.org/IanVG oy8r8lssr3b7oltfok2zjao648s5qzr 0 322688 2815012 2763425 2026-06-10T07:51:18Z Jtneill 10242 /* Online vs. Offline */ + Table 1 caption 2815012 wikitext text/x-wiki {{title|Music and social bonding:<br>How does shared musical experience foster emotional connection and group cohesion?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Music During the Social Hour (7255580614).jpg|thumb|'''Figure 1'''. Music brings people together, whether it's playing, listening or moving to the sound.]] ; Imagine this ... It's the height of COVID-19{{ic|add link to Wikipedia article}}, and you've been in isolation. Lonely and bored, you try to seek out some form of entertainment online. You come across a new band that makes music you really enjoy. You post about them, and have really insightful discussions with others online about their music. You share their songs with your friends, and find that you all have something new to talk about. You send their music to the family group chat, and strike up a conversation about how music has changed over the years. Music becomes a way to bond, to create new relationships and support old relationships (see Figure 1). {{RoundBoxBottom}} As human beings, social connection is an extremely important aspect of life. Social connectedness can be defined as a sense of confidence and comfort in a larger social group, and a sense of connection in social relationships (McLoughlin et. al., 2019). This is important for continued health and wellbeing outcomes, and a lack of social connection can become detrimental. Research has shown that social isolation and loneliness may increase chances of dying by about 30%, with no significant change when investigating outside factors such as age or sex (Bzdok et. al., 2020). More specifically in the case of cardiovascular-related deaths, the top three most impactful factors included whether the patient stopped smoking, how frequently they received social support, and how closely they were intertwined with their social groups. This just the physical impact, with an equally negative affect on mental health demonstrated as well. Research has found that without social connection, people may develop a lack of trust in others, low self-esteem, and may experience lower positive wellbeing such as depressive feelings and isolation (McLoughlin et. al., 2019). There is a clear need to promote social connectedness to improve the quality and length of life for everyone. Music is one such avenue to boost social connection. Several research articles, such as articles from Dieterich-Hartwell, and Ngobeni and Bopape {{f}} have shown that music can influence the emotional state of listeners, ranging from creating positive feelings and influencing emotional perception of certain environments (2019; 2024). This is reflected in certain therapy styles, including [[Music therapy|Music Therapy]] and [https://dtaa.org.au/therapy/#:~:text=DMT%20provides%20a%20range%20of,%3A%20attention%2C%20memory%2C%20executive%20function Dance Movement Therapy], in which interactions with music are used to allow patients to express and experience emotions in non-verbal ways (Dieterich-Hartwell, 2019). Music, then, is an avenue for self-exploration (Frith, 2007), a non-verbal therapeutic technique (Dieterich-Hartwell, 2019), and a way to influence emotional states (Ngobeni & Bopape, 2024). With all this in consideration, the following focus questions will outline the connection between social bonding and music and why this is important. {{RoundBoxTop|theme=3}} '''Focus questions''' * How does engaging in music promote positive mindsets and social bonding? * What are the benefits of contributing in some way to the music vs listening to the music and how does this promote community? * What is the difference between online and offline engagement with music? {{RoundBoxBottom}} == Promoting positive mindsets and social bonding == As per Savage et. al.'s definition, "music is for social bonding" is an equivalent statement to "vision is for seeing" (2021). Social bonding is an overarching theme that encompasses the many social-based concepts that music promotes. Even on the surface, consider how music can be delivered to the audience. Busking, festivals, and even social media are just a few examples, and there's an unending list of more avenues. Music is to be shared, which in turn, is facilitating a social encounter. Culturally, too, music is a shared social feature (Crooke et. al., 2023). Music is a universal common ground that can lead to social bonding across cultures, and is also a way in which people can learn to appreciate the uniqueness of individual cultures through a familiar lens. This can also be shown within smaller groups. Zaatar et. al. found that group Music Therapy assisted children with [[Autism spectrum|Autism Spectrum Disorder]] with improving their communication skills, and that there was improvements found in brain connectivity, directly improving their quality of life (Zaatar et. al., 2024). Interventions like this directly improves the social outcomes of people who would otherwise be at risk of potential isolation and difficulty bonding with others. Dance, a similarly cross-cultural and cross-group mode of expression, is strongly connected with music. Dancing itself can encourage social bonding as it is a common method of communication and exercise that is extremely group-friendly (Zaatar et. al., 2024). It is also a style of exercise that, common with other forms of exercise, produces a positive mood in people. Singing is another music-based activity that produces a positive mood when singing in a group, with research suggesting that amateurs experience higher emotional well-being singing when compared with professional singers (Dieterich-Hartwell, 2019). Being able to be in sync with a group and harmonise successfully also has a sense of social reward that benefits group members and facilitates social bonding (Savage et. al., 2021). This not only further supports the importance of engaging with music in group settings, but also implies that enjoying music with a group of like-minded individuals is more important than a high level of skill in the music-based activity of choice. <quiz display="simple"> {"Music is linked to social bonding" is an equivalent statement to "vision is linked to seeing": |type="()"} + True - False {Dancing and singing have what common links?: |type="()"} + Group-friendly activities that promote positive moods and allow for connection with like-minded people - Activities that promote exercise that leads to improved social bonding the better at the activity you get </quiz> == Contribution to music-making == Research suggests that there's a benefit to being an active participant, be it through making, performing or moving to the music (Dieterich-Hartwell, 2019; Frith, 2007). By playing a more active role, there's a deeper understanding and connection with the music and group members. However, this does not mean that there is zero benefit to just listening to music. === Contributor vs. audience === There are many different ways to actively contribute to and be an audience to music in some manner, however the most highly researched ways are through creating the music track itself and through dancing to the music. Dancing is a non-verbal communicator, using body language as an expressive tool that contributes to the positive feelings also associated with music in general (Dieterich-Hartwell, 2019). In a group setting, this allows people to communicate with each other and feel more connected using the powerful communication tool of body language. However, dancing with others is not the only way a person can benefit from dance. If a person sits and watches someone else dance, their brain is tracking the movements and will mimic them. Through this mechanism, the brain produces a very similar response in the audience that it does in the dancer (Dieterich-Hartwell, 2019). What this means is, an audience member can sense and empathise with the emotions expressed in the movements using a form of neurological mirroring. In terms of the emotional and social benefits of dance, a person does not need to actively take part in the movements to benefit from the activity.{{f}} Unlike dance, music production benefits are much higher as an active participant. Performing and producing the music itself, be it vocally or instrumentally, has a significant community-based reward that is not seen mimicked in audience members (Savage et. al., 2021; Levstek et. al., 2021). Singing in harmony with others, learning and developing repetition, and performing music in sync connects the group and facilitates social connectedness{{f}}. Furthermore, engaging creatively with music develops cognitive and motor skills, which contribute to wellness (Zaatar et. al., 2024). When comparing the neural pathways activated when engaged in music production and when engaging socially, research by Zaatar et. al. found that the neural processing was quite similar in nature (2024). What this means is, when creating music, the brain is processing the information in a similar way as to how it processes social activities. === Promoting community through creation === Music is a significant part of many cultures that has evolved throughout history as a way for humans to bond socially (Freeman, 1998). Anthropologists and psychologists have found evidence of historical rhythmic-based activities, that suggest an old connection of music-based sounds and dance as a community communication tool{{f}}. Today, this community tool is still used to bring people together. Being personally involved in music making not only encourages social bonding but also helps with self-expression, emotion management and feelings of belongingness{{f}}. It helps create a community that mutually understands each other through creativity (Levstek et. al., 2021). Creation is a very personal experience, so by sharing it with others it allows for a greater sense of community bonding to occur. Mirroring someone else's creation through instrument playing, movement, and even by mirroring facial expression allows for a higher sense of empathy and sensing of the emotions felt by the original creator (Dieterich-Hartwell, 2019). Understanding is a key bridge to community. A clear example of a community built on creation is members of a music production group{{example}}. A literature review from 2024 condensed the majority of research on the benefits of music groups on social connection and fostering of community and found a consistent theme of increased wellbeing of members in music groups (Zaatar et. al.). Researchers theorised that an increase in endorphins, oxytocin and other hormones could be attributed to the increase in mood and social outcomes. In particular, oxytocin is important for social bonding as it supports eye contact, cooperation and communication skills. The literature review also found that music-based group activities, such as group singing, could potentially be correlated with an increase in pain tolerance for group members. Not only are music creation group important for social outcomes, but there is potential for general quality of life and wellbeing improvements. <quiz display="simple"> {When engaging in music, both dance and music production benefit audience members and contributors equally: |type="()"} - True + False {Social bonding through music has been shown as a tool evolved throughout history: |type="()"} + True - False </quiz> == Online vs. Offline == Music can be presented and enjoyed both live and pre-recorded, and the always evolving online world supports the sharing of music without having to go watch a performer live. However, as there is a difference in experience between taking an active role in the music versus being an audience member, whether or not online and offline music experience is different in effectiveness comes into question{{rewrite}}. A study done to investigate the role of music groups in isolation outcomes during Covid-19 found that the music groups were extremely important in nurturing connection and community during what would otherwise be an isolating time (Levstek et. al., 2021). However, when compared to the role of in-person music groups, the online group could not replicate the direct social connection outcomes. While only one aspect of the many positive aspects of music groups, social connection is highly important to all people and the direct benefits are just as important. A potential reason for this difference is the impact of facial expression and body language as communication tools. A study found that, when participants were asked to rate their social experiences, face-to-face interaction was consistently scored higher than text-based interactions (Bzdok et. al., 2020). This included both in person meetings and video calls, which were rated evenly. The importance of engaging with music in real life is also evident at live performances versus listening to a recorded music track. Live music is uniquely important for in-person connection, self-expression being recognised in a social setting, and to foster a true exploration into how performance works on an individual level (Frith, 2007). While recorded music can emulate the emotional response expected from listening to music, only live performance can be in essence a conversation between performer and audience (Trost et. al., 2024). At a live show, a performer can respond to the audience using unique and dynamic musical modulation using techniques such as tempo changing, loudness fluctuations and simple improvisation to express emotion in the moment (Czepiel et. al., 2021). Two separate studies from 2021 and 2024 studied the physiological and neurological reaction respectively of audience members at a live performance when comparing to listening to a recorded performance and found significant fluctuations. In 2021, they found that audience members' physiological reactions such as cardiorespiratory and facial responses would be similar across the entire audience during notable moments of the performance (Czepiel et. al., 2021). These moments included sudden volume changes and tempo shifts. In 2024, the left amygdala was found to have significantly higher activity during live, emotive and adaptive performances (Trost et. al., 2024). The study found that only live music could produce this emotional response in audiences, which is key to the social bonding aspect of music. <quiz display="simple"> {There is no benefit to listening to live music performances over recordings: |type="()"} - True + False {What is key to the difference between working in a music group online and in person? |type="()"} + Direct communicative tools such as facial expression and body language, that are only present face-to-face - Online groups are better as they can achieve a wider group of people that cannot be achieved in person </quiz> '''Table 1.''' Neurological and Physiological Responses to Live and Pre-recorded Music {| class="wikitable" style="margin: auto; |- ! !! Live Shows !! Pre-recorded Music |- | '''Neurological Response'''|| Amygdala reacts || No notable shift |- | '''Physiological Response''' || Reactions are mimicked amongst the audience || Reactions are unique to the singular audience member |} ==Conclusion== Social bonding and music are both closely connected concepts, {{g}}one without the other being akin to vision without sight. Without social cohesion, people are about 30% more likely to die, with cardiovascular disease outcomes being predicted almost entirely by the social connectedness of the patient (Bzdok et. al., 2020). Without social connections, people are more likely to experience low self-esteem, lower positive wellbeing, and isolation from peers (McLoughlin et. al., 2019). Music is an avenue for self-exploration that helps to promote group cohesion and improves social outcomes. Music promotes a positive mindset in that music is proven to have positive influence over the emotions of listeners (Savage et. al., 2021). With a rich history of community cohesion and cross-cultural connection, music is a shared social feature that can enhance and promote group cohesion across multiple cultures (Crooke et. al., 2023). Dance is one way of interacting with music, which has been shown to promote social bonding through a group-friendly mode of non-verbal communication as well as promoting a positive mood due to it being a form of exercise (Zaatar et. al., 2024). Singing has the same outcomes, with research showing that a group of amateurs had a more positive outcome over professionals, suggesting that the important part is not skill, but rather bonding with a group of like-minded individuals of a similar skill level (Dieterich-Hartwell, 2019). Rather than just listening to music, actively engaging with music is important to both receiving higher wellbeing outcomes as well as superior community building. From the perspective of dance, body language allows people to express and communicate with each other in a way that may be more open and comfortable, and allows for better social cohesion (Dieterich-Hartwell, 2019). However, even audience members can benefit due to the brain mirroring the movements mentally, which produces a very similar response in the audience member as the dancer. Producing the music in a group, however, is associated with significantly better outcomes that can not be mimicked by viewership (Levstek et. al., 2021). Repetition, successful harmonisation, and performance all contribute to social connectedness and promotes community through the creation of something unique and personal. Music can be engaged with offline and online, which is important in a digital age. Face-to-face communication, whether in person or on video call, is important for social bonding and can not be replicated through alternative means (Levstek et. al., 2021; Bzdok et. al., 2020). Live performance art is also associated with far more emotive responses in people than the recorded alternative of music listening. Studies showed that the amygdala uniquely reacts to emotive performances (Trost et. al., 2024), and that physiologically audience members react in extremely similar ways to sudden notable shifts in the performance (Czepiel et. al., 2021). These results could not be replicated using music recordings. Music is an important way in which to promote community and social bonding in a multitude of ways, whether through listening, dancing, or performing. With a long history, music proves to be extremely effective in many ways both as a facilitator and direct catalyst for social connectedness and development of community with which to better support the people in them. Knowing this, better quality of life can be maintained through understanding how these mechanisms work and how to apply them to each individual. ==See also== * [[Motivation and emotion/Book/2022/Motivational music and exercise]] (Book chapter, 2022) * [[wikipedia:music_theory|Music theory]] (Wikipedia) ==References== {{Hanging indent|1= Bzdok, D. & Dunbar, R. I. M. (2020). The neurobiology of social distance. ''Trends in Cognitive Sciences, 24''(9), 717–733. https://doi.org/10.1016/j.tics.2020.05.016. Crooke, A. H. D., Thompson, W. F., Fraser, T., & Davidson, J. (2023). Music, social cohesion, and intercultural understanding: a conceptual framework for intercultural music engagement. ''Musicae Scientiae, 28''(1), 18–38. https://doi.org/10.1177/10298649231157099 Czepiel, A., Fink, L.K., Fink, L.T., Wald-Fuhrmann, M., Tröndle, M., & Merrill, J. (2021). Synchrony in the periphery: inter-subject correlation of physiological responses during live music concerts. ''Scientific Reports, 11''. https://doi.org/10.1038/s41598-021-00492-3 Dieterich-Hartwell, R. M. (2019). Music, movement, and emotions: an inquiry with suggestions for the practice of dance/movement therapy. ''Body, Movement and Dance in Psychotherapy, 14''(4), 249–263. https://doi.org/10.1080/17432979.2019.1676310 Freeman, W. J. (1998). A neurobiological role of music in social bonding. ''The Origins of Music, 22'', 411–424. https://escholarship.org/uc/item/9025x8rt Frith, S. (2007). Live music matters. ''Scottish Music Review, 1''(1), 1–17. Levstek, M., Barnby R. M., Pocock, K. L., & Banerjee, R. (2021). “It all makes us feel together”: young people's experiences of virtual group music-making during the covid-19 pandemic. ''Frontiers in Psychology, 12''. https://doi.org/10.3389/fpsyg.2021.703892 McLoughlin, L. T., Spears, B. A., Taddeo, C. M., & Hermens, D. F. (2019). Remaining connected in the face of cyberbullying: why social connectedness is important for mental health. ''Psychology in the Schools, 56''(6): 945–958. https://doi.org/10.1002/pits.22232 Ngobeni, N. C., & Bopape, M. H. (2024). Music as a strategic instrument for enhancing social coherence - a comprehensive need. ''Jurnal Seni Musik, 13''(2), 90–99. https://doi.org/10.15294/jsm.v13i2.12260 Savage, P. E., Loui, P., Tarr, B., Schachner, A., Glowacki, L., Mithen, S., & Fitch, W. T. (2021). Music as a coevolved system for social bonding. ''Behavioral and Brain Sciences, 44''(59). https://doi.org/10.1017/S0140525X20000333 Trost, W., Trevor, C., Fernandez, N., Steiner, F., & Frühholz, S. (2024). Live music stimulates the affective brain and emotionally entrains listeners in real time. ''Proceedings of the National Academy of Sciences, 121''(10). https://doi.org/10.1073/pnas.2316306121 Zaatar, M. T., Alhakim, K., Enayeh, M., & Tamer, R. (2024). The transformative power of music: insights into neuroplasticity, health, and disease. ''Brain, Behavior, & Immunity - Health, 35''. https://doi.org/10.1016/j.bbih.2023.100716. }} ==External links== * [https://www.youtube.com/watch?v=evVRxrOo5iw How Music Can Break Social Barriers | Dr. David Greenberg] (TEDx Talks, Youtube) * [https://www.youtube.com/watch?v=Y7tiIv_3yho How We Process Music | Neuroscience for Musicians] (Danny Li, Youtube) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Music]] [[Category:Motivation and emotion/Book/Social psychology]] sfwil19zrq85qlc8az9ovmxzt1tsjak 0 322693 2814974 2762297 2026-06-10T06:58:22Z Jtneill 10242 + more specific category 2814974 wikitext text/x-wiki {{title|Self-disclosure motivation:<br>What motivates people to share personal information with others?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} [[File:Janet Fisher Distant thoughts.jpg|thumb|160px|'''Figure 1'''. A sense of calm and reflection, symbolising Mia’s moment after leaving the bus — processing what she has shared and the unexpected openness of her conversation]] ;Imagine this ... It is late in the afternoon, and Mia is sitting on a crowded bus after a long day. A stranger sits next to her, and within minutes, they are deep in conversations {{g}} By the time Mia reaches her stop, she has shared details about her recent breakup, her struggles with balancing study and work, and even her fear of failing her degree. When she steps off the bus, walks for three mins {{g}} finds herself sitting in a field of flowers. Mia feels a rush of relief—it’s as if a weight has been lifted just by speaking the words aloud (see Figure 1). Why did Mia feel comfortable enough to open up? Was it the anonymity of the stranger. {{g}} the relief of being listened to, or the emotional weight she had been carrying all day? {{RoundBoxBottom}} This moment illustrates the power of self-disclosure: sharing personal thoughts, feelings, and experiences with others. Disclosure is not only a personal act but also a deeply social one. People reveal themselves to seek comfort, build trust, or even manage impressions in relationships. Self-disclosure can bring closeness and support, but it also carries risk-being judged, rejected or misunderstood. This chapter explores the motivations behind such moment of self-disclosure and the psychological, social and situational factors that drive people to share personal information with others. {{RoundBoxTop|theme=3}} '''Focus questions''' * Why do people disclose personal information even when it can be risky? * How psychological theory explains what motivates self-disclosure{{g}}? * What are the main reasons people choose to share personal information? * How digital communication platforms changed the motivations and patterns of self disclosure{{g}}? {{RoundBoxBottom}} == What is self-disclosure? == * '''[[wikipedia:Self-disclosure|Self-disclosure]]''' is a process of communication by which one person reveals information about themselves to another. The information can be descriptive or evaluative, and can include thoughts, feelings, aspirations, goals, failures, successes, fears, and dreams, as well as one's likes, dislikes, and favorites (wikipedia description{{f}}). Self-disclosure is a fundamental process that contributes to the development, maintenance and deepening of close relationships. The motivation to disclose stems partly from the human need to form and maintain enduring, positive, and meaningful relationships, satisfying a basic drive for belonging within both intimate dyads and broader social groups such as families, workplaces, and cultural communities ''(Rimé, 2016)''. The nature of disclosure can range from superficial exchanges to highly intimate revelations, depending on the depth of the relationship and the context of communication ''(Rimé, 2016)''. Altman and Taylor’s ''(1973)'' Social Penetration Theory explains that disclosure is a central mechanism through which individuals transition from acquaintances to close relational partners. At a surface level, disclosure may involve relatively inconsequential topics such as food preferences, daily activities, or recent events. However, it can also extend to highly personal and sensitive information, such as fears, religious convictions, or potentially stigmatizing experiences including abortion, sexual orientation, or illness ''(Rimé, 2016)''. == Theoretical foundations == * [[wikipedia:Social_penetration_theory|Social Penetration theory]] == Costs and benefits of disclosure == *trust and relationship satisfaction *vulnerability and regret *oversharing and privacy risks == Applications and implications == *self-disclosure in therapy and support groups *social media *impacts on mental health and wellbeing ==Perpectives== [[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]'''Evolutionary''' From an evolutionary perspective, self-disclosure is considered both a valuable and controversial intervention within the therapeutic process because it engages and supports interpersonal motivational systems; attachment, caregiving, ranking and sexual system central to human survival and bonding, key drives identified by the Evolutionary Theory of Motivation (ETM). Approximately 90% of therapists report using self-disclosure, which generally involves sharing personal information with patients{{f}}. Patients tend to respond more positively to therapists who employ disclosure in a balanced way, as this activates affiliative motives, strengthens trust, and reduces anxiety by signalling warmth and authenticity{{f}}. Monticelli et al. ''(2022)'', introduces motivational monitoring to evaluate how therapist and client motivational systems interact during clinical sessions. Research conducted by Simonds and Spokes ''(2017)''<ref>{{Cite web|url=https://pubmed.ncbi.nlm.nih.gov/28060578/|title=Therapist self-disclosure and the therapeutic alliance in the treatment of eating problems}}</ref> found that in patients with eating disorders, therapists{{g}} self-disclosure helped regulate the social emotion of shame and encouraged disclosure, enhancing therapeutic relationship and improving treatment outcomes. ETM highlighted that such outcomes reflect evolved needs for affiliation and emotional regulation, showing how self-disclosure can help restore therapeutic bond when it is threatened by patient withdrawal or criticism. Monticelli et al. (2022) caution that disclosure must remain patient-focused; when driven by the therapist’s needs, it risks undermining the alliance and shifting attention away from the patient’s adaptive goals. ETM emphasizes ongoing monitoring of interpersonal motivational systems—both in transcripts and in real time—to ensure that disclosure interventions remain aligned with evolved mechanisms for connection, trust, and healing. '''Biological''' Biological perspectives highlight self-disclosure as an intrinsically rewarding behaviour supported by neural reward systems. Tamir and Mitchell ''(2012)'',{{g}} found in their study that individuals are often willing to forgo monetary rewards to disclose information about themselves, suggesting that disclosure has inherent subjective value comparable to primary rewards such as food or sex. Their neuroimaging findings revealed strong activation in the mesolimbic dopamine system, including the nucleus accumbens and ventral tegmental area, during acts of self-disclosure. This system, which is typically engaged by primary rewards and secondary reinforcers, also responds to social rewards such as agreement, humour, and attraction. Interestingly, both components of self-disclosure—the act of self-reflection and the act of communicating one’s thoughts to others—elicited reward-related neural activity, with effects magnified when disclosures were shared interpersonally. Participants consistently chose options where they could disclose information about themselves even when it meant earning less money. These results support the hypothesis that humans possess a species-specific motivation to broadcast their beliefs and knowledge, a tendency that may confer adaptive advantages by strengthening social bonds, eliciting feedback, and enhancing collective learning. Thus, from a biological standpoint, disclosure is motivated by evolved reward mechanisms that sustain the cooperative and highly social nature of human groups. ''(Tamir & Mitchell, 2012)''. '''Social Cognitive''' From a social cognitive perspective, self-disclosure is shaped by reinforcement dynamics: when individuals receive positive feedback in response to sharing personal information, they are motivated to disclose more deeply over time ''(Baack, Fogliasso, & Harris, 2000)''. Disclosure is not random but is influenced by three key factors: personal characteristics, reward–cost assessments, and situational context. People disclose peripheral aspects of themselves more readily, while the innermost “core” of personality—encompassing self-concept, values, and vulnerabilities—is reserved for those with whom trust has been established. Social Penetration Theory ''(Altman & Taylor, 1973)'' describes this process as gradual, moving through “layers” of the self, much like peeling an onion. Early stages of relationships involve rapid exchange of surface-level information, but progress slows as disclosure moves toward central values and self-defining traits, which carry higher risks of rejection. Whether this deeper penetration occurs depends on the balance of rewards and costs: when both partners perceive benefits as outweighing risks, disclosure accelerates, intimacy deepens, and individuals are more willing to reveal sensitive aspects of their identities. Conversely, when perceived costs outweigh rewards, disclosure stalls, and relational growth is limited ''(Baack, Fogliasso, & Harris, 2000)''. Self-disclosure is seen as an indicator of the level of sincerity of expression and the level of trust and love in bilateral relations ''(Arslan, 2021)''. Gender differences in self-disclosure have long been debated, {{g}} research on gender and self-disclosure shows that men and women differ not simply in how much they share, but in the types of experiences they are willing to discuss through meta-analyses{{rewrite}}. Recent work by Carbone et al. ''(2024)'' provides clearer evidence by examining not only disclosure behaviour but also the psychological desire to disclose. Across three studies, they found a robust interaction between gender and information valence: men and women were equally likely to want to share positive information, but men were significantly less inclined than women to disclose negative experiences. Importantly, the research also highlighted different underlying motivations, with men more likely to disclose as a means of self-enhancement and image management, while women more often disclosed to seek comfort and social support. These findings suggest that gender differences in disclosure cannot be reduced to overall frequency but instead depend on the type of information shared and the social functions disclosure serves ''(Carbone et al., 2024)''. ==Learning features== ;Quiz <quiz display=simple> {Which of the following is a common motivation for self-disclosure?: |type="()"} - To increase stress + To build trust and intimacy - To avoid social interaction - To remain anonymous {Trust plays a key role in whether people choose to share personal information |type="()"} + True - False </quiz> ==Conclusion== Self-disclosure is a multifaceted process motivated by psychological, social, and biological factors that both enrich and complicate human relationships. It can strengthen intimacy, foster trust, and enhance well-being, yet it also carries risks such as vulnerability, regret, and privacy loss. Evolutionary and biological perspectives emphasize disclosure as rooted in adaptive survival needs and neural reward mechanisms, while social cognitive approaches and Social Penetration Theory highlight reinforcement dynamics, reward–cost assessments, and the gradual movement from superficial to intimate layers of the self. Individual differences, including personality traits and gender, further shape the motivations and patterns of disclosure, demonstrating that disclosure is not uniform but context-dependent. In applied contexts, disclosure plays a critical role in therapeutic settings, interpersonal relationships, and digital communication. In therapy, balanced disclosure by both clients and clinicians can regulate emotions, strengthen alliances, and improve outcomes. In everyday relationships, disclosure maintains trust and provides emotional relief, while in digital environments it offers opportunities for connection alongside heightened risks to privacy and authenticity. Ultimately, the motivation to disclose reflects enduring human needs for belonging, authenticity, and emotional regulation, illustrating why people continue to share personal information even when potential costs are apparent. Research on self-disclosure motivation remains limited in several ways. Much of the existing evidence relies on self-report data, which may not accurately capture the depth or context of disclosure. Cross-cultural studies are still underdeveloped, leaving open questions about how cultural norms influence disclosure practices and motivations. {{tip|Self-disclosure is motivated by the need for connection, authenticity, and emotional relief. While sharing personal information can create risks, it also strengthens trust, builds intimacy, and supports well-being across relationships, therapy, and digital spaces.}} ==See also== * [[wikipedia:Social_penetration_theory|Social Penetration Theory]] ==References== {{Hanging indent|1= Altman, I., & Taylor, D. (1973). Social penetration: The development of interpersonal relationships. Holt, Rinehart & Winston. Arslan, S. (2021). Self-disclosure: A psychological analysis. International Journal of Educational Research Review, 6(4), 453–455. https://doi.org/10.24331/ijere.1010942 Baack, D. W., Fogliasso, C. E., & Harris, J. (2000). The personal impact of ethical decisions: A social penetration theory. Journal of Business Ethics, 24(1), 39–49. https://doi.org/10.1023/A:1006037510116 Carbone, E., Loewenstein, G., Scopelliti, I., & Vosgerau, J. (2024). He said, she said: Gender differences in the disclosure of positive and negative information. Journal of Experimental Social Psychology, 110, Article 104525. https://doi.org/10.1016/j.jesp.2023.104525 Luo, M., & Hancock, J. T. (2020). Self-disclosure and social media: Motivations, mechanisms and psychological well-being. Current Opinion in Psychology, 31, 110–115. https://doi.org/10.1016/j.copsyc.2019.08.019 Monticelli, F., Tombolini, L., Guerra, F., Liotti, M., Monticelli, C., Gasperini, E., Russo, M., Novaretto, S., La Vista, L., Mallozzi, P., Imperatori, C., & Del Brutto, C. (2022). Using motivational monitoring to evaluate the efficacy of self-disclosure and self-involving interventions. Journal of Contemporary Psychotherapy, 52(3), 217–225. https://doi.org/10.1007/s10879-022-09533-y Pratiwi, N. I., Maharani, D. P., Arniti, N. K., Anggreswari, N. P. Y., Suparna, P., & Haes, P. E. (2025). Interpersonal communication in the dynamics of “friends with benefits” among Generation Z. Jurnal Komunikatio, 11(1), 1–14. https://doi.org/10.30997/jk.v11i1.16394 Rimé, B. (2016). Self-disclosure. In J. D. Wright (Ed.), International encyclopedia of the social & behavioral sciences (2nd ed., Vol. 21, pp. 348–354). Elsevier. https://doi.org/10.1016/B978-0-12-397045-9.00075-6 Simonds, L. M., & Spokes, N. (2017). Therapist self-disclosure and the therapeutic alliance in the treatment of eating problems. Eating Disorders, 25(2), 151–164. https://doi.org/10.1080/10640266.2016.1269557 Tamir, D. I., & Mitchell, J. P. (2012). Disclosing information about the self is intrinsically rewarding. Proceedings of the National Academy of Sciences, 109(21), 8038–8043. https://doi.org/10.1073/pnas.1202129109 }} ==External links== {{expand}} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Communication]] [[Category:Motivation and emotion/Book/Self-disclosure]] adc6d6vv5pa9ihuw8qct583nofvxgnc 0 322972 2814969 2762275 2026-06-10T06:47:57Z Jtneill 10242 Fix category 2814969 wikitext text/x-wiki {{title|Role-playing game motivation:<br>What are the key psychosocial elements of RPGs that enhance player engagement?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} ;Scenario Toby has work early in the morning tomorrow but is currently playing his favourite massively multiplayer online role-playing game called World of Warcraft, {{g}} he has recently joined a guild in this game and has made some new friends in it who have invited him to join a raid they have planned for later that night. Toby has accepted the invitation even though this raid may keep Toby up late into the night and affect his work tomorrow {{g}} he agreed to join his guildmates in the difficult and time-consuming activity hoping to get some high-level rewards for completing it. Which intrinsic and extrinsic motivational factors may be at play in this scenario? Are there any conflicting motivations between playing his game and being rested for work the next day? {{RoundBoxBottom}} [[w:role-playing game|Role-playing games]] are a popular form of entertainment and have evolved into several different mediums from computer games to live action, allowing players to create characters and narratives to explore in fantasy settings. These allow players an outlet to explore different identities, perspectives and, social interactions within a safe environment. But what motivates players to get into the hobby and continue to engage with it? Through the [[w:self-determination theory|self-determination theory]] (SDT; Deci, & Ryan, 1985) some understanding of what motivates these players are {{g}} and the psychosocial elements present which can affect this motivation. This chapter focuses on understanding the psychosocial elements of role-playing games that motivate and enhance player engagement. This is done through the lens of SDT, as it provides a framework to view how players can be motivated through intrinsic and extrinsic motivational factors to continue playing RPGs and how understanding the factors of motivation involved may be applied to other areas such as education or the workforce. {{RoundBoxTop|theme=3}} '''Focus questions:''' # What are RPGs and the different forms they come in? # What key theories can explain player motivation in RPGs? # What sort of psychosocial factors affect player engagement? # Are there any possible applications for this type of motivation in other areas?{{ic|Use open-ended questions}} {{RoundBoxBottom}} ==What are role-playing games?== [[w:Role-playing game|Role-playing games]] have developed into several different formats over the course of their time as a popular form of entertainment, {{g}} such varieties include Live-action, Tabletop, and video game RPGs which can be single player or massively multiplayer online. These different forms allow for variations in how the role-playing can be played out, {{g}} often a video game will provide less creativity for actions to the players than a tabletop game would.{{f}} [[File:Role playing gamers (II).jpg|right|140px|thumb|'''Figure 2'''. A group playing a Tabletop Role-Playing game.]] ====LARP – Live action role-playing==== [[w:Live action role-playing game|Live action Role-playing]], also known as LARP is like improvisational theatre. Players must physically act out their characters actions rather than describe them, and similarly the real environment is used to represent the fictional setting of the game world. LARP typically uses costumes and props in addition to venues that have been prepared to reflect fictional or fantasy settings. In live-action pole-play participants also act out combat scenarios with prop weapons such as foam swords (Cyril et al., 2019). ====TTRPG – Tabletop role-playing games ==== [[w:Tabletop role-playing game|Tabletop role-playing games]] are arguably the most recognisable format due to the popularity of TTRPGs such as [[w:Dungeons and Dragons|Dungeons & Dragons]] which was the first commercially available RPG published during the 1970s. {{g}}while also being closely related to pen-and-paper RPGs in their mechanics and how they are played. TTRPGs have rulebooks that can be followed by the gamemaster and players to inform them on what, enemies, weapons, abilities, classes or spells that can be used within a playthrough. TTRPGs typically have a [[w:Gamemaster|Gamemaster]] (GM) who will guide the narrative and the players towards their goal, the GM also controls [[w:Non-player character|non-player characters]] (NPCs) within the story this is done through verbally describing the world and any situation that the players may find themselves within it and how NPCs react to any actions taken by the players. The players in TTRPGs create characters which fit into a class such as a mage or warrior, which often will help the player decide how they will act or perform within the story (Liapis & Denisova, 2023). ====Massively multiplayer online RPGs and single player RPGs==== As computer games have become popularised the ideas of TTRPGs been translated to electronic media in various forms. There are two broadly defined types of video game RPGs, those being single player role-playing games and [[w:Massively multiplayer online role-playing game|massively multiplayer online role-playing games]]. Single Player RPGs are a loosely define{{g}} genre of games which share similarities to role-playing games such as D&D, basing much of their mechanics, settings and the terminology used on those found in TTRPGs such as character and stat creation and interactive narratives which can change depending on player actions{{rewrite}}. This adaptation provides players with a visual and interactive form of role-playing but can provide limits to how a player can interact with the game due to constraints of the games themselves which may not allow complete freedom in every facet otherwise available in TTRPGs (Ryan et al., 2006). Another popular form of video game RPG are MMORPGs, which are different from single player experiences by the number of players who can interact with one another creating large social interactions between players through a persistent world which can change even if a player is offline and away from the game, {{g}} this is hosted and maintained on servers owned by the game’s publisher. MMORPGs typically allow for players to form and join groups together to work towards a common goal or reward, or simply to establish social interactions and communities within the game world. ==Self-determination theory== [[w:Self-determination theory|Self-determination theory]] (SDT) is a macro theory of human motivation from the humanistic psychology perspective which consists of mini theories proposed by Edward L. Deci and Richard Ryan in 1985. While the theory was later expanded upon, further separating the concepts of [[w:intrinsic motivation|intrinsic]] and [[w:extrinsic motivation|extrinsic]] motivation and the addition of three basic psychological needs which motivate behaviour which are [[w:autonomy|autonomy]], competence, and relatedness. ====Mini theories of SDT==== There are six mini theories currently seen in the modern version of SDT, which address one facet of motivation or personality functioning, attempting to explain different motivationally based behaviours and phenomena. [[File:SDT_macrotheory.png|alt=|left|thumb|550x550px|''Figure 2.'' Five of the macro-theories which constitute the SDT, the last theory called Relationship motivation theory is not included in the graph ]] ====1. Cognitive evaluation theory (CET)==== This theory focuses on intrinsic motivation and how competence, autonomy and, social context influences motivation, in other words CET attempts to explain how extrinsic factors can affect intrinsic motivation (Deci & Ryan, 2012) CET describes two processes by which intrinsic motivation can be affected. The first of which describes that events such as a reward from completing a behaviour or if there is a feeling of having a choice to complete a behaviour, can either undermine or enhance an individual’s intrinsic motivation depending on whether this event leads to a perceived lack of autonomy and an external locus of causality or if there is a perceived level of autonomy and an internal one. The second process proposes that events such as positive feedback leads to increasing an individual’s perceived competence by supporting the competence need and enhancing the intrinsic motivation of that individual, it should be noted that for the positive feedback to enhance motivation it must be for activities or behaviours that are autonomously motivated or within an autonomy supportive context (Deci & Ryan, 2012). In contrast negative feedback that leads to a perceived incompetence has the opposite effect and will lead to undermining intrinsic motivation (Deci & Ryan, 2012). ====2. Organismic integration theory (OIT)==== The OIT focuses on extrinsic motivation and proposes that humans are growth orientated which manifests as the concept of internalisation. OIT suggests different types of extrinsic motivation fall upon a continuum of internalisation, which consists of external regulation, introjection, identification, and integration. The more internalised the extrinsic motivation the higher the level of autonomy is felt when a behaviour is acted out(Reeve, 2012). OIT also looks at the social contexts and how it can impact an individual in deeply accepting or resisting values or goals and how that can enhance or forestall internalisation. OIT ties in with the concepts of autonomy and relatedness in affecting motivation (Ryan & Deci, 2002, Deci & Ryan, 2012). ====3. Causality orientations theory (COT)==== COT describes three types of causality orientations to explain the differences between how people regulate behaviour and orient toward environments in varying ways. These orientation types include the autonomy orientation in which an individual will act or complete a behaviour out of interest and perceived value in the act; the control orientation in which an individual is focused on any reward, gains, or approval that may come from the act; and lastly the impersonal orientation which is when an individual has anxiety caused by their perceived lack of competency (Deci & Ryan, 2012). ====4. Basic Needs theory (BNT)==== BNT proposes that the three psychological needs of autonomy, competence, and relatedness are predictive and essential to psychological well-being and optimal functioning and if one is facet is being affected there will be negative functional costs associated with it(Reeve, 2012). From this it is argued that situations and environments that support these three factors will impact well-being (Deci & Ryan, 2012, Reeve, 2012). ====5. Goals contents theory (GCT)==== GCT focuses on the impact on motivation and wellness from intrinsic and extrinsic goals and the distinctions between the two. As goals can be viewed as differentially impacting basic need satisfactions, in turn also being associated with psychological well-being(Reeve, 2012). Goals like financial success or popularity as extrinsic goals compared to intrinsic goals such as personal growth and developing close and meaningful relationships (Deci & Ryan, 2012, Reeve, 2012). ====6. Relationship motivation theory (RMT)==== RMT focuses on relatedness and how developing and maintain close personal relationships or belonging to groups as well as other relationships and how these interactions are essential for an individual’s well-being as these relationships provide satisfaction of the need for relatedness (Deci & Ryan, 2014). ====Basic Needs==== [[File:Basic needs.png|200px|right|thumb|'''Figure 3'''. The three basic psychological needs proposed by [[w:Self-determination theory|self-determination theory]] are considered essential components for intrinsic motivation and psychological well-being.]] *Autonomy – The need for an individual to feel that they are a causal agent in their life, to act on their own volition and experience freedom of will rather than feeling like they are forced to behave or act but that they feel a sense of willingness and choice in acting (Deci & Vansteenkiste, 2004). *Competence – The need for an individual to feel they are capable of effectively completing tasks and be efficient in dealing with their surrounding environment. This ability to control the outcome of situations they find themselves in such as work or school and in turn mastering these situations is key in maintaining a sense of competence (Deci & Vansteenkiste, 2004). *Relatedness – The need to interact with, be connected to, and experience caring for other people. This desire to create emotional connections with others is evident how people interact in many situations and why they seek out joining groups and activities with others (Deci & Vansteenkiste, 2004). ==Types of motivation in SDT== There are two types of motivation are proposed by SDT: extrinsic motivation and intrinsic motivation which creates a taxonomy of motivation which can be used to visually represent levels of motivation across the three orientations [[File:Taxonomy of motivation.png|thumb|600x600px|''Figure 4.'' Taxonomy of motivation: continuum of autonomy.]] ====Intrinsic motivation==== [[w:intrinsic motivation|Intrinsic motivation]] is the internal, natural drive to seek out challenges and development which is associated with optimal cognitive and social development according to SDT (Di Domenico & Ryan, 2017). This form of motivation is the most autonomous orientation as it behaviour that is intrinsically motivated are inherently interesting or enjoyable and are completed in the absence of external reward or punishment (Niemiec & Ryan 2009). The cognitive evaluation theory describes this form of motivation and how social and environmental factors help or hinder it; CET focuses on the SDT needs of competence and autonomy. Also tries to explain how motivation “crowding out” and how it can occur by the intrinsic motivation of an individual being overtaken by extrinsic motivation such as monetary rewards (Deci, E. L., & Ryan, R. M.,1985). ====Extrinsic mootivation==== [[w:extrinsic motivation|Extrinsic motivation]] comes from external sources, and the different ways this motivation can be regulated is explained by the OIT, one of the mini theories of self-determination theory (Deci, E. L., & Ryan, R. M., 1985). The four types of extrinsic motivation are as follows: '''1. Externally regulated behaviour''' The least autonomous of the 4 types and is when an act is performed due to external demand or a reward. Actions of this type have an externally perceived locus of control '''2. Introjected regulation of behaviour''' When an individual takes on regulations to behaviour that they do not fully accept as their own. This behaviour is seen in people motivated to demonstrate their ability to maintain self-worth which while an internally driven motivation, introjected behaviour is seen as having an externally perceived locus of causality. (Deci & Ryan, 1995). '''3. Regulation through identification''' More autonomously driven compared to the previous two types, this form of extrinsic motivation is when an individual sees value in the extrinsically motivated behaviour and willingly internalises the behaviour so that a sense of choice and personal commitment is felt, this regulation type has an internally perceived locus of causality (Reeve, 2012). '''4. Integrated Regulation''' The most autonomous of the 4 types, occurs when regulations are fully integrated within an individual’s beliefs and values, while this type shares similarities to intrinsic motivation it is still classed as extrinsic but also has an internally perceived locus of causality (Reeve, 2012). '''Table 1.''' Examples of intrinsic and extrinsic motivation {| class="wikitable" style="margin: auto; |- | '''Intrinsic Motivation''' || Reading a book because you enjoy the story |- | '''Extrinsic Motivation''' || Studying so you get a good grade |} ==Key Psychosocial elements within RPGs== Some of the psychosocial elements found within role-playing games are the social connections and relationships that can be fostered by them, in addition to providing a space to improve collaborative skills within safe environments (Higgins, 2023). RPGs also have been found to provide players an environment that supports autonomy and promotes competence through engaging and overcoming challenges presented within the games’ narrative (Higgins, 2023). TTRPG have also been found to support connectedness to others, and foster cooperation whilst playing through the narrative experience of these as well as an environment that allows them to explore their personality through role-play. Role-playing games have also been found to positively affect critical thinking and problem solving (Higgins, 2023, Yuliawati et al., 2024). ==How they increase engagement and motivation to play == The literature on motivation to play MMORPGs point towards factors such as social interaction, {{g}} if the game is challenging or competitive, as well as immersion or the ability to escape into the fantasy world provided by the game (Chen, V. H et.al2006). It was also found that one of the most important factors to both enjoyment and the level of engagement was social interaction within MMORPGs, in addition to the skills and challenges that a player acquires and must face (Ryan et al., 2006). The link between social interaction and engagement found that non-engaged gamers were disinterested in social interaction and viewed it as a waste of and hindering their game play while in contrast engaged gamers valued social interactions and reported it to be both important and enjoyable in their experience playing. This study concluded that a balance between perceived challenges and skills and the types of social interaction within the game where influential in both enjoyment and engagement in playing MMORPGs, (Chen, V. H et.al, 2006). this may be linked with SDT’s concept of relatedness as well as competency as players felt greater enjoyment and engagement when they possessed adequate skills to play the game and if they had the ability to socially interact with other players (Higgins, C. D. 2023). Further studies on MMORPG motivation, which sampled a Spanish player base of the popular game World of Warcraft {{g}} found similar importance in the social aspects of online role-playing games. Additional subscales of motivation which also were found to correlate was exploration, which had a considerable link to motivation while, {{g}} there was a low interest in escaping from reality and identification with the player avatar. {{g}} (Fuster et.al 2013) Another study which applied the SDT framework on the factors affecting motivation of MMORPG players found game enjoyment and intentions for future play were significantly related to measurers{{s}} of autonomy, competence and relatedness in addition to autonomy and competence being positively related to post-play mood which can further impact on continued engagement as players will want to revisit that increase in mood (Ryan et al., 2006). Similar studies on motivation for playing MMORPGs found three components which explained player motivation these were the achievement, social and, immersion (Yee, 2006), {{g}} these components consist of several subcomponents; achievement consists of advancement, mechanics and competition and links can be linked to competence{{awkward}}. Social consists of socialising, relationship, and teamwork which can be linked with relatedness, and the final component of immersion which includes discovery, role-playing, customisation, and escapism links to the need for autonomy (Schoenau-Fog, H. 2011., Yee, 2006,). These {{what}} support other findings and can be linked with CET concepts such as autonomy linked to, competence and relatedness and how they can affect facilitate or impair both intrinsic and extrinsic motivation (Ryan et al., 2006). ==Applications== A scoping review of the potential benefits that TTRPG interventions could provide found while the field is still in an early stage of research, {{g}} it shows promising signs of effectiveness as a beneficial psychological intervention (Yuliawati et al., 2024). As TTRPGs were shown to promote psychosocial and cognitive skills in their players in addition to preventing negative effects and stress due to how TTRPGs can give players a sense of agency through perceived control over major parts of the game, this can be linked to the need for autonomy found with the SDT framework and other studies looking at player motivation (Schoenau-Fog, H. 2011., Yee, 2006, Yuliawati et al., 2024){{rewrite}}. Additional applications of RPG-based motivation may be found in education, as several studies have investigated the positive motivation outcomes from taking elements of TTRPGs and other forms into an education context. These studies have found that role-playing games and gamification can have a positive effect on student academic skills as well as provide opportunity for safe social interactions and collaboration (Almén, B., Ekman, J. 2025, Rosselet & Stauffer, 2013). ==Conclusion== Using the self-determination theory to provide insight into how intrinsic and extrinsic motivation can be used to explain player engagement with role-playing games in their various forms as these genres of games provide players with opportunity to fulfill the basic needs of autonomy, competence, and relatedness through the social interactions facilitated by RPGs{{rewrite}}. Challenges found within these games which require players to adapt and overcome with developed skills and knowledge while also providing players the chance to explore their personalities, ideas, and beliefs in a safe environment also provide motivation to continued engagement with RPGs(Higgins, C. D. 2023){{rewrite}}. In addition {{g}} while the games may provide intrinsic motivation through being an enjoyable experience, extrinsic motivational factors such as rewards and approval of fellow players can also contribute to player engagement and present possible applications within fields such as education as recent studies have shown signs that RPG based interventions in education may be effective in increasing learning motivation outcomes (Cyril, et al., 2019, Niemiec & Ryan, 2009, Reeve, 2012){{rewrite}}{{g}} ==See also== * [[Motivation and emotion/Book/2023/Role-playing games, motivation, and emotion]] (Book chapter, 2023) * [[w:Role-playing game|Role-playing game]] (Wikipedia) * [[w:Self determination theory|Self determination theory]] (Wikipedia) ==References== {{Hanging indent|1= Almén, B., & Ekman, J. (2025). Roll For Initiative: EFL Game Masters and Players’ Perceptions of Using TTRPGs in an Extramural English and Classroom Setting. Cyril, B., Dobrovolný, V., Děchtěrenko, F., Stárková, T., & Bromová, E. (2019). It’s Better to Enjoy Learning than Playing: Motivational Effects of an Educational Live Action Role-playing Game. Frontline Learning Research, 7(3), 64–90. https://doi.org/10.14786/flr.v7i3.459 Deci, E. L., & Ryan, R. M. (1985). Intrinsic motivation and self-determination in human behavior. New York, NY: Plenum. Deci, E. L., & Ryan, R. M. (1995). Human autonomy: The basis for true self-esteem. In M. Kemis (Ed.), Efficacy, Agency, and Self-esteem (Pp. 31–49). New York: Plenum. https://psycnet.apa.org/record/1995-97476-002 Deci, E. L., & Ryan, R. M. (2012). Self-Determination Theory. In SAGE Publications Ltd eBooks (pp. 416–437). https://doi.org/10.4135/9781446249215.n21 Deci, E. L., & Ryan, R. M. (2014). Autonomy and need satisfaction in close relationships: Relationships Motivation Theory. In Springer eBooks (pp. 53–73). https://doi.org/10.1007/978-94-017-8542-6_3 Deci, E. L., & Vansteenkiste, M. (2004). Self-determination theory and basic need satisfaction: Understanding human development in positive psychology. RICERCHE DI PSICOLOGIA, 27, 17–34. https://lirias.kuleuven.be/handle/123456789/113911 Di Domenico, S. I., & Ryan, R. M. (2017). The Emerging Neuroscience of Intrinsic Motivation: A new frontier in Self-Determination research. Frontiers in Human Neuroscience, 11. https://doi.org/10.3389/fnhum.2017.00145 Higgins, C. D. (2023). Leveling up: Table-top role-playing games as resources for emerging adult development during COVID-19 (Doctoral dissertation). Liapis, A., & Denisova, A. (2023). The Challenge of Evaluating Player Experience in Tabletop Role-Playing Games. ., 1–10. https://doi.org/10.1145/3582437.3582457 Niemiec, C. P., & Ryan, R. M. (2009). Autonomy, competence, and relatedness in the classroom. Theory and Research in Education, 7(2), 133–144. https://doi.org/10.1177/1477878509104318 Reeve, J. (2012). A Self-determination Theory Perspective on Student Engagement. In Springer eBooks (pp. 149–172). https://doi.org/10.1007/978-1-4614-2018-7_7 Rosselet, J. G., & Stauffer, S. D. (2013). Using group role-playing games with gifted children and adolescents: A psychosocial intervention model. International Journal of Play Therapy, 22(4), 173–192. https://doi.org/10.1037/a0034557 Ryan, R. M., & Deci, E. L. (2002). Overview of self-determination theory: An organismic-dialectical perspective. . https://psycnet.apa.org/record/2002-01702-001 Ryan, R. M., Rigby, C. S., & Przybylski, A. (2006). The Motivational Pull of Video Games: A Self-Determination Theory Approach. Motivation and Emotion, 30(4), 344–360. https://doi.org/10.1007/s11031-006-9051-8 Yee, N. (2006). Motivations for play in online games. CyberPsychology & Behavior, 9(6), 772–775. https://doi.org/10.1089/cpb.2006.9.772 Yuliawati, L., Wardhani, P. A. P., & Ng, J. H. (2024). A scoping review of tabletop role-playing game (TTRPG) as psychological intervention: potential benefits and future directions. Psychology research and behavior management, 2885-2903. }} == External Links:== * [https://positivepsychology.com/motivation-theories-psychology/ 20 Most Popular Theories of Motivation in Psychology] * [https://www.britannica.com/topic/role-playing-video-game Britannica entry on role-playing video games and their history] * [https://selfdeterminationtheory.org/theory/ Self-determination theory website] [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Groups]] [[Category:Motivation and emotion/Book/Role-playing]] 0at71ifw6dwpwfwhj4l3c7nzlw57xai 2814970 2814969 2026-06-10T06:49:14Z Jtneill 10242 2814970 wikitext text/x-wiki {{title|Role-playing game motivation:<br>What are the key psychosocial elements of RPGs that enhance player engagement?}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} ;Scenario Toby has work early in the morning tomorrow but is currently playing his favourite massively multiplayer online role-playing game called World of Warcraft, {{g}} he has recently joined a guild in this game and has made some new friends in it who have invited him to join a raid they have planned for later that night. Toby has accepted the invitation even though this raid may keep Toby up late into the night and affect his work tomorrow {{g}} he agreed to join his guildmates in the difficult and time-consuming activity hoping to get some high-level rewards for completing it. Which intrinsic and extrinsic motivational factors may be at play in this scenario? Are there any conflicting motivations between playing his game and being rested for work the next day? {{RoundBoxBottom}} [[w:role-playing game|Role-playing games]] are a popular form of entertainment and have evolved into several different mediums from computer games to live action, allowing players to create characters and narratives to explore in fantasy settings. These allow players an outlet to explore different identities, perspectives and, social interactions within a safe environment. But what motivates players to get into the hobby and continue to engage with it? Through the [[w:self-determination theory|self-determination theory]] (SDT; Deci, & Ryan, 1985) some understanding of what motivates these players are {{g}} and the psychosocial elements present which can affect this motivation. This chapter focuses on understanding the psychosocial elements of role-playing games that motivate and enhance player engagement. This is done through the lens of SDT, as it provides a framework to view how players can be motivated through intrinsic and extrinsic motivational factors to continue playing RPGs and how understanding the factors of motivation involved may be applied to other areas such as education or the workforce. {{RoundBoxTop|theme=3}} '''Focus questions:''' # What are RPGs and the different forms they come in? # What key theories can explain player motivation in RPGs? # What sort of psychosocial factors affect player engagement? # Are there any possible applications for this type of motivation in other areas?{{ic|Use open-ended questions}} {{RoundBoxBottom}} ==What are role-playing games?== [[w:Role-playing game|Role-playing games]] have developed into several different formats over the course of their time as a popular form of entertainment, {{g}} such varieties include Live-action, Tabletop, and video game RPGs which can be single player or massively multiplayer online. These different forms allow for variations in how the role-playing can be played out, {{g}} often a video game will provide less creativity for actions to the players than a tabletop game would.{{f}} [[File:Role playing gamers (II).jpg|right|140px|thumb|'''Figure 2'''. A group playing a Tabletop Role-Playing game.]] ====LARP – Live action role-playing==== [[w:Live action role-playing game|Live action Role-playing]], also known as LARP is like improvisational theatre. Players must physically act out their characters actions rather than describe them, and similarly the real environment is used to represent the fictional setting of the game world. LARP typically uses costumes and props in addition to venues that have been prepared to reflect fictional or fantasy settings. In live-action pole-play participants also act out combat scenarios with prop weapons such as foam swords (Cyril et al., 2019). ====TTRPG – Tabletop role-playing games ==== [[w:Tabletop role-playing game|Tabletop role-playing games]] are arguably the most recognisable format due to the popularity of TTRPGs such as [[w:Dungeons and Dragons|Dungeons & Dragons]] which was the first commercially available RPG published during the 1970s. {{g}}while also being closely related to pen-and-paper RPGs in their mechanics and how they are played. TTRPGs have rulebooks that can be followed by the gamemaster and players to inform them on what, enemies, weapons, abilities, classes or spells that can be used within a playthrough. TTRPGs typically have a [[w:Gamemaster|Gamemaster]] (GM) who will guide the narrative and the players towards their goal, the GM also controls [[w:Non-player character|non-player characters]] (NPCs) within the story this is done through verbally describing the world and any situation that the players may find themselves within it and how NPCs react to any actions taken by the players. The players in TTRPGs create characters which fit into a class such as a mage or warrior, which often will help the player decide how they will act or perform within the story (Liapis & Denisova, 2023). ====Massively multiplayer online RPGs and single player RPGs==== As computer games have become popularised the ideas of TTRPGs been translated to electronic media in various forms. There are two broadly defined types of video game RPGs, those being single player role-playing games and [[w:Massively multiplayer online role-playing game|massively multiplayer online role-playing games]]. Single Player RPGs are a loosely define{{g}} genre of games which share similarities to role-playing games such as D&D, basing much of their mechanics, settings and the terminology used on those found in TTRPGs such as character and stat creation and interactive narratives which can change depending on player actions{{rewrite}}. This adaptation provides players with a visual and interactive form of role-playing but can provide limits to how a player can interact with the game due to constraints of the games themselves which may not allow complete freedom in every facet otherwise available in TTRPGs (Ryan et al., 2006). Another popular form of video game RPG are MMORPGs, which are different from single player experiences by the number of players who can interact with one another creating large social interactions between players through a persistent world which can change even if a player is offline and away from the game, {{g}} this is hosted and maintained on servers owned by the game’s publisher. MMORPGs typically allow for players to form and join groups together to work towards a common goal or reward, or simply to establish social interactions and communities within the game world. ==Self-determination theory== [[w:Self-determination theory|Self-determination theory]] (SDT) is a macro theory of human motivation from the humanistic psychology perspective which consists of mini theories proposed by Edward L. Deci and Richard Ryan in 1985. While the theory was later expanded upon, further separating the concepts of [[w:intrinsic motivation|intrinsic]] and [[w:extrinsic motivation|extrinsic]] motivation and the addition of three basic psychological needs which motivate behaviour which are [[w:autonomy|autonomy]], competence, and relatedness. ====Mini theories of SDT==== There are six mini theories currently seen in the modern version of SDT, which address one facet of motivation or personality functioning, attempting to explain different motivationally based behaviours and phenomena. [[File:SDT_macrotheory.png|alt=|left|thumb|550x550px|''Figure 2.'' Five of the macro-theories which constitute the SDT, the last theory called Relationship motivation theory is not included in the graph ]] ====1. Cognitive evaluation theory (CET)==== This theory focuses on intrinsic motivation and how competence, autonomy and, social context influences motivation, in other words CET attempts to explain how extrinsic factors can affect intrinsic motivation (Deci & Ryan, 2012) CET describes two processes by which intrinsic motivation can be affected. The first of which describes that events such as a reward from completing a behaviour or if there is a feeling of having a choice to complete a behaviour, can either undermine or enhance an individual’s intrinsic motivation depending on whether this event leads to a perceived lack of autonomy and an external locus of causality or if there is a perceived level of autonomy and an internal one. The second process proposes that events such as positive feedback leads to increasing an individual’s perceived competence by supporting the competence need and enhancing the intrinsic motivation of that individual, it should be noted that for the positive feedback to enhance motivation it must be for activities or behaviours that are autonomously motivated or within an autonomy supportive context (Deci & Ryan, 2012). In contrast negative feedback that leads to a perceived incompetence has the opposite effect and will lead to undermining intrinsic motivation (Deci & Ryan, 2012). ====2. Organismic integration theory (OIT)==== The OIT focuses on extrinsic motivation and proposes that humans are growth orientated which manifests as the concept of internalisation. OIT suggests different types of extrinsic motivation fall upon a continuum of internalisation, which consists of external regulation, introjection, identification, and integration. The more internalised the extrinsic motivation the higher the level of autonomy is felt when a behaviour is acted out(Reeve, 2012). OIT also looks at the social contexts and how it can impact an individual in deeply accepting or resisting values or goals and how that can enhance or forestall internalisation. OIT ties in with the concepts of autonomy and relatedness in affecting motivation (Ryan & Deci, 2002, Deci & Ryan, 2012). ====3. Causality orientations theory (COT)==== COT describes three types of causality orientations to explain the differences between how people regulate behaviour and orient toward environments in varying ways. These orientation types include the autonomy orientation in which an individual will act or complete a behaviour out of interest and perceived value in the act; the control orientation in which an individual is focused on any reward, gains, or approval that may come from the act; and lastly the impersonal orientation which is when an individual has anxiety caused by their perceived lack of competency (Deci & Ryan, 2012). ====4. Basic Needs theory (BNT)==== BNT proposes that the three psychological needs of autonomy, competence, and relatedness are predictive and essential to psychological well-being and optimal functioning and if one is facet is being affected there will be negative functional costs associated with it(Reeve, 2012). From this it is argued that situations and environments that support these three factors will impact well-being (Deci & Ryan, 2012, Reeve, 2012). ====5. Goals contents theory (GCT)==== GCT focuses on the impact on motivation and wellness from intrinsic and extrinsic goals and the distinctions between the two. As goals can be viewed as differentially impacting basic need satisfactions, in turn also being associated with psychological well-being(Reeve, 2012). Goals like financial success or popularity as extrinsic goals compared to intrinsic goals such as personal growth and developing close and meaningful relationships (Deci & Ryan, 2012, Reeve, 2012). ====6. Relationship motivation theory (RMT)==== RMT focuses on relatedness and how developing and maintain close personal relationships or belonging to groups as well as other relationships and how these interactions are essential for an individual’s well-being as these relationships provide satisfaction of the need for relatedness (Deci & Ryan, 2014). ====Basic Needs==== [[File:Basic needs.png|200px|right|thumb|'''Figure 3'''. The three basic psychological needs proposed by [[w:Self-determination theory|self-determination theory]] are considered essential components for intrinsic motivation and psychological well-being.]] *Autonomy – The need for an individual to feel that they are a causal agent in their life, to act on their own volition and experience freedom of will rather than feeling like they are forced to behave or act but that they feel a sense of willingness and choice in acting (Deci & Vansteenkiste, 2004). *Competence – The need for an individual to feel they are capable of effectively completing tasks and be efficient in dealing with their surrounding environment. This ability to control the outcome of situations they find themselves in such as work or school and in turn mastering these situations is key in maintaining a sense of competence (Deci & Vansteenkiste, 2004). *Relatedness – The need to interact with, be connected to, and experience caring for other people. This desire to create emotional connections with others is evident how people interact in many situations and why they seek out joining groups and activities with others (Deci & Vansteenkiste, 2004). ==Types of motivation in SDT== There are two types of motivation are proposed by SDT: extrinsic motivation and intrinsic motivation which creates a taxonomy of motivation which can be used to visually represent levels of motivation across the three orientations [[File:Taxonomy of motivation.png|thumb|600x600px|''Figure 4.'' Taxonomy of motivation: continuum of autonomy.]] ====Intrinsic motivation==== [[w:intrinsic motivation|Intrinsic motivation]] is the internal, natural drive to seek out challenges and development which is associated with optimal cognitive and social development according to SDT (Di Domenico & Ryan, 2017). This form of motivation is the most autonomous orientation as it behaviour that is intrinsically motivated are inherently interesting or enjoyable and are completed in the absence of external reward or punishment (Niemiec & Ryan 2009). The cognitive evaluation theory describes this form of motivation and how social and environmental factors help or hinder it; CET focuses on the SDT needs of competence and autonomy. Also tries to explain how motivation “crowding out” and how it can occur by the intrinsic motivation of an individual being overtaken by extrinsic motivation such as monetary rewards (Deci, E. L., & Ryan, R. M.,1985). ====Extrinsic mootivation==== [[w:extrinsic motivation|Extrinsic motivation]] comes from external sources, and the different ways this motivation can be regulated is explained by the OIT, one of the mini theories of self-determination theory (Deci, E. L., & Ryan, R. M., 1985). The four types of extrinsic motivation are as follows: '''1. Externally regulated behaviour''' The least autonomous of the 4 types and is when an act is performed due to external demand or a reward. Actions of this type have an externally perceived locus of control '''2. Introjected regulation of behaviour''' When an individual takes on regulations to behaviour that they do not fully accept as their own. This behaviour is seen in people motivated to demonstrate their ability to maintain self-worth which while an internally driven motivation, introjected behaviour is seen as having an externally perceived locus of causality. (Deci & Ryan, 1995). '''3. Regulation through identification''' More autonomously driven compared to the previous two types, this form of extrinsic motivation is when an individual sees value in the extrinsically motivated behaviour and willingly internalises the behaviour so that a sense of choice and personal commitment is felt, this regulation type has an internally perceived locus of causality (Reeve, 2012). '''4. Integrated Regulation''' The most autonomous of the 4 types, occurs when regulations are fully integrated within an individual’s beliefs and values, while this type shares similarities to intrinsic motivation it is still classed as extrinsic but also has an internally perceived locus of causality (Reeve, 2012). '''Table 1.''' Examples of intrinsic and extrinsic motivation {| class="wikitable" style="margin: auto; |- | '''Intrinsic Motivation''' || Reading a book because you enjoy the story |- | '''Extrinsic Motivation''' || Studying so you get a good grade |} ==Key Psychosocial elements within RPGs== Some of the psychosocial elements found within role-playing games are the social connections and relationships that can be fostered by them, in addition to providing a space to improve collaborative skills within safe environments (Higgins, 2023). RPGs also have been found to provide players an environment that supports autonomy and promotes competence through engaging and overcoming challenges presented within the games’ narrative (Higgins, 2023). TTRPG have also been found to support connectedness to others, and foster cooperation whilst playing through the narrative experience of these as well as an environment that allows them to explore their personality through role-play. Role-playing games have also been found to positively affect critical thinking and problem solving (Higgins, 2023, Yuliawati et al., 2024). ==How they increase engagement and motivation to play == The literature on motivation to play MMORPGs point towards factors such as social interaction, {{g}} if the game is challenging or competitive, as well as immersion or the ability to escape into the fantasy world provided by the game (Chen, V. H et.al2006). It was also found that one of the most important factors to both enjoyment and the level of engagement was social interaction within MMORPGs, in addition to the skills and challenges that a player acquires and must face (Ryan et al., 2006). The link between social interaction and engagement found that non-engaged gamers were disinterested in social interaction and viewed it as a waste of and hindering their game play while in contrast engaged gamers valued social interactions and reported it to be both important and enjoyable in their experience playing. This study concluded that a balance between perceived challenges and skills and the types of social interaction within the game where influential in both enjoyment and engagement in playing MMORPGs, (Chen, V. H et.al, 2006). this may be linked with SDT’s concept of relatedness as well as competency as players felt greater enjoyment and engagement when they possessed adequate skills to play the game and if they had the ability to socially interact with other players (Higgins, C. D. 2023). Further studies on MMORPG motivation, which sampled a Spanish player base of the popular game World of Warcraft {{g}} found similar importance in the social aspects of online role-playing games. Additional subscales of motivation which also were found to correlate was exploration, which had a considerable link to motivation while, {{g}} there was a low interest in escaping from reality and identification with the player avatar. {{g}} (Fuster et.al 2013) Another study which applied the SDT framework on the factors affecting motivation of MMORPG players found game enjoyment and intentions for future play were significantly related to measurers{{s}} of autonomy, competence and relatedness in addition to autonomy and competence being positively related to post-play mood which can further impact on continued engagement as players will want to revisit that increase in mood (Ryan et al., 2006). Similar studies on motivation for playing MMORPGs found three components which explained player motivation these were the achievement, social and, immersion (Yee, 2006), {{g}} these components consist of several subcomponents; achievement consists of advancement, mechanics and competition and links can be linked to competence{{awkward}}. Social consists of socialising, relationship, and teamwork which can be linked with relatedness, and the final component of immersion which includes discovery, role-playing, customisation, and escapism links to the need for autonomy (Schoenau-Fog, H. 2011., Yee, 2006,). These {{what}} support other findings and can be linked with CET concepts such as autonomy linked to, competence and relatedness and how they can affect facilitate or impair both intrinsic and extrinsic motivation (Ryan et al., 2006). ==Applications== A scoping review of the potential benefits that TTRPG interventions could provide found while the field is still in an early stage of research, {{g}} it shows promising signs of effectiveness as a beneficial psychological intervention (Yuliawati et al., 2024). As TTRPGs were shown to promote psychosocial and cognitive skills in their players in addition to preventing negative effects and stress due to how TTRPGs can give players a sense of agency through perceived control over major parts of the game, this can be linked to the need for autonomy found with the SDT framework and other studies looking at player motivation (Schoenau-Fog, H. 2011., Yee, 2006, Yuliawati et al., 2024){{rewrite}}. Additional applications of RPG-based motivation may be found in education, as several studies have investigated the positive motivation outcomes from taking elements of TTRPGs and other forms into an education context. These studies have found that role-playing games and gamification can have a positive effect on student academic skills as well as provide opportunity for safe social interactions and collaboration (Almén, B., Ekman, J. 2025, Rosselet & Stauffer, 2013). ==Conclusion== Using the self-determination theory to provide insight into how intrinsic and extrinsic motivation can be used to explain player engagement with role-playing games in their various forms as these genres of games provide players with opportunity to fulfill the basic needs of autonomy, competence, and relatedness through the social interactions facilitated by RPGs{{rewrite}}. Challenges found within these games which require players to adapt and overcome with developed skills and knowledge while also providing players the chance to explore their personalities, ideas, and beliefs in a safe environment also provide motivation to continued engagement with RPGs(Higgins, C. D. 2023){{rewrite}}. In addition {{g}} while the games may provide intrinsic motivation through being an enjoyable experience, extrinsic motivational factors such as rewards and approval of fellow players can also contribute to player engagement and present possible applications within fields such as education as recent studies have shown signs that RPG based interventions in education may be effective in increasing learning motivation outcomes (Cyril, et al., 2019, Niemiec & Ryan, 2009, Reeve, 2012){{rewrite}}{{g}} ==See also== * [[Motivation and emotion/Book/2023/Role-playing games, motivation, and emotion]] (Book chapter, 2023) * [[w:Role-playing game|Role-playing game]] (Wikipedia) * [[w:Self determination theory|Self determination theory]] (Wikipedia) ==References== {{Hanging indent|1= Almén, B., & Ekman, J. (2025). Roll For Initiative: EFL Game Masters and Players’ Perceptions of Using TTRPGs in an Extramural English and Classroom Setting. Cyril, B., Dobrovolný, V., Děchtěrenko, F., Stárková, T., & Bromová, E. (2019). It’s Better to Enjoy Learning than Playing: Motivational Effects of an Educational Live Action Role-playing Game. Frontline Learning Research, 7(3), 64–90. https://doi.org/10.14786/flr.v7i3.459 Deci, E. L., & Ryan, R. M. (1985). Intrinsic motivation and self-determination in human behavior. New York, NY: Plenum. Deci, E. L., & Ryan, R. M. (1995). Human autonomy: The basis for true self-esteem. In M. Kemis (Ed.), Efficacy, Agency, and Self-esteem (Pp. 31–49). New York: Plenum. https://psycnet.apa.org/record/1995-97476-002 Deci, E. L., & Ryan, R. M. (2012). Self-Determination Theory. In SAGE Publications Ltd eBooks (pp. 416–437). https://doi.org/10.4135/9781446249215.n21 Deci, E. L., & Ryan, R. M. (2014). Autonomy and need satisfaction in close relationships: Relationships Motivation Theory. In Springer eBooks (pp. 53–73). https://doi.org/10.1007/978-94-017-8542-6_3 Deci, E. L., & Vansteenkiste, M. (2004). Self-determination theory and basic need satisfaction: Understanding human development in positive psychology. RICERCHE DI PSICOLOGIA, 27, 17–34. https://lirias.kuleuven.be/handle/123456789/113911 Di Domenico, S. I., & Ryan, R. M. (2017). The Emerging Neuroscience of Intrinsic Motivation: A new frontier in Self-Determination research. Frontiers in Human Neuroscience, 11. https://doi.org/10.3389/fnhum.2017.00145 Higgins, C. D. (2023). Leveling up: Table-top role-playing games as resources for emerging adult development during COVID-19 (Doctoral dissertation). Liapis, A., & Denisova, A. (2023). The Challenge of Evaluating Player Experience in Tabletop Role-Playing Games. ., 1–10. https://doi.org/10.1145/3582437.3582457 Niemiec, C. P., & Ryan, R. M. (2009). Autonomy, competence, and relatedness in the classroom. Theory and Research in Education, 7(2), 133–144. https://doi.org/10.1177/1477878509104318 Reeve, J. (2012). A Self-determination Theory Perspective on Student Engagement. In Springer eBooks (pp. 149–172). https://doi.org/10.1007/978-1-4614-2018-7_7 Rosselet, J. G., & Stauffer, S. D. (2013). Using group role-playing games with gifted children and adolescents: A psychosocial intervention model. International Journal of Play Therapy, 22(4), 173–192. https://doi.org/10.1037/a0034557 Ryan, R. M., & Deci, E. L. (2002). Overview of self-determination theory: An organismic-dialectical perspective. . https://psycnet.apa.org/record/2002-01702-001 Ryan, R. M., Rigby, C. S., & Przybylski, A. (2006). The Motivational Pull of Video Games: A Self-Determination Theory Approach. Motivation and Emotion, 30(4), 344–360. https://doi.org/10.1007/s11031-006-9051-8 Yee, N. (2006). Motivations for play in online games. CyberPsychology & Behavior, 9(6), 772–775. https://doi.org/10.1089/cpb.2006.9.772 Yuliawati, L., Wardhani, P. A. P., & Ng, J. H. (2024). A scoping review of tabletop role-playing game (TTRPG) as psychological intervention: potential benefits and future directions. Psychology research and behavior management, 2885-2903. }} == External Links:== * [https://positivepsychology.com/motivation-theories-psychology/ 20 Most Popular Theories of Motivation in Psychology] * [https://www.britannica.com/topic/role-playing-video-game Britannica entry on role-playing video games and their history] * [https://selfdeterminationtheory.org/theory/ Self-determination theory website] [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Groups]] [[Category:Motivation and emotion/Book/Games/Role-playing]] aop2a8jf4w1emv22467b5hu0dc83tyu 0 323040 2814973 2760943 2026-06-10T06:57:29Z Jtneill 10242 + more specific category 2814973 wikitext text/x-wiki {{title|Self-disclosure and well-being:<br>What is the relationship between self-disclosure and psychological well-being?}} __TOC__ ==Overview== {{RoundBoxTop|theme=1}}[[File:The people in the train of Line 2.jpg|thumb|'''Figure 1.''' Strangers on a train]] ;Scenario Imagine you are sitting on a train when a stranger takes the seat beside you. You nod, smile politely, and offer a quiet hello. The stranger notices your Jack Skellington hoodie and shares that The Nightmare Before Christmas was their favourite film to watch with their sister each Christmas. They go on to reveal that their sister passed away last year, and their family has been struggling with the loss. This kind of unexpected and intimate disclosure is known as the “stranger-on-the-train” phenomenon, the sharing of deeply personal information with someone one is unlikely to meet again (Rubin, 1975). {{RoundBoxBottom}} This scenario highlights a common human tendency: the need to share personal experiences, even with those outside our closest circles. Self-disclosure is more than small talk; it can be a way to process emotions, seek support, and feel connected. However, not all disclosure is beneficial. Sometimes, revealing too much or sharing in the wrong context can leave people feeling vulnerable, misunderstood, or even rejected. Research shows that people often underestimate the risks of online self-disclosure, with frequent sharing linked to a reduced perception of potential harm{{f}}. In many cases, users prioritise the value of disclosure over concerns about privacy or data sensitivity (Fejes-Vékássy, Ujhelyi, & Lantos, 2024). This makes the relationship between self-disclosure and psychological well-being both fascinating and complex. Psychological science helps explain why people disclose, how disclosure affects relationships, and under what conditions it improves or undermines well-being. {{RoundBoxTop|theme=3}} '''Focus questions''' *What is self-disclosure, and why do people engage in it? *How does self-disclosure influence psychological well-being? *Under what conditions does self-disclosure enhance well-being, and when might it be harmful? *How can psychological research help individuals use disclosure to improve their lives? {{RoundBoxBottom}} == What is Self-disclosure: == [[wikipedia:Self-disclosure|Self-disclosure]] is the act of expressing personal thoughts, feelings, or experiences to others. It can range from surface-level information, such as hobbies or opinions, to more intimate details, including fears, dreams, or painful experiences. Self-disclosure plays an essential role in relationships as it fosters authenticity, trust, and closeness. [[wikipedia:Social_penetration_theory|Social Penetration Theory]] (Altman & Taylor, 1973) explains disclosure as occurring in layers, moving from basic details toward more intimate ones. However, disclosure does not always follow a linear path. Petronio’s Communication Privacy Management Theory (2002) highlights that people act as “managers” of their personal information, deciding what to reveal, when, and to whom. The Johari Window model (Luft & Ingham, 1955) suggests disclosure expands self-awareness and mutual understanding, showing that openness not only deepens relationships but also increases insight into the self. Other theories that look at how disclosure works in different contexts. Uncertainty Reduction Theory (Berger & Calabrese, 1975) suggests that people disclose personal information during initial encounters to make interactions more predictable. In the case of strangers, disclosure reduces the unknown by providing context about who they are, what they value, or what they are experiencing, which can make the interaction feel less awkward and more meaningful. The Interpersonal Process Model of Intimacy (Reis & Shaver, 1988) emphasises that intimacy develops when disclosures are met with empathy and validation, showing that the quality of response is just as important as the act of sharing. These key theories and their practical applications are summarised in Table 1. Recent studies confirm that disclosure has both personal and relational benefits. For instance, it has been linked to greater resilience through self-esteem and self-compassion (Harvey & Boynton, 2021) and to improved well-being during stressful times such as the COVID-19 pandemic (Matthes et al., 2021). Emotional disclosures appear particularly impactful compared to factual ones (Wu et al., 2022), and sharing negative experiences can even increase empathy and prosocial behaviour (Lyyra et al., 2023). '''Table 1.''' ''Theories of self-disclosure, key ideas, and everyday examples''. {| class="wikitable" style="width:100%; font-size:85%; margin:auto;" |- ! Theory!! Key Idea !! Example |- |Social Penetration Theory (Altman & Taylor, 1973) |Disclosure deepens in layers from superficial to intimate |Starting by sharing your favourite music with a new friend, later confiding in them about family struggles |- |Communication Privacy Management (Petronio, 2002) |People manage boundaries around personal info |Choosing to tell a close friend about a health issue but keeping it private from colleagues |- |Johari Window (Luft & Ingham, 1955) |Disclosure moves information from the “hidden self” to the “open self,” increasing mutual understanding |Revealing to your partner that you are unhappy in your current job, which they did not know before |- | Uncertainty Reduction Theory (Berger & Calabrese, 1975) || Disclosure reduces ambiguity in new interactions || On meeting a classmate for the first time, sharing where you grew up to find common ground |- |Interpersonal Process Model of Intimacy (Reis & Shaver, 1988) |Intimacy depends on supportive responses to disclosure |Telling a friend you are feeling lonely, and them responding with empathy and an invitation to spend time together |} == What is Well-Being? == Psychological well-being refers to how a person evaluates and experiences their {{missing}}. It extends beyond momentary happiness to include deeper aspects of functioning, such as meaning, growth, and positive relationships. This makes well-being a multidimensional construct rather than a single measure of “feeling good.” One influential framework is Ryff’s model of psychological well-being (1989), also known as the six-factor model, which identifies six core dimensions: self-acceptance, positive relations with others, autonomy, environmental mastery, purpose in life, and personal growth. Together, these dimensions highlight that well-being is not only about the presence of positive emotions but also about navigating life’s challenges and the pursuit of meaningful goals. More recent research continues to emphasise this multidimensional view. For example, Diener et al. (2020) distinguish between hedonic well-being (pleasure, happiness, life satisfaction) and [[Motivation and emotion/Book/2011/Eudaimonic well-being|eudaimonic well-being]] (meaning, fulfilment, personal growth), showing that both are essential for psychological health. Hendriks et al. (2020) found that positive relationships and a strong sense of purpose are particularly consistent predictors of life satisfaction across cultures. More recently, Satici et al. (2023) also highlight the importance of psychological flexibility, showing that individuals who can adapt to challenges while maintaining their values report higher well-being. Taken together, these perspectives suggest that psychological well-being is a dynamic process shaped by emotional experiences, personal strengths, and social connections. This complexity makes it a valuable counterpart to examine alongside self-disclosure, which can influence many of these same domains. == Relationship Between Self-Disclosure and Psychological Well-Being == Self-disclosure and psychological well-being are closely linked, as the act of sharing personal experiences can influence multiple aspects of mental health and life satisfaction. At its core, disclosure allows individuals to feel understood and validated, which strengthens social bonds and provides emotional relief. These experiences directly align with well-being dimensions such as positive relationships, self-acceptance, and personal growth (Ryff, 1989). One pathway linking disclosure to well-being is its role in [[wikipedia:Emotional_self-regulation|emotion regulation]]. By expressing personal struggles or feelings, people are able to process difficult experiences, reduce stress, and gain perspective. Research by Aldahadha (2023) found that disclosure, particularly when combined with [[wikipedia:Mindfulness|mindfulness]], was associated with higher happiness and overall well-being. Similarly, Harvey and Boynton (2021) demonstrated that disclosure enhances resilience partly through increases in self-esteem and self-compassion. These findings show that disclosure can serve as a coping strategy, helping individuals adapt to life’s challenges. Another pathway is through [[wikipedia:Social_connection|social connection]]. Disclosure fosters intimacy and trust in relationships, which in turn predict higher well-being. A meta-analysis by Chu, Sun, and Jiang (2023) {{ic|not in References}} confirmed that self-disclosure on social media was positively associated with psychological well-being, especially when it facilitated supportive interactions. During the COVID-19 pandemic, online disclosure provided a sense of connection and improved happiness and well-being despite physical isolation (Matthes et al., 2021). These findings illustrate how disclosure can maintain well-being in both face-to-face and digital contexts. Cultural factors also shape the relationship. While Western contexts often view disclosure as a sign of openness and authenticity, other cultural settings place greater value on discretion. Duan et al. (2022) observed that resilience and personal growth contribute to well-being, and it is plausible that in different cultural settings, how disclosure interacts with these traits may vary depending on social norms. This highlights that disclosure’s benefits are not universal but context-dependent. Taken together, evidence suggests that self-disclosure contributes to psychological well-being by supporting emotional regulation, strengthening relationships, and fostering growth. At the same time, disclosure carries risks such as co-rumination, oversharing, or negative social responses, which can undermine these benefits{{f}}. The overall impact therefore depends on the quality of what is shared, the context in which it occurs, and the responses it receives{{f}}. == Applications of Psychological Research == Research suggests that self-disclosure can be deliberately used as a tool to promote psychological well-being, provided it is applied thoughtfully and contextually. One key pathway is through emotional processing and regulation. Aldahadha (2023) showed that disclosure, particularly when combined with mindfulness, was associated with greater happiness and well-being, highlighting its value as a coping strategy. Similarly, Harvey and Boynton (2021) found that disclosure strengthened resilience by increasing self-esteem and self-compassion, suggesting that supportive environments for sharing can help individuals adapt to stress. Self-disclosure can also be used to build and maintain positive relationships, a central component of psychological health. Sharing personal experiences promotes intimacy and trust (Altman & Taylor, 1973; Reis & Shaver, 1988), which align with Ryff’s (1989) six dimensions of well-being, particularly positive relationships and self-acceptance. Online contexts may extend these benefits: Chu, Sun, and Jiang’s (2023) meta-analysis found that social media disclosure was positively associated with well-being when it generated supportive interactions, while Matthes et al. (2021) demonstrated that online disclosure maintained happiness and connection during the COVID-19 pandemic. At the same time, intentional self-disclosure requires managing boundaries to protect well-being. Petronio’s (2002) Communication Privacy Management theory underscores the importance of making strategic decisions about what to share and with whom. Over-disclosure or poorly timed disclosure may result in rejection, loss of privacy, or social strain (Rubin, 1975). More recently, Fejes-Vékássy, Ujhelyi, and Lantos (2024) found that frequent online sharers often underestimate risks, prioritising disclosure over concerns about privacy or data sensitivity. This highlights the need for awareness of potential downsides, particularly in digital contexts. == Quiz == <quiz display=simple> {Self-disclosure always leads to positive outcomes for psychological well-being. |type="()"} - True + False {According to Social Penetration Theory, disclosure tends to move from surface-level details to more intimate information. |type="()"} + True - False {Sharing personal experiences online during the COVID-19 pandemic had no measurable impact on well-being. |type="()"} - True + False {Ryff’s (1989) model of psychological well-being includes six dimensions, such as self-acceptance and positive relationships. |type="()"} + True - False {People who frequently disclose online often perceive sharing as less risky than it really is. |type="()"} + True - False </quiz> == Conclusion == Self-disclosure is a powerful psychological process that can foster resilience, authenticity, and connection. Evidence shows it enhances well-being by supporting emotion regulation (Aldahadha, 2023; Harvey & Boynton, 2021), building positive relationships (Reis & Shaver, 1988; Ryff, 1989), and strengthening social bonds both offline and online (Chu et al., 2023; Matthes et al., 2021). However, disclosure also carries risks, particularly when it involves oversharing, poor boundary management, or unsafe contexts (Petronio, 2002; Fejes-Vékássy et al., 2024). The key to harnessing disclosure lies in balance: sharing in ways that invite empathy and support while protecting privacy and respecting cultural expectations. By approaching disclosure with mindfulness and intention, individuals can maximise its benefits and reduce potential harms. Ultimately, thoughtful self-disclosure can play a central role in promoting psychological well-being, supporting not only personal growth but also stronger, healthier relationships. Cultural factors also play a role in how disclosure supports well-being. Duan, Guan, and Bu (2022) demonstrated that resilience and personal growth contribute to well-being across diverse contexts, yet the value of disclosure may vary according to cultural norms. In settings where openness is encouraged, disclosure may foster belonging and support; in cultures where discretion is prized, it may expose individuals to stigma or misunderstanding. Overall, psychological research demonstrates that self-disclosure can be used strategically to enhance well-being, provided individuals balance openness with appropriate boundaries, cultural sensitivity, and attention to context. ==See also== {{expand}} ==References== {{Hanging indent|1= Aldahadha, B. (2023). Self-disclosure, mindfulness, and their relationships with happiness and well-being. ''Middle East Current Psychiatry, 30''(1), 7. https://doi.org/10.1186/s43045-023-00278-5 Altman, I., & Taylor, D. (1973). ''Social penetration: The development of interpersonal relationships''. Holt, Rinehart & Winston. Berger, C. R., & Calabrese, R. J. (1975). Some explorations in initial interaction and beyond: Toward a developmental theory of interpersonal communication. ''Human Communication Research, 1''(2), 99–112. https://doi.org/10.1111/j.1468-2958.1975.tb00258.x Diener, E., Lucas, R. E., & Oishi, S. (2018). Advances and open questions in the science of subjective well-being. ''Collabra: Psychology, 4''(1), 15. https://doi.org/10.1525/collabra.115 Duan, W., Guan, Y., & Bu, H. (2022). The role of resilience and personal growth in psychological well-being during life challenges. ''Journal of Happiness Studies, 23''(5), 2235–2252. https://doi.org/10.1007/s10902-021-00462-2 Fejes-Vékássy, L., Ujhelyi, A., & Lantos, N. A. (2024). I don’t care, I share! – The importance of self-disclosure overwrites the risks of sharing on social media. ''Current Psychology, 43'', 30120–30134. https://doi.org/10.1007/s12144-024-06496-2 Harvey, J., & Boynton, K. (2021). Self-disclosure and psychological resilience: The mediating roles of self-esteem and self-compassion. ''Interpersona, 15''(1), 90–104. https://doi.org/10.5964/ijpr.4533 Hendriks, T., Schotanus-Dijkstra, M., Hassankhan, A., De Jong, J., & Bohlmeijer, E. (2020). The efficacy of multi-component positive psychology interventions for well-being: A meta-analysis. ''Journal of Happiness Studies, 21''(1), 357–390. https://doi.org/10.1007/s10902-019-00082-1 Luft, J., & Ingham, H. (1955). The Johari window, a graphic model of interpersonal awareness. ''Proceedings of the Western Training Laboratory in Group Development''. Los Angeles: UCLA. Lyyra, N., Sormunen, M., & Keltikangas-Järvinen, L. (2023). The impact of self-disclosure of negative experiences on prosociality. ''Social Cognitive and Affective Neuroscience, 19''(1), nsae003. https://doi.org/10.1093/scan/nsae003 Matthes, J., Koban, K., Neureiter, A., & Stevic, A. (2021). Longitudinal relationships among fear of COVID-19, smartphone online self-disclosure, happiness, and psychological well-being. ''Journal of Medical Internet Research, 23''(9), e28700. https://doi.org/10.2196/28700 Petronio, S. (2002). ''Boundaries of privacy: Dialectics of disclosure''. SUNY Press. Reis, H. T., & Shaver, P. (1988). Intimacy as an interpersonal process. In S. W. Duck (Ed.), ''Handbook of personal relationships'' (pp. 367–389). Wiley. Rubin, Z. (1975). Disclosing oneself to a stranger: Reciprocity and its limits. ''Journal of Experimental Social Psychology, 11''(3), 233–260. https://doi.org/10.1016/S0022-1031(75)80025-4 Ryff, C. D. (1989). Happiness is everything, or is it? Explorations on the meaning of psychological well-being. ''Journal of Personality and Social Psychology, 57''(6), 1069–1081. https://doi.org/10.1037/0022-3514.57.6.1069 Satici, S. A., Przepiorka, A., & Diener, E. (2023). Psychological flexibility and subjective well-being: A multi-country study. ''Current Psychology, 42''(2), 1337–1348. https://doi.org/10.1007/s12144-021-01519-8 Wu, X., Li, Y., et al. (2022). The effects of emotional disclosure on psychological outcomes. ''Frontiers in Psychology, 13'', 989826. https://doi.org/10.3389/fpsyg.2022.989826 }} ==External links== [https://www.youtube.com/watch?v=pgl42XtFb0M Self Disclosure: Explained] (YouTube) [https://www.youtube.com/watch?v=mxz8KyV3Ydc What is Well-Being?] (YouTube) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Communication]] [[Category:Motivation and emotion/Book/Self-disclosure]] [[Category:Motivation and emotion/Book/Wellbeing]] 598jitmpgjwexvj9v3lj5x5g83txwpi 2814979 2814973 2026-06-10T07:11:34Z Jtneill 10242 2814979 wikitext text/x-wiki {{title|Self-disclosure and well-being:<br>What is the relationship between self-disclosure and psychological well-being?}} __TOC__ ==Overview== {{RoundBoxTop|theme=1}}[[File:The people in the train of Line 2.jpg|thumb|'''Figure 1.''' Strangers on a train]] ;Scenario Imagine you are sitting on a train when a stranger takes the seat beside you. You nod, smile politely, and offer a quiet hello. The stranger notices your Jack Skellington hoodie and shares that The Nightmare Before Christmas was their favourite film to watch with their sister each Christmas. They go on to reveal that their sister passed away last year, and their family has been struggling with the loss. This kind of unexpected and intimate disclosure is known as the “stranger-on-the-train” phenomenon, the sharing of deeply personal information with someone one is unlikely to meet again (Rubin, 1975). {{RoundBoxBottom}} This scenario highlights a common human tendency: the need to share personal experiences, even with those outside our closest circles. Self-disclosure is more than small talk; it can be a way to process emotions, seek support, and feel connected. However, not all disclosure is beneficial. Sometimes, revealing too much or sharing in the wrong context can leave people feeling vulnerable, misunderstood, or even rejected. Research shows that people often underestimate the risks of online self-disclosure, with frequent sharing linked to a reduced perception of potential harm{{f}}. In many cases, users prioritise the value of disclosure over concerns about privacy or data sensitivity (Fejes-Vékássy, Ujhelyi, & Lantos, 2024). This makes the relationship between self-disclosure and psychological well-being both fascinating and complex. Psychological science helps explain why people disclose, how disclosure affects relationships, and under what conditions it improves or undermines well-being. {{RoundBoxTop|theme=3}} '''Focus questions''' *What is self-disclosure, and why do people engage in it? *How does self-disclosure influence psychological well-being? *Under what conditions does self-disclosure enhance well-being, and when might it be harmful? *How can psychological research help individuals use disclosure to improve their lives? {{RoundBoxBottom}} == What is Self-disclosure: == [[wikipedia:Self-disclosure|Self-disclosure]] is the act of expressing personal thoughts, feelings, or experiences to others. It can range from surface-level information, such as hobbies or opinions, to more intimate details, including fears, dreams, or painful experiences. Self-disclosure plays an essential role in relationships as it fosters authenticity, trust, and closeness. [[wikipedia:Social_penetration_theory|Social Penetration Theory]] (Altman & Taylor, 1973) explains disclosure as occurring in layers, moving from basic details toward more intimate ones. However, disclosure does not always follow a linear path. Petronio’s Communication Privacy Management Theory (2002) highlights that people act as “managers” of their personal information, deciding what to reveal, when, and to whom. The Johari Window model (Luft & Ingham, 1955) suggests disclosure expands self-awareness and mutual understanding, showing that openness not only deepens relationships but also increases insight into the self. Other theories that look at how disclosure works in different contexts. Uncertainty Reduction Theory (Berger & Calabrese, 1975) suggests that people disclose personal information during initial encounters to make interactions more predictable. In the case of strangers, disclosure reduces the unknown by providing context about who they are, what they value, or what they are experiencing, which can make the interaction feel less awkward and more meaningful. The Interpersonal Process Model of Intimacy (Reis & Shaver, 1988) emphasises that intimacy develops when disclosures are met with empathy and validation, showing that the quality of response is just as important as the act of sharing. These key theories and their practical applications are summarised in Table 1. Recent studies confirm that disclosure has both personal and relational benefits. For instance, it has been linked to greater resilience through self-esteem and self-compassion (Harvey & Boynton, 2021) and to improved well-being during stressful times such as the COVID-19 pandemic (Matthes et al., 2021). Emotional disclosures appear particularly impactful compared to factual ones (Wu et al., 2022), and sharing negative experiences can even increase empathy and prosocial behaviour (Lyyra et al., 2023). '''Table 1.''' ''Theories of self-disclosure, key ideas, and everyday examples''. {| class="wikitable" style="width:100%; font-size:85%; margin:auto;" |- ! Theory!! Key Idea !! Example |- |Social Penetration Theory (Altman & Taylor, 1973) |Disclosure deepens in layers from superficial to intimate |Starting by sharing your favourite music with a new friend, later confiding in them about family struggles |- |Communication Privacy Management (Petronio, 2002) |People manage boundaries around personal info |Choosing to tell a close friend about a health issue but keeping it private from colleagues |- |Johari Window (Luft & Ingham, 1955) |Disclosure moves information from the “hidden self” to the “open self,” increasing mutual understanding |Revealing to your partner that you are unhappy in your current job, which they did not know before |- | Uncertainty Reduction Theory (Berger & Calabrese, 1975) || Disclosure reduces ambiguity in new interactions || On meeting a classmate for the first time, sharing where you grew up to find common ground |- |Interpersonal Process Model of Intimacy (Reis & Shaver, 1988) |Intimacy depends on supportive responses to disclosure |Telling a friend you are feeling lonely, and them responding with empathy and an invitation to spend time together |} == What is Well-Being? == Psychological well-being refers to how a person evaluates and experiences their {{missing}}. It extends beyond momentary happiness to include deeper aspects of functioning, such as meaning, growth, and positive relationships. This makes well-being a multidimensional construct rather than a single measure of “feeling good.” One influential framework is Ryff’s model of psychological well-being (1989), also known as the six-factor model, which identifies six core dimensions: self-acceptance, positive relations with others, autonomy, environmental mastery, purpose in life, and personal growth. Together, these dimensions highlight that well-being is not only about the presence of positive emotions but also about navigating life’s challenges and the pursuit of meaningful goals. More recent research continues to emphasise this multidimensional view. For example, Diener et al. (2020) distinguish between hedonic well-being (pleasure, happiness, life satisfaction) and [[Motivation and emotion/Book/2011/Eudaimonic well-being|eudaimonic well-being]] (meaning, fulfilment, personal growth), showing that both are essential for psychological health. Hendriks et al. (2020) found that positive relationships and a strong sense of purpose are particularly consistent predictors of life satisfaction across cultures. More recently, Satici et al. (2023) also highlight the importance of psychological flexibility, showing that individuals who can adapt to challenges while maintaining their values report higher well-being. Taken together, these perspectives suggest that psychological well-being is a dynamic process shaped by emotional experiences, personal strengths, and social connections. This complexity makes it a valuable counterpart to examine alongside self-disclosure, which can influence many of these same domains. == Relationship Between Self-Disclosure and Psychological Well-Being == Self-disclosure and psychological well-being are closely linked, as the act of sharing personal experiences can influence multiple aspects of mental health and life satisfaction. At its core, disclosure allows individuals to feel understood and validated, which strengthens social bonds and provides emotional relief. These experiences directly align with well-being dimensions such as positive relationships, self-acceptance, and personal growth (Ryff, 1989). One pathway linking disclosure to well-being is its role in [[wikipedia:Emotional_self-regulation|emotion regulation]]. By expressing personal struggles or feelings, people are able to process difficult experiences, reduce stress, and gain perspective. Research by Aldahadha (2023) found that disclosure, particularly when combined with [[wikipedia:Mindfulness|mindfulness]], was associated with higher happiness and overall well-being. Similarly, Harvey and Boynton (2021) demonstrated that disclosure enhances resilience partly through increases in self-esteem and self-compassion. These findings show that disclosure can serve as a coping strategy, helping individuals adapt to life’s challenges. Another pathway is through [[wikipedia:Social_connection|social connection]]. Disclosure fosters intimacy and trust in relationships, which in turn predict higher well-being. A meta-analysis by Chu, Sun, and Jiang (2023) {{ic|not in References}} confirmed that self-disclosure on social media was positively associated with psychological well-being, especially when it facilitated supportive interactions. During the COVID-19 pandemic, online disclosure provided a sense of connection and improved happiness and well-being despite physical isolation (Matthes et al., 2021). These findings illustrate how disclosure can maintain well-being in both face-to-face and digital contexts. Cultural factors also shape the relationship. While Western contexts often view disclosure as a sign of openness and authenticity, other cultural settings place greater value on discretion. Duan et al. (2022) observed that resilience and personal growth contribute to well-being, and it is plausible that in different cultural settings, how disclosure interacts with these traits may vary depending on social norms. This highlights that disclosure’s benefits are not universal but context-dependent. Taken together, evidence suggests that self-disclosure contributes to psychological well-being by supporting emotional regulation, strengthening relationships, and fostering growth. At the same time, disclosure carries risks such as co-rumination, oversharing, or negative social responses, which can undermine these benefits{{f}}. The overall impact therefore depends on the quality of what is shared, the context in which it occurs, and the responses it receives{{f}}. == Applications of Psychological Research == Research suggests that self-disclosure can be deliberately used as a tool to promote psychological well-being, provided it is applied thoughtfully and contextually. One key pathway is through emotional processing and regulation. Aldahadha (2023) showed that disclosure, particularly when combined with mindfulness, was associated with greater happiness and well-being, highlighting its value as a coping strategy. Similarly, Harvey and Boynton (2021) found that disclosure strengthened resilience by increasing self-esteem and self-compassion, suggesting that supportive environments for sharing can help individuals adapt to stress. Self-disclosure can also be used to build and maintain positive relationships, a central component of psychological health. Sharing personal experiences promotes intimacy and trust (Altman & Taylor, 1973; Reis & Shaver, 1988), which align with Ryff’s (1989) six dimensions of well-being, particularly positive relationships and self-acceptance. Online contexts may extend these benefits: Chu, Sun, and Jiang’s (2023) meta-analysis found that social media disclosure was positively associated with well-being when it generated supportive interactions, while Matthes et al. (2021) demonstrated that online disclosure maintained happiness and connection during the COVID-19 pandemic. At the same time, intentional self-disclosure requires managing boundaries to protect well-being. Petronio’s (2002) Communication Privacy Management theory underscores the importance of making strategic decisions about what to share and with whom. Over-disclosure or poorly timed disclosure may result in rejection, loss of privacy, or social strain (Rubin, 1975). More recently, Fejes-Vékássy, Ujhelyi, and Lantos (2024) found that frequent online sharers often underestimate risks, prioritising disclosure over concerns about privacy or data sensitivity. This highlights the need for awareness of potential downsides, particularly in digital contexts. == Quiz == <quiz display=simple> {Self-disclosure always leads to positive outcomes for psychological well-being. |type="()"} - True + False {According to Social Penetration Theory, disclosure tends to move from surface-level details to more intimate information. |type="()"} + True - False {Sharing personal experiences online during the COVID-19 pandemic had no measurable impact on well-being. |type="()"} - True + False {Ryff’s (1989) model of psychological well-being includes six dimensions, such as self-acceptance and positive relationships. |type="()"} + True - False {People who frequently disclose online often perceive sharing as less risky than it really is. |type="()"} + True - False </quiz> == Conclusion == Self-disclosure is a powerful psychological process that can foster resilience, authenticity, and connection. Evidence shows it enhances well-being by supporting emotion regulation (Aldahadha, 2023; Harvey & Boynton, 2021), building positive relationships (Reis & Shaver, 1988; Ryff, 1989), and strengthening social bonds both offline and online (Chu et al., 2023; Matthes et al., 2021). However, disclosure also carries risks, particularly when it involves oversharing, poor boundary management, or unsafe contexts (Petronio, 2002; Fejes-Vékássy et al., 2024). The key to harnessing disclosure lies in balance: sharing in ways that invite empathy and support while protecting privacy and respecting cultural expectations. By approaching disclosure with mindfulness and intention, individuals can maximise its benefits and reduce potential harms. Ultimately, thoughtful self-disclosure can play a central role in promoting psychological well-being, supporting not only personal growth but also stronger, healthier relationships. Cultural factors also play a role in how disclosure supports well-being. Duan, Guan, and Bu (2022) demonstrated that resilience and personal growth contribute to well-being across diverse contexts, yet the value of disclosure may vary according to cultural norms. In settings where openness is encouraged, disclosure may foster belonging and support; in cultures where discretion is prized, it may expose individuals to stigma or misunderstanding. Overall, psychological research demonstrates that self-disclosure can be used strategically to enhance well-being, provided individuals balance openness with appropriate boundaries, cultural sensitivity, and attention to context. ==See also== {{expand}} ==References== {{Hanging indent|1= Aldahadha, B. (2023). Self-disclosure, mindfulness, and their relationships with happiness and well-being. ''Middle East Current Psychiatry, 30''(1), 7. https://doi.org/10.1186/s43045-023-00278-5 Altman, I., & Taylor, D. (1973). ''Social penetration: The development of interpersonal relationships''. Holt, Rinehart & Winston. Berger, C. R., & Calabrese, R. J. (1975). Some explorations in initial interaction and beyond: Toward a developmental theory of interpersonal communication. ''Human Communication Research, 1''(2), 99–112. https://doi.org/10.1111/j.1468-2958.1975.tb00258.x Diener, E., Lucas, R. E., & Oishi, S. (2018). Advances and open questions in the science of subjective well-being. ''Collabra: Psychology, 4''(1), 15. https://doi.org/10.1525/collabra.115 Duan, W., Guan, Y., & Bu, H. (2022). The role of resilience and personal growth in psychological well-being during life challenges. ''Journal of Happiness Studies, 23''(5), 2235–2252. https://doi.org/10.1007/s10902-021-00462-2 Fejes-Vékássy, L., Ujhelyi, A., & Lantos, N. A. (2024). I don’t care, I share! – The importance of self-disclosure overwrites the risks of sharing on social media. ''Current Psychology, 43'', 30120–30134. https://doi.org/10.1007/s12144-024-06496-2 Harvey, J., & Boynton, K. (2021). Self-disclosure and psychological resilience: The mediating roles of self-esteem and self-compassion. ''Interpersona, 15''(1), 90–104. https://doi.org/10.5964/ijpr.4533 Hendriks, T., Schotanus-Dijkstra, M., Hassankhan, A., De Jong, J., & Bohlmeijer, E. (2020). The efficacy of multi-component positive psychology interventions for well-being: A meta-analysis. ''Journal of Happiness Studies, 21''(1), 357–390. https://doi.org/10.1007/s10902-019-00082-1 Luft, J., & Ingham, H. (1955). The Johari window, a graphic model of interpersonal awareness. ''Proceedings of the Western Training Laboratory in Group Development''. Los Angeles: UCLA. Lyyra, N., Sormunen, M., & Keltikangas-Järvinen, L. (2023). The impact of self-disclosure of negative experiences on prosociality. ''Social Cognitive and Affective Neuroscience, 19''(1), nsae003. https://doi.org/10.1093/scan/nsae003 Matthes, J., Koban, K., Neureiter, A., & Stevic, A. (2021). Longitudinal relationships among fear of COVID-19, smartphone online self-disclosure, happiness, and psychological well-being. ''Journal of Medical Internet Research, 23''(9), e28700. https://doi.org/10.2196/28700 Petronio, S. (2002). ''Boundaries of privacy: Dialectics of disclosure''. SUNY Press. Reis, H. T., & Shaver, P. (1988). Intimacy as an interpersonal process. In S. W. Duck (Ed.), ''Handbook of personal relationships'' (pp. 367–389). Wiley. Rubin, Z. (1975). Disclosing oneself to a stranger: Reciprocity and its limits. ''Journal of Experimental Social Psychology, 11''(3), 233–260. https://doi.org/10.1016/S0022-1031(75)80025-4 Ryff, C. D. (1989). Happiness is everything, or is it? Explorations on the meaning of psychological well-being. ''Journal of Personality and Social Psychology, 57''(6), 1069–1081. https://doi.org/10.1037/0022-3514.57.6.1069 Satici, S. A., Przepiorka, A., & Diener, E. (2023). Psychological flexibility and subjective well-being: A multi-country study. ''Current Psychology, 42''(2), 1337–1348. https://doi.org/10.1007/s12144-021-01519-8 Wu, X., Li, Y., et al. (2022). The effects of emotional disclosure on psychological outcomes. ''Frontiers in Psychology, 13'', 989826. https://doi.org/10.3389/fpsyg.2022.989826 }} ==External links== [https://www.youtube.com/watch?v=pgl42XtFb0M Self Disclosure: Explained] (YouTube) [https://www.youtube.com/watch?v=mxz8KyV3Ydc What is Well-Being?] (YouTube) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Communication]] [[Category:Motivation and emotion/Book/Self-disclosure]] [[Category:Motivation and emotion/Book/Well-being]] 30gxm9p5egz82ndfxhu2m0of9yo5sdn 2814980 2814979 2026-06-10T07:13:23Z Jtneill 10242 /* External links */ + bullet-points 2814980 wikitext text/x-wiki {{title|Self-disclosure and well-being:<br>What is the relationship between self-disclosure and psychological well-being?}} __TOC__ ==Overview== {{RoundBoxTop|theme=1}}[[File:The people in the train of Line 2.jpg|thumb|'''Figure 1.''' Strangers on a train]] ;Scenario Imagine you are sitting on a train when a stranger takes the seat beside you. You nod, smile politely, and offer a quiet hello. The stranger notices your Jack Skellington hoodie and shares that The Nightmare Before Christmas was their favourite film to watch with their sister each Christmas. They go on to reveal that their sister passed away last year, and their family has been struggling with the loss. This kind of unexpected and intimate disclosure is known as the “stranger-on-the-train” phenomenon, the sharing of deeply personal information with someone one is unlikely to meet again (Rubin, 1975). {{RoundBoxBottom}} This scenario highlights a common human tendency: the need to share personal experiences, even with those outside our closest circles. Self-disclosure is more than small talk; it can be a way to process emotions, seek support, and feel connected. However, not all disclosure is beneficial. Sometimes, revealing too much or sharing in the wrong context can leave people feeling vulnerable, misunderstood, or even rejected. Research shows that people often underestimate the risks of online self-disclosure, with frequent sharing linked to a reduced perception of potential harm{{f}}. In many cases, users prioritise the value of disclosure over concerns about privacy or data sensitivity (Fejes-Vékássy, Ujhelyi, & Lantos, 2024). This makes the relationship between self-disclosure and psychological well-being both fascinating and complex. Psychological science helps explain why people disclose, how disclosure affects relationships, and under what conditions it improves or undermines well-being. {{RoundBoxTop|theme=3}} '''Focus questions''' *What is self-disclosure, and why do people engage in it? *How does self-disclosure influence psychological well-being? *Under what conditions does self-disclosure enhance well-being, and when might it be harmful? *How can psychological research help individuals use disclosure to improve their lives? {{RoundBoxBottom}} == What is Self-disclosure: == [[wikipedia:Self-disclosure|Self-disclosure]] is the act of expressing personal thoughts, feelings, or experiences to others. It can range from surface-level information, such as hobbies or opinions, to more intimate details, including fears, dreams, or painful experiences. Self-disclosure plays an essential role in relationships as it fosters authenticity, trust, and closeness. [[wikipedia:Social_penetration_theory|Social Penetration Theory]] (Altman & Taylor, 1973) explains disclosure as occurring in layers, moving from basic details toward more intimate ones. However, disclosure does not always follow a linear path. Petronio’s Communication Privacy Management Theory (2002) highlights that people act as “managers” of their personal information, deciding what to reveal, when, and to whom. The Johari Window model (Luft & Ingham, 1955) suggests disclosure expands self-awareness and mutual understanding, showing that openness not only deepens relationships but also increases insight into the self. Other theories that look at how disclosure works in different contexts. Uncertainty Reduction Theory (Berger & Calabrese, 1975) suggests that people disclose personal information during initial encounters to make interactions more predictable. In the case of strangers, disclosure reduces the unknown by providing context about who they are, what they value, or what they are experiencing, which can make the interaction feel less awkward and more meaningful. The Interpersonal Process Model of Intimacy (Reis & Shaver, 1988) emphasises that intimacy develops when disclosures are met with empathy and validation, showing that the quality of response is just as important as the act of sharing. These key theories and their practical applications are summarised in Table 1. Recent studies confirm that disclosure has both personal and relational benefits. For instance, it has been linked to greater resilience through self-esteem and self-compassion (Harvey & Boynton, 2021) and to improved well-being during stressful times such as the COVID-19 pandemic (Matthes et al., 2021). Emotional disclosures appear particularly impactful compared to factual ones (Wu et al., 2022), and sharing negative experiences can even increase empathy and prosocial behaviour (Lyyra et al., 2023). '''Table 1.''' ''Theories of self-disclosure, key ideas, and everyday examples''. {| class="wikitable" style="width:100%; font-size:85%; margin:auto;" |- ! Theory!! Key Idea !! Example |- |Social Penetration Theory (Altman & Taylor, 1973) |Disclosure deepens in layers from superficial to intimate |Starting by sharing your favourite music with a new friend, later confiding in them about family struggles |- |Communication Privacy Management (Petronio, 2002) |People manage boundaries around personal info |Choosing to tell a close friend about a health issue but keeping it private from colleagues |- |Johari Window (Luft & Ingham, 1955) |Disclosure moves information from the “hidden self” to the “open self,” increasing mutual understanding |Revealing to your partner that you are unhappy in your current job, which they did not know before |- | Uncertainty Reduction Theory (Berger & Calabrese, 1975) || Disclosure reduces ambiguity in new interactions || On meeting a classmate for the first time, sharing where you grew up to find common ground |- |Interpersonal Process Model of Intimacy (Reis & Shaver, 1988) |Intimacy depends on supportive responses to disclosure |Telling a friend you are feeling lonely, and them responding with empathy and an invitation to spend time together |} == What is Well-Being? == Psychological well-being refers to how a person evaluates and experiences their {{missing}}. It extends beyond momentary happiness to include deeper aspects of functioning, such as meaning, growth, and positive relationships. This makes well-being a multidimensional construct rather than a single measure of “feeling good.” One influential framework is Ryff’s model of psychological well-being (1989), also known as the six-factor model, which identifies six core dimensions: self-acceptance, positive relations with others, autonomy, environmental mastery, purpose in life, and personal growth. Together, these dimensions highlight that well-being is not only about the presence of positive emotions but also about navigating life’s challenges and the pursuit of meaningful goals. More recent research continues to emphasise this multidimensional view. For example, Diener et al. (2020) distinguish between hedonic well-being (pleasure, happiness, life satisfaction) and [[Motivation and emotion/Book/2011/Eudaimonic well-being|eudaimonic well-being]] (meaning, fulfilment, personal growth), showing that both are essential for psychological health. Hendriks et al. (2020) found that positive relationships and a strong sense of purpose are particularly consistent predictors of life satisfaction across cultures. More recently, Satici et al. (2023) also highlight the importance of psychological flexibility, showing that individuals who can adapt to challenges while maintaining their values report higher well-being. Taken together, these perspectives suggest that psychological well-being is a dynamic process shaped by emotional experiences, personal strengths, and social connections. This complexity makes it a valuable counterpart to examine alongside self-disclosure, which can influence many of these same domains. == Relationship Between Self-Disclosure and Psychological Well-Being == Self-disclosure and psychological well-being are closely linked, as the act of sharing personal experiences can influence multiple aspects of mental health and life satisfaction. At its core, disclosure allows individuals to feel understood and validated, which strengthens social bonds and provides emotional relief. These experiences directly align with well-being dimensions such as positive relationships, self-acceptance, and personal growth (Ryff, 1989). One pathway linking disclosure to well-being is its role in [[wikipedia:Emotional_self-regulation|emotion regulation]]. By expressing personal struggles or feelings, people are able to process difficult experiences, reduce stress, and gain perspective. Research by Aldahadha (2023) found that disclosure, particularly when combined with [[wikipedia:Mindfulness|mindfulness]], was associated with higher happiness and overall well-being. Similarly, Harvey and Boynton (2021) demonstrated that disclosure enhances resilience partly through increases in self-esteem and self-compassion. These findings show that disclosure can serve as a coping strategy, helping individuals adapt to life’s challenges. Another pathway is through [[wikipedia:Social_connection|social connection]]. Disclosure fosters intimacy and trust in relationships, which in turn predict higher well-being. A meta-analysis by Chu, Sun, and Jiang (2023) {{ic|not in References}} confirmed that self-disclosure on social media was positively associated with psychological well-being, especially when it facilitated supportive interactions. During the COVID-19 pandemic, online disclosure provided a sense of connection and improved happiness and well-being despite physical isolation (Matthes et al., 2021). These findings illustrate how disclosure can maintain well-being in both face-to-face and digital contexts. Cultural factors also shape the relationship. While Western contexts often view disclosure as a sign of openness and authenticity, other cultural settings place greater value on discretion. Duan et al. (2022) observed that resilience and personal growth contribute to well-being, and it is plausible that in different cultural settings, how disclosure interacts with these traits may vary depending on social norms. This highlights that disclosure’s benefits are not universal but context-dependent. Taken together, evidence suggests that self-disclosure contributes to psychological well-being by supporting emotional regulation, strengthening relationships, and fostering growth. At the same time, disclosure carries risks such as co-rumination, oversharing, or negative social responses, which can undermine these benefits{{f}}. The overall impact therefore depends on the quality of what is shared, the context in which it occurs, and the responses it receives{{f}}. == Applications of Psychological Research == Research suggests that self-disclosure can be deliberately used as a tool to promote psychological well-being, provided it is applied thoughtfully and contextually. One key pathway is through emotional processing and regulation. Aldahadha (2023) showed that disclosure, particularly when combined with mindfulness, was associated with greater happiness and well-being, highlighting its value as a coping strategy. Similarly, Harvey and Boynton (2021) found that disclosure strengthened resilience by increasing self-esteem and self-compassion, suggesting that supportive environments for sharing can help individuals adapt to stress. Self-disclosure can also be used to build and maintain positive relationships, a central component of psychological health. Sharing personal experiences promotes intimacy and trust (Altman & Taylor, 1973; Reis & Shaver, 1988), which align with Ryff’s (1989) six dimensions of well-being, particularly positive relationships and self-acceptance. Online contexts may extend these benefits: Chu, Sun, and Jiang’s (2023) meta-analysis found that social media disclosure was positively associated with well-being when it generated supportive interactions, while Matthes et al. (2021) demonstrated that online disclosure maintained happiness and connection during the COVID-19 pandemic. At the same time, intentional self-disclosure requires managing boundaries to protect well-being. Petronio’s (2002) Communication Privacy Management theory underscores the importance of making strategic decisions about what to share and with whom. Over-disclosure or poorly timed disclosure may result in rejection, loss of privacy, or social strain (Rubin, 1975). More recently, Fejes-Vékássy, Ujhelyi, and Lantos (2024) found that frequent online sharers often underestimate risks, prioritising disclosure over concerns about privacy or data sensitivity. This highlights the need for awareness of potential downsides, particularly in digital contexts. == Quiz == <quiz display=simple> {Self-disclosure always leads to positive outcomes for psychological well-being. |type="()"} - True + False {According to Social Penetration Theory, disclosure tends to move from surface-level details to more intimate information. |type="()"} + True - False {Sharing personal experiences online during the COVID-19 pandemic had no measurable impact on well-being. |type="()"} - True + False {Ryff’s (1989) model of psychological well-being includes six dimensions, such as self-acceptance and positive relationships. |type="()"} + True - False {People who frequently disclose online often perceive sharing as less risky than it really is. |type="()"} + True - False </quiz> == Conclusion == Self-disclosure is a powerful psychological process that can foster resilience, authenticity, and connection. Evidence shows it enhances well-being by supporting emotion regulation (Aldahadha, 2023; Harvey & Boynton, 2021), building positive relationships (Reis & Shaver, 1988; Ryff, 1989), and strengthening social bonds both offline and online (Chu et al., 2023; Matthes et al., 2021). However, disclosure also carries risks, particularly when it involves oversharing, poor boundary management, or unsafe contexts (Petronio, 2002; Fejes-Vékássy et al., 2024). The key to harnessing disclosure lies in balance: sharing in ways that invite empathy and support while protecting privacy and respecting cultural expectations. By approaching disclosure with mindfulness and intention, individuals can maximise its benefits and reduce potential harms. Ultimately, thoughtful self-disclosure can play a central role in promoting psychological well-being, supporting not only personal growth but also stronger, healthier relationships. Cultural factors also play a role in how disclosure supports well-being. Duan, Guan, and Bu (2022) demonstrated that resilience and personal growth contribute to well-being across diverse contexts, yet the value of disclosure may vary according to cultural norms. In settings where openness is encouraged, disclosure may foster belonging and support; in cultures where discretion is prized, it may expose individuals to stigma or misunderstanding. Overall, psychological research demonstrates that self-disclosure can be used strategically to enhance well-being, provided individuals balance openness with appropriate boundaries, cultural sensitivity, and attention to context. ==See also== {{expand}} ==References== {{Hanging indent|1= Aldahadha, B. (2023). Self-disclosure, mindfulness, and their relationships with happiness and well-being. ''Middle East Current Psychiatry, 30''(1), 7. https://doi.org/10.1186/s43045-023-00278-5 Altman, I., & Taylor, D. (1973). ''Social penetration: The development of interpersonal relationships''. Holt, Rinehart & Winston. Berger, C. R., & Calabrese, R. J. (1975). Some explorations in initial interaction and beyond: Toward a developmental theory of interpersonal communication. ''Human Communication Research, 1''(2), 99–112. https://doi.org/10.1111/j.1468-2958.1975.tb00258.x Diener, E., Lucas, R. E., & Oishi, S. (2018). Advances and open questions in the science of subjective well-being. ''Collabra: Psychology, 4''(1), 15. https://doi.org/10.1525/collabra.115 Duan, W., Guan, Y., & Bu, H. (2022). The role of resilience and personal growth in psychological well-being during life challenges. ''Journal of Happiness Studies, 23''(5), 2235–2252. https://doi.org/10.1007/s10902-021-00462-2 Fejes-Vékássy, L., Ujhelyi, A., & Lantos, N. A. (2024). I don’t care, I share! – The importance of self-disclosure overwrites the risks of sharing on social media. ''Current Psychology, 43'', 30120–30134. https://doi.org/10.1007/s12144-024-06496-2 Harvey, J., & Boynton, K. (2021). Self-disclosure and psychological resilience: The mediating roles of self-esteem and self-compassion. ''Interpersona, 15''(1), 90–104. https://doi.org/10.5964/ijpr.4533 Hendriks, T., Schotanus-Dijkstra, M., Hassankhan, A., De Jong, J., & Bohlmeijer, E. (2020). The efficacy of multi-component positive psychology interventions for well-being: A meta-analysis. ''Journal of Happiness Studies, 21''(1), 357–390. https://doi.org/10.1007/s10902-019-00082-1 Luft, J., & Ingham, H. (1955). The Johari window, a graphic model of interpersonal awareness. ''Proceedings of the Western Training Laboratory in Group Development''. Los Angeles: UCLA. Lyyra, N., Sormunen, M., & Keltikangas-Järvinen, L. (2023). The impact of self-disclosure of negative experiences on prosociality. ''Social Cognitive and Affective Neuroscience, 19''(1), nsae003. https://doi.org/10.1093/scan/nsae003 Matthes, J., Koban, K., Neureiter, A., & Stevic, A. (2021). Longitudinal relationships among fear of COVID-19, smartphone online self-disclosure, happiness, and psychological well-being. ''Journal of Medical Internet Research, 23''(9), e28700. https://doi.org/10.2196/28700 Petronio, S. (2002). ''Boundaries of privacy: Dialectics of disclosure''. SUNY Press. Reis, H. T., & Shaver, P. (1988). Intimacy as an interpersonal process. In S. W. Duck (Ed.), ''Handbook of personal relationships'' (pp. 367–389). Wiley. Rubin, Z. (1975). Disclosing oneself to a stranger: Reciprocity and its limits. ''Journal of Experimental Social Psychology, 11''(3), 233–260. https://doi.org/10.1016/S0022-1031(75)80025-4 Ryff, C. D. (1989). Happiness is everything, or is it? Explorations on the meaning of psychological well-being. ''Journal of Personality and Social Psychology, 57''(6), 1069–1081. https://doi.org/10.1037/0022-3514.57.6.1069 Satici, S. A., Przepiorka, A., & Diener, E. (2023). Psychological flexibility and subjective well-being: A multi-country study. ''Current Psychology, 42''(2), 1337–1348. https://doi.org/10.1007/s12144-021-01519-8 Wu, X., Li, Y., et al. (2022). The effects of emotional disclosure on psychological outcomes. ''Frontiers in Psychology, 13'', 989826. https://doi.org/10.3389/fpsyg.2022.989826 }} ==External links== * [https://www.youtube.com/watch?v=pgl42XtFb0M Self disclosure: Explained] (YouTube) * [https://www.youtube.com/watch?v=mxz8KyV3Ydc What is well-being?] (YouTube) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Communication]] [[Category:Motivation and emotion/Book/Self-disclosure]] [[Category:Motivation and emotion/Book/Well-being]] ivtzxpzg9c9yioho4rxamimufncy1dc 0 323061 2814898 2762610 2026-06-09T18:54:16Z ~2026-34180-25 3091622 I added one source (Lehtivuori, 2023) to the reference list. 2814898 wikitext text/x-wiki {{title|Incentive theory of motivation:<br>How do external incentives influence approach and avoidance behaviour?}} __TOC__ ==Overview== {{RoundBoxTop|theme=1}} [[File:Deadline ! (7320523564).jpg|thumb|250x250px|'''Figure 1.''' Student working towards deadline]] '''Scenario''' Maddy is a university student who has an essay due on the weekend. She knows she should start working on it early so that she has time to do her best work, but instead, she puts it off. Throughout the week, she spends her free time binging shows and scrolling through Instagram reels. Finally, the day before her essay is due, she realises that if she doesn’t start now, she is going to fail her course. Fuelled by the fear of failing, she spends her weekend frantically writing. Jamie, Maddy’s classmate, is a high achiever and hopes to get a dean’s excellence award for her marks this semester. Jamie started researching and planning her essay weeks before the deadline. She set aside time each day to work on her essay, meaning she finished a draft early in the week and was able to get feedback. She handed in her final, polished essay before the weekend, feeling confident in her work and excited for her result. {{RoundBoxBottom}} In this scenario, two students are working towards the same deadline (see Figure 1); however, each one is fuelled by a different type of motivation. Maddy is driven by a desire to avoid a negative outcome (e.g., failing her course); this is an example of [[Motivation and emotion/Book/2013/Avoidance motivation|avoidance]] behaviour (Elliot, 2013). Meanwhile, Jamie is acting to gain a positive outcome (e.g., achieving a high mark), an example of [[Motivation and emotion/Book/2022/Approach motivation|approach]] behaviour (Elliot, 2013). Both Maddy's and Jamie’s behaviours are driven by external incentives. Incentive theory suggests that human behaviour is driven by external factors, made up of both positive incentives (rewards) and negative incentives (punishments). These incentives can be tangible (like money or food) or intangible (like praise or recognition) (Silverman et al., 2016). The strength of an incentive impacts the likelihood of an individual acting in response. However, the same incentive can have varying power depending on the person or situation (Hagger et al., 2015). Other factors, including the delivery time of rewards, reward salience, and choice, can also impact how effective an incentive is (Lehtivuori, 2023). Some studies have suggested that external incentives can undermine intrinsic motivation (motivation driven by personal values and enjoyment) (Deci & Ryan, 2000). This is particularly significant as intrinsic motivation is linked to better outcomes in the performance and well-being of individuals, compared with externally controlled behaviour (Deci & Ryan, 2000). However, different theorists hold contentious views on the issue, and it remains the topic of debate (Lehtivuori, 2023). Understanding motivation has major real-world significance in that motivation is essential for producing behaviour. Knowing how external factors influence behaviour enables environments to be manipulated to promote desirable behaviours (Hagger et al., 2020). Managers, teachers, coaches, health care providers, parents, and anyone else needing to mobilise others to action, benefit from a knowledge of motivation and how to harness it (Deci & Ryan, 2000). {{RoundBoxTop|theme=3}} '''Focus questions''' # What are approach and avoidance behaviours? # What impacts the strength of an incentive? # How do external incentives impact intrinsic motivation? # What are the practical applications of incentive theory? {{RoundBoxBottom}} == Approach and avoidance behaviour== The incentive theory of motivation is underpinned by the concepts of approach and avoidance behaviour. The approach-avoidance distinction has appeared in scholarly writing for thousands of years, with the earliest reference being from the Greek philosopher Democritus (460–370 B.C.E.), who first described the human tendency to pursue immediate pleasure and avoid pain (Elliot, 2006). This concept continued to hold significance in the work of early psychologist William James in the late nineteenth century and later in Freud’s psychodynamic theory (Elliot, 2006). In the 1960s and 1970s, cognitive and social-cognitive theorists moved away from affective explanations of motivation; however, in the 1990s, approach-avoidance theories returned to prominence as theorists acknowledged the interconnected nature of cognition and motivation. In the last several decades, the approach-avoidance distinction has been used and applied in many ways (Elliot, 2006). === Appetitive vs aversive stimuli === Approach motivation is generally associated with appetitive stimuli, rewards, and incentives, while avoidance is connected with aversive stimuli, punishments, and threats (Elliot et al., 2013). It is expected that individuals with normal adaptive functioning will orient themselves towards rewards and incentives, and away from punishments and threats (Elliot et al., 2013). Approach and avoidance can be literal or figurative, {{g}} for example, stepping back from a painting to view it better is an approach behaviour, despite involving physically moving away from it (Elliot, 2006). Additionally, approach and avoidance behaviour do not just include new movement toward or away from stimuli, but also maintaining current orientation toward or away from a stimulus (Elliot et al., 2013). === Emotional valence === Approach and avoidance behaviours have traditionally been linked to specific emotions (Elliot, 2013). Particularly, they are connected to emotional valence, that is, positive or negative feelings (Elliot, 2013). Approach behaviour is linked with positive valence, and avoidance behaviour is linked with negative valence (Elliot, 2013). However, this concept does not apply to all emotions; anger, for example, has a negative valence but is often linked to approach motivation (Elliot, 2013). === Stimulus evaluation === [[File:Cake Dessert.jpg|thumb|'''Figure 2.''' Dessert can have positive or negative motivational valence|294x294px]] When presented with a stimulus, people engage in cognitive appraisals to determine if the stimulus is beneficial or detrimental (Elliot, 2013). These stimuli can vary from concrete (e.g., physical objects) to abstract (e.g., memories) (Elliot, 2013). Stimulus evaluation occurs at multiple levels across the neuroaxis, including rudimentary exteroceptive reflexes, subcortical computations, and higher-order cortical processing (Elliot, 2013). These evaluations involve a contextual lens, meaning an individual's current needs are considered when appraising a stimulus. For example, a dessert (see Figure 2) that usually holds positive valence may have negative motivational valence when an individual is full or is on a diet (Elliot, 2013). Understanding that the same incentive may hold different value depending on the context is crucial in applying incentive motivation effectively to real-world scenarios. == Strength of incentives == Not all incentives are created equal. Some incentives have the power to motivate significant behaviour change, while others lead to no change at all. Further, one incentive could motivate one person while the same incentive could have no influence on another person. So {{g}} what are the factors that impact the strength and effectiveness of an incentive? === Delivery time === In studies looking at the impact of external rewards on intrinsic motivation, the delivery time of the rewards is significant. Immediate, as opposed to delayed delivery of rewards has a strong positive impact on intrinsic motivation (Woolley & Fishbach, 2018). This may be due to [[w:Temporal_discounting|temporal discounting]], meaning we value immediate rewards more highly than later ones (Liu et al., 2022). Liu et al. (2022) specifically improved upon prior research by removing a possible misattribution effect of extrinsic motivation{{explain}}. This is important because, unlike extrinsic motivation, which is often short in duration and linked to more passive behaviours, intrinsic motivation is connected to higher performance and longer-lasting behaviour change (Liu et al., 2022). === Reward salience and choice === Hendijani & Steel (2020) studied the interaction between reward salience and choice on motivation and performance and noted interesting connections. In many contexts{{f}}, the provision of choice has been seen to improve motivation and performance. Similarly, external rewards have been utilised across a range of domains to enhance performance. In the study, Hendijani & Steel examined the interaction between these two variables. It was found that salient rewards in a no-choice condition had a positive effect on motivation, as did non-salient rewards in a choice condition. Meanwhile, salient rewards in a choice condition reduced motivation and performance. It is theorised these results are due to personal loci of control. In the salient reward and no-choice condition, a consistent external locus of control is established, promoting extrinsic motivation. In the non-salient, choice condition, an internal locus of control emerges. However, the salient reward, choice condition generates a confused locus of control, diminishing motivation and performance. === '''Fostering competence and autonomy''' === Rewards that foster a sense of competence and autonomy in an individual are likely to result in stronger incentives{{f}}. This is based on meeting two of the basic psychological needs outlined in self-determination theory{{f}}. For example, a teacher providing meaningful feedback outlining what was done well and what could be improved on an assignment could be viewed as informative and thus foster a sense of competence (Hagger et al., 2015). However, external contingencies that are perceived as controlling will diminish feelings of autonomy and decrease motivation (Hagger et al., 2015). == Incentive theory and intrinsic motivation == Motivation can broadly be divided into two distinct types: intrinsic and extrinsic motivation. Intrinsic motivation is characterised by behaviour born out of personal interest and values (e.g., a personal commitment to excel)(Deci & Ryan, 2000). {{g}} While extrinsic motivation relates to behaviours driven by reasons external to one’s self (e.g., external pressure to participate) (Deci & Ryan, 2000). Incentive theory primarily focuses on extrinsic motivation, considering how external incentives influence behaviour. However, intrinsic motivation is linked to better outcomes in the performance and well-being of individuals, compared with externally controlled behaviour (Deci & Ryan, 1991; Deci & Ryan, 1997; Deci & Ryan, 1995). Therefore, understanding ways in which external incentives can promote intrinsic motivation rather than undermine it is crucial. The following sections will unpack intrinsic and extrinsic motivation interactions through the lens of self-determination theory (SDT). === Self-determination theory === Self-determination theory (SDT) centres around the objective of achieving self-motivation and personality integration. SDT identifies three primary needs that facilitate optimal functioning, {{g}} these needs are: competence, relatedness, and autonomy (Deci & Ryan, 2000). Meeting these needs is considered essential for constructive social development and personal well-being. Understanding SDT is relevant in many domains, including: for parents and educators concerned with the cognitive and personality development of children, for promoting learning and positive behaviour, for managers wanting to promote motivation and commitment in the workplace, and for psychotherapists and health professionals wanting to support maintained change in patients (Deci & Ryan, 2000). === Intrinsic motivation === SDT suggests that conditions which support autonomy and competence promote intrinsic motivation, while conditions that control behaviour and reduce perceived effectance undermine intrinsic motivation (Deci & Ryan, 2000). External social-contextual events, such as feedback, communication, or rewards, that lead to feelings of competence enhance intrinsic motivation. For example, intrinsic motivation was found to increase as a result of positive performance feedback and decrease as a result of negative feedback (Deci, 1975). This was due to its impact on perceived competence (Vallerand and Reid, 1984). However, feelings of competence only lead to enhanced intrinsic motivation when they are accompanied by a sense of autonomy, that is, when behaviour is perceived as being self-determined (Deci & Ryan, 2000). Autonomy supportive environments lead to increased intrinsic motivation, creativity, curiosity, desire to learn, and a desire to be challenged (Deci & Ryan, 2000). Understanding this is crucial in motivating others to commitment, effort, and high-quality performance (Deci & Ryan, 2000). In contrast, methods that employ excessive control, nonoptimal challenges, and lack connectedness result in diminished intrinsic motivation and can lead to negative impacts, including a lack of intuitive taking and responsibility as well as distress and psychopathology (Deci & Ryan, 2000). Despite the power of intrinsic motivation, people will only be intrinsically motivated in tasks that hold intrinsic interest for them (e.g., novelty, challenge, aesthetic value); extrinsic motivation is essential for tasks that hold no intrinsic interest (Deci & Ryan, 2000). === Cognitive evaluation theory === Cognitive Evaluation Theory (CET) is a subtheory within SDT that studies the factors that undermine or promote intrinsic motivation (Deci & Ryan, 2000). Research looking at the impact of extrinsic rewards on intrinsic motivation has resulted in much debate. Deci, Koestner & Ryan's 1999 meta-analysis found that tangible rewards contingent on task performance undermined intrinsic motivation, which was interpreted as resulting from a diminished sense of autonomy. Tang & Hall (1996) also suggest that task-contingent rewards lead to reduced intrinsic motivation through the overjustification effect. However, other meta-analyses, particularly from Cameron and colleagues, dispute the idea that there is a negative impact of extrinsic rewards on intrinsic motivation (Cameron & Pierce, 1994). A recent meta-analysis suggests extrinsic rewards can negatively impact intrinsic motivation, particularly when motivation is measured behaviourally after contingent rewards are no longer given (Lehtivuori, 2023). While different viewpoints on the issue remain, research suggests the relationship between extrinsic rewards and motivation is complex and that the type of reward, how the reward is earned, and under what conditions and context motivation is examined, impacts results (Lehtivuori, 2023). == Incentive interventions == A practical application of the incentive theory of motivation for addressing social issues is incentive interventions. Incentive interventions are an effective way of promoting socially valuable behaviours and leading to positive behaviour change in a range of settings (Silverman et al., 2016). They can be used both to create desirable/adaptive habits and to break undesirable/maladaptive habits (Gneezy et al., 2020). Incentives can reduce barriers involved in behaviour change, including counteracting [[w:_Present_bias|present bias]] (Gneezy et al., 2020). Within incentive interventions, tangible and desirable consequences (e.g., monetary rewards) are given to individuals upon performing an observable and verifiable behaviour (Silverman et al., 2016). Incentive interventions are built around the operant principle of reinforcement, that is, that consequences that affect behaviour are reinforcers. Reinforcers are used to increase the behaviours they are contingent on (reinforcement) (Silverman et al., 2016). The following table outlines some of the many applications of incentive interventions. {| class="wikitable" |Increasing physical activity in sedentary adults |A study using a deposit contract design found monetary incentive interventions successfully increased physical activity in sedentary and underactive adults (Washington et al., 2016). |- |Smoking cessation in smokers with ADHD |Individuals with attention-deficit/hyperactivity disorder (ADHD) are twice as likely to be smokers as the general population. As such, an incentive intervention study was designed to reduce smoking with monetary vouchers. This intervention, facilitated via an app, was found to be effective (Dan et al., 2016). |- |Promoting health behaviour change |A meta-analysis where participant-level data from seven studies were pooled, found financial incentives effective in promoting health behaviour change (Haff, 2015). |- |Remotely implemented interventions |A systematic review of financial incentive interventions targeting health behaviors found remote implementation of interventions using technology to be effective. In these studies, monitoring of behavior and/or delivery of incentives was done remotely (Kurti, 2016). |- |Work productivity |Incentive interventions have been found effective in studies looking at monetary incentives for work performance. Ratio schedules for reward delivery were found to be the most significant determinant of performance (Bucklin, 2001). |} ==Conclusion== Incentive theory of motivation is the concept that behaviours are driven by external factors or incentives. Generally, positive incentives drive approach behaviour, while negative incentives lead to avoidance behaviour. Incentives can vary from tangible to intangible and can vary in strength due to many factors related to the specific context and individual. Delivery time, reward salience, choice, and feelings of autonomy and competence are particularly significant in determining the strength of incentives. The exact manner in which extrinsic rewards interact with intrinsic motivation remains a matter of debate, with some theorists arguing for an undermining effect while others argue that no negative effect exists. Incentive theory has many practical applications, particularly in the fields of health, education, and management. Understanding how motivation is impacted by external incentives enables the promotion of desirable behaviours. A key application of this theory is incentive interventions for behaviour change, which can be utilised across health, workplace, and educational settings. ==See also== * [[Motivation and emotion/Book/2022/Approach motivation|Approach motivation]] (Book chapter, 2022) * [[Motivation and emotion/Book/2013/Avoidance motivation|Avoidance motivation]] (Book chapter, 2013) * [[Motivation and emotion/Book/2013/Extrinsic motivation|Extrinsic motivation]] (Book chapter, 2013) * [[Motivation and emotion/Book/2022/Hidden costs of reward|Hidden costs of reward]] (Book chapter, 2022) * [[w:Incentives|Incentives]] (Wikipedia) * [[Motivation and emotion/Book/2019/Rewards, punishments, and social cooperation|Rewards, punishments, and social cooperation]] (Book chapter, 2019) ==References== {{Hanging indent|1= Bucklin, B. R., & Dickinson, A. M. (2001). Individual monetary incentives: A review of different types of arrangements between performance and pay. ''Journal of Organizational Behavior Management, 21''(3), 45–137. https://doi.org/10.1300/J075v21n03_03 Cameron, J., & Pierce, W. D. (1994). Reinforcement, Reward, and Intrinsic Motivation: A Meta-Analysis. ''Review of Educational Research, 64''(3), 363–423. https://doi.org/10.3102/00346543064003363 Dan, M., Grabinski, M. J., & Raiff, B. R. (2016). Smartphone-based contingency management for smoking cessation with smokers diagnosed with attention-deficit/hyperactivity disorder. ''Translational Issues in Psychological Science, 2''(2), 116–127. https://doi.org/10.1037/tps0000062 Deci, E. L. (1975). ''Intrinsic motivation''. Plenum Press. https://doi.org/10.1007/978-1-4613-4446-9 Deci, E. L., Koestner, R., & Ryan, R. M. (1999). A Meta-Analytic Review of Experiments Examining the Effects of Extrinsic Rewards on Intrinsic Motivation. ''Psychological Bulletin, 125''(6), 627–668. https://doi.org/10.1037/0033-2909.125.6.627 Elliot, A. J. (2006). The Hierarchical Model of Approach-Avoidance Motivation. ''Motivation and Emotion, 30''(2), 111–116. https://doi.org/10.1007/s11031-006-9028-7 Elliot, A. J., Eder, A. B., & Harmon-Jones, E. (2013). Approach–Avoidance Motivation and Emotion: Convergence and Divergence. ''Emotion Review, 5''(3), 308–311. https://doi.org/10.1177/1754073913477517 Gneezy, U., Kajackaite, A., Meier, S., & Hagger, M. S. (2020). Incentive-based interventions. In ''The Handbook of Behavior Change'' (pp. 523–536). Cambridge University Press. https://doi.org/10.1017/9781108677318.036 Gneezy, U., Meier, S., & Rey-Biel, P. (2011). ''When and why incentives (don’t) work to modify behavior''. The Journal of Economic Perspectives, 25(4), 191–210. https://doi.org/10.1257/jep.25.4.191 Haff, N., Patel, M. S., Lim, R., Zhu, J., Troxel, A. B., Asch, D. A., & Volpp, K. G. (2015). The role of behavioral economic incentive design and demographic characteristics in financial incentive-based approaches to changing health behaviors: A meta-analysis. ''American Journal of Health Promotion, 29''(5), 314–323. https://doi.org/10.4278/ajhp.140714-LIT-333 Hagger, M., Cameron, L. D., Hamilton, K., Hankonen, N., & Lintunen, T. (Eds.). (2020). The handbook of behavior change. Cambridge University Press. https://doi.org/10.1017/9781108677318 Hagger, M. S., Koch, S., & Chatzisarantis, N. L. D. (2015). The effect of causality orientations and positive competence-enhancing feedback on intrinsic motivation: A test of additive and interactive effects. ''Personality and Individual Differences, 72'', 107–111. https://doi.org/10.1016/j.paid.2014.08.012 Hendijani, R., & Steel, P. (2020). Motivational congruence effect: How reward salience and choice influence motivation and performance. ''Cogent Business & Management, 7''(1). https://doi.org/10.1080/23311975.2020.1791444 Kurti, A. N., Davis, D., Redner, R., Jarvis, B., Zvorsky, I., Keith, D. R., Bolivar, H., White, T. J., Rippberger, P., Markeish, C., Atwood, G., & Higgins, S. T. (2016). A review of the literature on remote monitoring technology in incentive-based interventions for health-related behavior change. ''Translational Issues in Psychological Science, 2''(2), 128–152. https://doi.org/10.1037/tps0000067 Lehtivuori, A. (2023). When do extrinsic rewards undermine intrinsic motivation? A meta-analysis [Published doctoral thesis, University of Turku]. https://urn.fi/URN:ISBN:978-951-29-9132-7 Liu, Y., Yang, Y., Bai, X., Chen, Y., & Mo, L. (2022). Do immediate external rewards really enhance intrinsic motivation? ''Frontiers in Psychology, 13'', 853879. https://doi.org/10.3389/fpsyg.2022.853879 Ryan, R. M., & Deci, E. L. (2000). Self-Determination Theory and the Facilitation of Intrinsic Motivation, Social Development, and Well-Being. ''The American Psychologist, 55''(1), 68–78. https://doi.org/10.1037/0003-066X.55.1.68 Silverman, K., Jarvis, B. P., Jessel, J., & Lopez, A. A. (2016). Incentives and motivation.''Translational Issues in Psychological Science, 2''(2), 97–100. https://doi.org/10.1037/tps0000073 Tang, S.-H., & Hall, V. C. (1995). The overjustification effect: A meta-analysis. ''Applied Cognitive Psychology, 9''(5), 365–404. https://doi.org/10.1002/acp.2350090502 Vallerand, R. J., & Reid, G. (1984). On the Causal Effects of Perceived Competence on Intrinsic Motivation: A Test of Cognitive Evaluation Theory. ''Journal of Sport Psychology, 6''(1), 94–102. https://doi.org/10.1123/jsp.6.1.94 Washington, D. W., McMullen, D., & Devoto, A. (2016). A matched deposit contract intervention to increase physical activity in underactive and sedentary adults. ''Translational Issues in Psychological Science, 2''(2), 101–115. https://doi.org/10.1037/tps0000069 Woolley, K., & Fishbach, A. (2018). It’s About Time: Earlier Rewards Increase Intrinsic Motivation. ''Journal of Personality and Social Psychology, 114''(6), 877–890. https://doi.org/10.1037/pspa0000116 }} ==External links== * [https://selfdeterminedlife.substack.com/p/external-rewards-and-motivation The impact of external rewards on motivation] (The Self Determined Life) * [https://www.thebehavioralscientist.com/glossary/incentive-theory? What is incentive theory in behavioral economics?] (The Behavioral Scientist) * [https://www.workhuman.com/blog/incentive-theory-of-motivation/ What is the incentive theory of motivation?] (workhuman.com) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Incentive]] [[Category:Motivation and emotion/Book/Reward]] [[Category:Motivation and emotion/Book/Motivation]] ntle0vnlkprbcdntfvuhygh30vg558q 0 323153 2814933 2814827 2026-06-10T00:07:17Z Jtneill 10242 /* Emotion */ [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? 2814933 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}} # [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}} # [[/Attachment styles and relatedness motivation/]] - What is the relationship between attachment styles and pursuit of social connection? {{ME-By|User Name}} # [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}} # [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}} # [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}} # [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}} # [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}} # [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}} # [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}} # [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}} # [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}} # [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}} # [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}} # [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}} # [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}} # [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}} # [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}} # [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}} # [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}} # [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}} # [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}} # [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}} # [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}} # [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}} # [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}} # [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}} # [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}} # [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}} # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}} # [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}} # [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}} # [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}} # [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}} # [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}} # [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}} # [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}} # [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}} # [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}} # [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}} # [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}} # [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}} # [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}} # [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}} # [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}} # [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}} # [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}} # [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}} # [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}} # [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}} # [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}} # [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}} ==Emotion== # [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}} # [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}} # [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}} # [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}} # [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}} # [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}} # [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}} # [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}} # [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}} # [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}} # [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}} # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}} # [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}} # [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}} # [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}} # [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}} # [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}} # [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}} # [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}} # [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}} # [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}} # [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}} # [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}} # [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}} # [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}} # [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}} # [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}} # [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}} # [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}} # [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}} # [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}} # [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}} # [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}} # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}} # [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}} # [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}} # [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}} # [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}} # [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}} # [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}} # [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}} # [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}} # [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}} # [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}} ==Motivation and emotion== # [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}} # [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}} [[Category:Motivation and emotion/Book]] oipi7a08wl0tq8vz5blwibqwsjsnzmc 2814934 2814933 2026-06-10T00:09:38Z Jtneill 10242 /* Emotion */ # [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}} 2814934 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}} # [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}} # [[/Attachment styles and relatedness motivation/]] - What is the relationship between attachment styles and pursuit of social connection? {{ME-By|User Name}} # [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}} # [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}} # [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}} # [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}} # [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}} # [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}} # [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}} # [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}} # [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}} # [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}} # [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}} # [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}} # [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}} # [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}} # [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}} # [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}} # [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}} # [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}} # [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}} # [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}} # [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}} # [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}} # [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}} # [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}} # [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}} # [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}} # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}} # [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}} # [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}} # [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}} # [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}} # [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}} # [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}} # [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}} # [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}} # [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}} # [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}} # [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}} # [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}} # [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}} # [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}} # [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}} # [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}} # [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}} # [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}} # [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}} # [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}} # [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}} # [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}} ==Emotion== # [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}} # [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}} # [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}} # [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}} # [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}} # [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}} # [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}} # [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}} # [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}} # [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}} # [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}} # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}} # [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}} # [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}} # [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}} # [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}} # [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}} # [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}} # [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}} # [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}} # [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}} # [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}} # [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}} # [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}} # [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}} # [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}} # [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}} # [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}} # [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}} # [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}} # [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}} # [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}} # [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}} # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}} # [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}} # [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}} # [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}} # [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}} # [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}} # [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}} # [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}} # [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}} # [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}} # [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}} # [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}} ==Motivation and emotion== # [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}} # [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}} [[Category:Motivation and emotion/Book]] oyg03cf2h9krg8cfqqqqz1poyvdxfiy 2814944 2814934 2026-06-10T00:18:23Z Jtneill 10242 /* Motivation */ + [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? 2814944 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}} # [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}} # [[/Attachment styles and relatedness motivation/]] - What is the relationship between attachment styles and pursuit of social connection? {{ME-By|User Name}} # [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}} # [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}} # [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}} # [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}} # [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}} # [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}} # [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}} # [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}} # [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}} # [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}} # [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}} # [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}} # [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}} # [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}} # [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}} # [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}} # [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}} # [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}} # [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}} # [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}} # [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}} # [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}} # [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}} # [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}} # [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}} # [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}} # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}} # [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}} # [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}} # [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}} # [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}} # [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}} # [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}} # [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}} # [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}} # [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}} # [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}} # [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}} # [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}} # [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}} # [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}} # [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}} # [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}} # [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}} # [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}} # [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}} # [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}} # [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}} # [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}} # [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}} ==Emotion== # [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}} # [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}} # [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}} # [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}} # [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}} # [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}} # [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}} # [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}} # [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}} # [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}} # [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}} # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}} # [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}} # [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}} # [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}} # [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}} # [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}} # [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}} # [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}} # [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}} # [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}} # [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}} # [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}} # [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}} # [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}} # [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}} # [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}} # [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}} # [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}} # [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}} # [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}} # [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}} # [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}} # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}} # [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}} # [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}} # [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}} # [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}} # [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}} # [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}} # [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}} # [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}} # [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}} # [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}} # [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}} ==Motivation and emotion== # [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}} # [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}} [[Category:Motivation and emotion/Book]] gczextxmz7bizn4lvr0t3px8ovrdo48 2814949 2814944 2026-06-10T00:22:04Z Jtneill 10242 /* Emotion */ [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? 2814949 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}} # [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}} # [[/Attachment styles and relatedness motivation/]] - What is the relationship between attachment styles and pursuit of social connection? {{ME-By|User Name}} # [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}} # [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}} # [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}} # [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}} # [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}} # [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}} # [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}} # [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}} # [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}} # [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}} # [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}} # [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}} # [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}} # [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}} # [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}} # [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}} # [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}} # [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}} # [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}} # [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}} # [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}} # [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}} # [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}} # [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}} # [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}} # [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}} # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}} # [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}} # [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}} # [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}} # [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}} # [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}} # [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}} # [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}} # [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}} # [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}} # [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}} # [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}} # [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}} # [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}} # [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}} # [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}} # [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}} # [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}} # [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}} # [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}} # [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}} # [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}} # [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}} # [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}} ==Emotion== # [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}} # [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}} # [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}} # [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}} # [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}} # [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}} # [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}} # [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}} # [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}} # [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}} # [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}} # [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}} # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}} # [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}} # [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}} # [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}} # [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}} # [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}} # [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}} # [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}} # [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}} # [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}} # [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}} # [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}} # [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}} # [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}} # [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}} # [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}} # [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}} # [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}} # [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}} # [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}} # [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}} # [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}} # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}} # [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}} # [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}} # [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}} # [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}} # [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}} # [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}} # [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}} # [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}} # [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}} # [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}} # [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}} ==Motivation and emotion== # [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}} # [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}} [[Category:Motivation and emotion/Book]] 14p9mih09jgdie08srfx1b7ld3q293u 2814952 2814949 2026-06-10T00:26:08Z Jtneill 10242 /* Emotion */ + [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? 2814952 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}} # [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}} # [[/Attachment styles and relatedness motivation/]] - What is the relationship between attachment styles and pursuit of social connection? {{ME-By|User Name}} # [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}} # [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}} # [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}} # [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}} # [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}} # [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}} # [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}} # [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}} # [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}} # [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}} # [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}} # [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}} # [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}} # [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}} # [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}} # [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}} # [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}} # [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}} # [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}} # [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}} # [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}} # [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}} # [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}} # [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}} # [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}} # [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}} # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}} # [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}} # [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}} # [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}} # [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}} # [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}} # [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}} # [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}} # [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}} # [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}} # [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}} # [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}} # [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}} # [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}} # [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}} # [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}} # [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}} # [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}} # [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}} # [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}} # [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}} # [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}} # [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}} # [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}} ==Emotion== # [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}} # [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}} # [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}} # [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}} # [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}} # [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}} # [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}} # [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}} # [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}} # [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}} # [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}} # [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}} # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}} # [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}} # [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}} # [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}} # [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}} # [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}} # [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}} # [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}} # [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}} # [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}} # [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}} # [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}} # [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}} # [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}} # [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}} # [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}} # [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}} # [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}} # [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}} # [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}} # [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}} # [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}} # [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}} # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}} # [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}} # [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}} # [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}} # [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}} # [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}} # [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}} # [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}} # [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}} # [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}} # [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}} # [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}} ==Motivation and emotion== # [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}} # [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}} [[Category:Motivation and emotion/Book]] kj5i4o5vny8c1236cmr5kojp8t3cnvo 2814971 2814952 2026-06-10T06:52:59Z Jtneill 10242 /* Emotion */ [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? 2814971 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}} # [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}} # [[/Attachment styles and relatedness motivation/]] - What is the relationship between attachment styles and pursuit of social connection? {{ME-By|User Name}} # [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}} # [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}} # [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}} # [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}} # [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}} # [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}} # [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}} # [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}} # [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}} # [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}} # [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}} # [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}} # [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}} # [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}} # [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}} # [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}} # [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}} # [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}} # [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}} # [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}} # [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}} # [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}} # [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}} # [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}} # [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}} # [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}} # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}} # [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}} # [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}} # [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}} # [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}} # [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}} # [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}} # [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}} # [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}} # [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}} # [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}} # [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}} # [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}} # [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}} # [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}} # [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}} # [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}} # [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}} # [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}} # [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}} # [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}} # [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}} # [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}} # [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}} ==Emotion== # [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}} # [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}} # [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}} # [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}} # [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}} # [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}} # [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}} # [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}} # [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}} # [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}} # [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}} # [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}} # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}} # [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}} # [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}} # [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}} # [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}} # [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}} # [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}} # [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}} # [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}} # [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}} # [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}} # [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}} # [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}} # [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}} # [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}} # [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}} # [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}} # [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}} # [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}} # [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}} # [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}} # [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}} # [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}} # [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}} # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}} # [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}} # [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}} # [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}} # [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}} # [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}} # [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}} # [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}} # [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}} # [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}} # [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}} # [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}} ==Motivation and emotion== # [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}} # [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}} [[Category:Motivation and emotion/Book]] lv0jjiiinnq7jtnbkk51i1zkougnt0k 2814972 2814971 2026-06-10T06:56:21Z Jtneill 10242 /* Motivation */ [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? 2814972 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}} # [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}} # [[/Attachment styles and relatedness motivation/]] - What is the relationship between attachment styles and pursuit of social connection? {{ME-By|User Name}} # [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}} # [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}} # [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}} # [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}} # [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}} # [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}} # [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}} # [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}} # [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}} # [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}} # [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}} # [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}} # [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}} # [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}} # [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}} # [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}} # [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}} # [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}} # [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}} # [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}} # [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}} # [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}} # [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}} # [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}} # [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}} # [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}} # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}} # [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}} # [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}} # [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}} # [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}} # [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}} # [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}} # [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}} # [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}} # [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}} # [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}} # [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}} # [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}} # [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}} # [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}} # [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}} # [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}} # [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}} # [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}} # [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}} # [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}} # [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}} # [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}} # [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}} # [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}} ==Emotion== # [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}} # [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}} # [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}} # [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}} # [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}} # [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}} # [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}} # [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}} # [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}} # [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}} # [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}} # [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}} # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}} # [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}} # [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}} # [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}} # [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}} # [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}} # [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}} # [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}} # [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}} # [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}} # [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}} # [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}} # [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}} # [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}} # [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}} # [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}} # [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}} # [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}} # [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}} # [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}} # [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}} # [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}} # [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}} # [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}} # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}} # [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}} # [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}} # [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}} # [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}} # [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}} # [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}} # [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}} # [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}} # [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}} # [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}} # [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}} ==Motivation and emotion== # [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}} # [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}} [[Category:Motivation and emotion/Book]] po25ty8lk9rl7mdcgarnn6w18j0ndme 2814982 2814972 2026-06-10T07:16:34Z Jtneill 10242 /* Emotion */ + [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? 2814982 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}} # [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}} # [[/Attachment styles and relatedness motivation/]] - What is the relationship between attachment styles and pursuit of social connection? {{ME-By|User Name}} # [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}} # [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}} # [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}} # [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}} # [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}} # [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}} # [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}} # [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}} # [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}} # [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}} # [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}} # [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}} # [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}} # [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}} # [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}} # [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}} # [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}} # [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}} # [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}} # [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}} # [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}} # [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}} # [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}} # [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}} # [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}} # [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}} # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}} # [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}} # [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}} # [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}} # [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}} # [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}} # [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}} # [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}} # [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}} # [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}} # [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}} # [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}} # [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}} # [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}} # [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}} # [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}} # [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}} # [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}} # [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}} # [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}} # [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}} # [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}} # [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}} # [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}} # [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}} ==Emotion== # [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}} # [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}} # [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}} # [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}} # [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}} # [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}} # [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}} # [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}} # [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}} # [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}} # [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}} # [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}} # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}} # [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}} # [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}} # [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}} # [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}} # [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}} # [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}} # [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}} # [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}} # [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}} # [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}} # [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}} # [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}} # [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}} # [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}} # [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}} # [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}} # [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}} # [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}} # [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}} # [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}} # [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}} # [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}} # [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}} # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}} # [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}} # [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}} # [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}} # [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}} # [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}} # [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}} # [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}} # [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}} # [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}} # [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}} # [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}} # [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}} ==Motivation and emotion== # [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}} # [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}} [[Category:Motivation and emotion/Book]] p9fw0vuwvk2r5zosk0iejndjrg9biof 2814985 2814982 2026-06-10T07:21:16Z Jtneill 10242 /* Emotion */ + [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? 2814985 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}} # [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}} # [[/Attachment styles and relatedness motivation/]] - What is the relationship between attachment styles and pursuit of social connection? {{ME-By|User Name}} # [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}} # [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}} # [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}} # [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}} # [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}} # [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}} # [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}} # [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}} # [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}} # [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}} # [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}} # [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}} # [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}} # [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}} # [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}} # [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}} # [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}} # [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}} # [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}} # [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}} # [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}} # [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}} # [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}} # [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}} # [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}} # [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}} # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}} # [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}} # [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}} # [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}} # [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}} # [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}} # [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}} # [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}} # [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}} # [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}} # [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}} # [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}} # [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}} # [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}} # [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}} # [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}} # [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}} # [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}} # [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}} # [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}} # [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}} # [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}} # [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}} # [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}} # [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}} ==Emotion== # [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}} # [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}} # [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}} # [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}} # [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}} # [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}} # [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}} # [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}} # [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}} # [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}} # [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}} # [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}} # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}} # [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}} # [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}} # [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}} # [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}} # [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}} # [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}} # [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}} # [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}} # [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}} # [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}} # [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}} # [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}} # [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}} # [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}} # [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}} # [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}} # [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}} # [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}} # [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}} # [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}} # [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}} # [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}} # [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}} # [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}} # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}} # [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}} # [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}} # [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}} # [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}} # [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}} # [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}} # [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}} # [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}} # [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}} # [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}} # [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}} # [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}} ==Motivation and emotion== # [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}} # [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}} [[Category:Motivation and emotion/Book]] 1bvvv974l1wf0mcysl38j91qz6igb9x 2814987 2814985 2026-06-10T07:25:04Z Jtneill 10242 /* Emotion */ + [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? 2814987 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}} # [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}} # [[/Attachment styles and relatedness motivation/]] - What is the relationship between attachment styles and pursuit of social connection? {{ME-By|User Name}} # [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}} # [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}} # [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}} # [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}} # [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}} # [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}} # [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}} # [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}} # [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}} # [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}} # [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}} # [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}} # [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}} # [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}} # [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}} # [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}} # [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}} # [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}} # [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}} # [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}} # [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}} # [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}} # [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}} # [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}} # [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}} # [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}} # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}} # [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}} # [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}} # [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}} # [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}} # [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}} # [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}} # [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}} # [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}} # [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}} # [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}} # [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}} # [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}} # [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}} # [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}} # [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}} # [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}} # [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}} # [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}} # [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}} # [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}} # [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}} # [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}} # [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}} # [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}} ==Emotion== # [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}} # [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}} # [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}} # [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}} # [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}} # [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}} # [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}} # [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}} # [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}} # [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}} # [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}} # [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}} # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}} # [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}} # [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}} # [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}} # [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}} # [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}} # [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}} # [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}} # [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}} # [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}} # [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}} # [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}} # [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}} # [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}} # [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}} # [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}} # [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}} # [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}} # [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}} # [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}} # [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}} # [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}} # [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}} # [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}} # [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}} # [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}} # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}} # [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}} # [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}} # [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}} # [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}} # [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}} # [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}} # [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}} # [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}} # [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}} # [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}} # [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}} # [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}} ==Motivation and emotion== # [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}} # [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}} [[Category:Motivation and emotion/Book]] pio7c3mbukvxsfmiez6skzj4gqjvf1x 2814990 2814987 2026-06-10T07:30:14Z Jtneill 10242 + [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? 2814990 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}} # [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}} # [[/Attachment styles and relatedness motivation/]] - What is the relationship between attachment styles and pursuit of social connection? {{ME-By|User Name}} # [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}} # [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}} # [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}} # [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}} # [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}} # [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}} # [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}} # [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}} # [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}} # [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}} # [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}} # [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}} # [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}} # [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}} # [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}} # [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}} # [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}} # [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}} # [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}} # [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}} # [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}} # [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}} # [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}} # [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}} # [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}} # [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}} # [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}} # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}} # [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}} # [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}} # [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}} # [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}} # [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}} # [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}} # [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}} # [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}} # [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}} # [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}} # [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}} # [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}} # [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}} # [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}} # [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}} # [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}} # [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}} # [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}} # [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}} # [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}} # [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}} # [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}} # [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}} # [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}} ==Emotion== # [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}} # [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}} # [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}} # [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}} # [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}} # [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}} # [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}} # [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}} # [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}} # [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}} # [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}} # [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}} # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}} # [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}} # [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}} # [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}} # [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}} # [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}} # [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}} # [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}} # [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}} # [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}} # [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}} # [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}} # [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}} # [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}} # [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}} # [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}} # [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}} # [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}} # [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}} # [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}} # [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}} # [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}} # [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}} # [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}} # [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}} # [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}} # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}} # [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}} # [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}} # [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}} # [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}} # [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}} # [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}} # [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}} # [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}} # [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}} # [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}} # [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}} # [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}} ==Motivation and emotion== # [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}} # [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}} [[Category:Motivation and emotion/Book]] bqmhzrqvan9wet8r3h4a4nd4a7iu0ht 2814995 2814990 2026-06-10T07:35:12Z Jtneill 10242 /* Emotion */ + [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? 2814995 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}} # [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}} # [[/Attachment styles and relatedness motivation/]] - What is the relationship between attachment styles and pursuit of social connection? {{ME-By|User Name}} # [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}} # [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}} # [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}} # [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}} # [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}} # [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}} # [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}} # [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}} # [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}} # [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}} # [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}} # [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}} # [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}} # [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}} # [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}} # [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}} # [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}} # [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}} # [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}} # [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}} # [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}} # [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}} # [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}} # [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}} # [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}} # [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}} # [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}} # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}} # [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}} # [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}} # [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}} # [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}} # [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}} # [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}} # [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}} # [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}} # [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}} # [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}} # [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}} # [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}} # [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}} # [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}} # [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}} # [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}} # [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}} # [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}} # [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}} # [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}} # [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}} # [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}} # [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}} # [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}} ==Emotion== # [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}} # [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}} # [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}} # [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}} # [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}} # [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}} # [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}} # [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}} # [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}} # [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}} # [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}} # [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}} # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}} # [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}} # [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}} # [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}} # [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}} # [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}} # [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}} # [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}} # [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}} # [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}} # [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}} # [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}} # [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}} # [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}} # [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}} # [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}} # [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}} # [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}} # [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}} # [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}} # [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}} # [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}} # [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}} # [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}} # [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}} # [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}} # [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}} # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}} # [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}} # [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}} # [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}} # [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}} # [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}} # [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}} # [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}} # [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}} # [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}} # [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}} # [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}} # [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}} ==Motivation and emotion== # [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}} # [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}} [[Category:Motivation and emotion/Book]] 6sufa4zd255de91agdfxeputfsqlb5i 2814997 2814995 2026-06-10T07:37:20Z Jtneill 10242 /* Motivation */ + [[/Getting started/]] - Why is getting started hard and how can this be overcome? (from 2024) 2814997 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}} # [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}} # [[/Attachment styles and relatedness motivation/]] - What is the relationship between attachment styles and pursuit of social connection? {{ME-By|User Name}} # [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}} # [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}} # [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}} # [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}} # [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}} # [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}} # [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}} # [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}} # [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}} # [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}} # [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}} # [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}} # [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}} # [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}} # [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}} # [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}} # [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}} # [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}} # [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}} # [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}} # [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}} # [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}} # [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}} # [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}} # [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}} # [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}} # [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}} # [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}} # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}} # [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}} # [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}} # [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}} # [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}} # [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}} # [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}} # [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}} # [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}} # [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}} # [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}} # [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}} # [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}} # [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}} # [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}} # [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}} # [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}} # [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}} # [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}} # [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}} # [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}} # [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}} # [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}} # [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}} # [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}} ==Emotion== # [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}} # [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}} # [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}} # [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}} # [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}} # [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}} # [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}} # [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}} # [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}} # [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}} # [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}} # [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}} # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}} # [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}} # [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}} # [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}} # [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}} # [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}} # [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}} # [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}} # [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}} # [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}} # [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}} # [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}} # [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}} # [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}} # [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}} # [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}} # [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}} # [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}} # [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}} # [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}} # [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}} # [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}} # [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}} # [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}} # [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}} # [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}} # [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}} # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}} # [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}} # [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}} # [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}} # [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}} # [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}} # [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}} # [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}} # [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}} # [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}} # [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}} # [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}} # [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}} ==Motivation and emotion== # [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}} # [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}} [[Category:Motivation and emotion/Book]] 5rwbfwhnp58snli6ahe8popxy37z4j8 2815001 2814997 2026-06-10T07:40:30Z Jtneill 10242 /* Motivation */ + [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? 2815001 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}} # [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}} # [[/Attachment styles and relatedness motivation/]] - What is the relationship between attachment styles and pursuit of social connection? {{ME-By|User Name}} # [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}} # [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}} # [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}} # [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}} # [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}} # [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}} # [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}} # [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}} # [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}} # [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}} # [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}} # [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}} # [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}} # [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}} # [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}} # [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}} # [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}} # [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}} # [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}} # [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}} # [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}} # [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}} # [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}} # [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}} # [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}} # [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}} # [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}} # [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}} # [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}} # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}} # [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}} # [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}} # [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}} # [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}} # [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}} # [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}} # [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}} # [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}} # [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}} # [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}} # [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}} # [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}} # [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}} # [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}} # [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}} # [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}} # [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}} # [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}} # [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}} # [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}} # [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}} # [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}} # [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}} # [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}} ==Emotion== # [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}} # [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}} # [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}} # [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}} # [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}} # [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}} # [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}} # [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}} # [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}} # [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}} # [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}} # [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}} # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}} # [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}} # [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}} # [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}} # [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}} # [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}} # [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}} # [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}} # [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}} # [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}} # [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}} # [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}} # [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}} # [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}} # [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}} # [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}} # [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}} # [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}} # [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}} # [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}} # [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}} # [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}} # [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}} # [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}} # [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}} # [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}} # [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}} # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}} # [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}} # [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}} # [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}} # [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}} # [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}} # [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}} # [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}} # [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}} # [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}} # [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}} # [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}} # [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}} ==Motivation and emotion== # [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}} # [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}} [[Category:Motivation and emotion/Book]] 95rqqii3j3uzt6wvtyf6batg6chjl5q 2815005 2815001 2026-06-10T07:43:42Z Jtneill 10242 /* Motivation */ + [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? 2815005 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}} # [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}} # [[/Attachment styles and relatedness motivation/]] - What is the relationship between attachment styles and pursuit of social connection? {{ME-By|User Name}} # [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}} # [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}} # [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}} # [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}} # [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}} # [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}} # [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}} # [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}} # [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}} # [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}} # [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}} # [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}} # [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}} # [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}} # [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}} # [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}} # [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}} # [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}} # [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}} # [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}} # [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}} # [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}} # [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}} # [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}} # [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}} # [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}} # [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}} # [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}} # [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}} # [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}} # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}} # [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}} # [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}} # [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}} # [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}} # [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}} # [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}} # [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}} # [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}} # [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}} # [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}} # [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}} # [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}} # [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}} # [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}} # [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}} # [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}} # [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}} # [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}} # [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}} # [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}} # [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}} # [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}} # [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}} # [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}} ==Emotion== # [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}} # [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}} # [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}} # [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}} # [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}} # [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}} # [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}} # [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}} # [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}} # [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}} # [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}} # [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}} # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}} # [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}} # [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}} # [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}} # [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}} # [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}} # [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}} # [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}} # [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}} # [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}} # [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}} # [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}} # [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}} # [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}} # [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}} # [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}} # [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}} # [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}} # [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}} # [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}} # [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}} # [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}} # [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}} # [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}} # [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}} # [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}} # [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}} # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}} # [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}} # [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}} # [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}} # [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}} # [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}} # [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}} # [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}} # [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}} # [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}} # [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}} # [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}} # [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}} ==Motivation and emotion== # [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}} # [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}} [[Category:Motivation and emotion/Book]] 0u6yydjcmh8xzmc6xkijez6wrt43mov 2815008 2815005 2026-06-10T07:45:53Z Jtneill 10242 /* Motivation */ + {{ME-By|User Name}} # 2815008 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}} # [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}} # [[/Attachment styles and relatedness motivation/]] - What is the relationship between attachment styles and pursuit of social connection? {{ME-By|User Name}} # [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}} # [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}} # [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}} # [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}} # [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}} # [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}} # [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}} # [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}} # [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}} # [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}} # [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}} # [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}} # [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}} # [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}} # [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}} # [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}} # [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}} # [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}} # [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}} # [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}} # [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}} # [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}} # [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}} # [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}} # [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}} # [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}} # [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}} # [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}} # [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}} # [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}} # [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}} # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}} # [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}} # [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}} # [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}} # [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}} # [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}} # [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}} # [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}} # [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}} # [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}} # [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}} # [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}} # [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}} # [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}} # [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}} # [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}} # [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}} # [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}} # [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}} # [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}} # [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}} # [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}} # [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}} # [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}} # [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}} ==Emotion== # [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}} # [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}} # [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}} # [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}} # [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}} # [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}} # [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}} # [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}} # [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}} # [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}} # [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}} # [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}} # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}} # [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}} # [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}} # [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}} # [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}} # [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}} # [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}} # [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}} # [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}} # [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}} # [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}} # [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}} # [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}} # [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}} # [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}} # [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}} # [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}} # [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}} # [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}} # [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}} # [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}} # [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}} # [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}} # [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}} # [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}} # [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}} # [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}} # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}} # [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}} # [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}} # [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}} # [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}} # [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}} # [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}} # [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}} # [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}} # [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}} # [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}} # [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}} # [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}} ==Motivation and emotion== # [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}} # [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}} [[Category:Motivation and emotion/Book]] 8ztxjan6c7brnh6ai7oa2lpx47ogng3 2815015 2815008 2026-06-10T09:34:50Z Jtneill 10242 /* Emotion */ + [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? 2815015 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}} # [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}} # [[/Attachment styles and relatedness motivation/]] - What is the relationship between attachment styles and pursuit of social connection? {{ME-By|User Name}} # [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}} # [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}} # [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}} # [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}} # [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}} # [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}} # [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}} # [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}} # [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}} # [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}} # [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}} # [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}} # [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}} # [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}} # [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}} # [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}} # [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}} # [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}} # [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}} # [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}} # [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}} # [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}} # [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}} # [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}} # [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}} # [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}} # [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}} # [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}} # [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}} # [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}} # [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}} # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}} # [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}} # [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}} # [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}} # [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}} # [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}} # [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}} # [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}} # [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}} # [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}} # [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}} # [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}} # [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}} # [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}} # [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}} # [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}} # [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}} # [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}} # [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}} # [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}} # [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}} # [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}} # [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}} # [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}} # [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}} ==Emotion== # [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}} # [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}} # [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}} # [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}} # [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}} # [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}} # [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}} # [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}} # [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}} # [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}} # [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}} # [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}} # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}} # [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}} # [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}} # [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}} # [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}} # [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}} # [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}} # [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}} # [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}} # [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}} # [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}} # [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}} # [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}} # [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}} # [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}} # [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}} # [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}} # [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}} # [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}} # [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}} # [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}} # [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}} # [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}} # [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}} # [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}} # [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}} # [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}} # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}} # [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}} # [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}} # [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}} # [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}} # [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}} # [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}} # [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}} # [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}} # [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}} # [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}} # [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}} # [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}} # [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}} ==Motivation and emotion== # [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}} # [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}} [[Category:Motivation and emotion/Book]] o65g1h25ygupwsxfundcdc3sm0dnydx 2815016 2815015 2026-06-10T09:39:59Z Jtneill 10242 /* Emotion */ [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? 2815016 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}} # [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}} # [[/Attachment styles and relatedness motivation/]] - What is the relationship between attachment styles and pursuit of social connection? {{ME-By|User Name}} # [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}} # [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}} # [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}} # [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}} # [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}} # [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}} # [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}} # [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}} # [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}} # [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}} # [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}} # [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}} # [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}} # [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}} # [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}} # [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}} # [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}} # [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}} # [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}} # [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}} # [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}} # [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}} # [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}} # [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}} # [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}} # [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}} # [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}} # [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}} # [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}} # [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}} # [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}} # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}} # [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}} # [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}} # [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}} # [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}} # [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}} # [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}} # [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}} # [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}} # [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}} # [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}} # [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}} # [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}} # [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}} # [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}} # [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}} # [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}} # [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}} # [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}} # [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}} # [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}} # [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}} # [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}} # [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}} # [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}} ==Emotion== # [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}} # [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}} # [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}} # [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}} # [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}} # [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}} # [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}} # [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}} # [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}} # [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}} # [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}} # [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}} # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}} # [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}} # [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}} # [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}} # [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}} # [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}} # [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}} # [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}} # [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}} # [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}} # [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}} # [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}} # [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}} # [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}} # [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}} # [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}} # [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}} # [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}} # [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}} # [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}} # [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}} # [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}} # [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}} # [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}} # [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}} # [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}} # [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}} # [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}} # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}} # [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}} # [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}} # [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}} # [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}} # [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}} # [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}} # [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}} # [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}} # [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}} # [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}} # [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}} # [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}} # [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}} ==Motivation and emotion== # [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}} # [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}} [[Category:Motivation and emotion/Book]] 4yt8199medqv4l7o21b2jtbyl10cvie 2815017 2815016 2026-06-10T09:43:02Z Jtneill 10242 /* Motivation */ [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? 2815017 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}} # [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}} # [[/Attachment styles and relatedness motivation/]] - What is the relationship between attachment styles and pursuit of social connection? {{ME-By|User Name}} # [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}} # [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}} # [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}} # [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}} # [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}} # [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}} # [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}} # [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}} # [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}} # [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}} # [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}} # [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}} # [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}} # [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}} # [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}} # [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}} # [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}} # [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}} # [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}} # [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}} # [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}} # [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}} # [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}} # [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}} # [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}} # [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}} # [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}} # [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}} # [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}} # [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}} # [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}} # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}} # [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}} # [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}} # [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}} # [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}} # [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}} # [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}} # [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}} # [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}} # [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}} # [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}} # [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}} # [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}} # [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}} # [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}} # [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}} # [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}} # [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}} # [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}} # [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}} # [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}} # [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}} # [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}} # [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}} # [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}} # [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}} ==Emotion== # [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}} # [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}} # [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}} # [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}} # [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}} # [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}} # [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}} # [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}} # [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}} # [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}} # [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}} # [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}} # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}} # [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}} # [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}} # [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}} # [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}} # [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}} # [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}} # [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}} # [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}} # [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}} # [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}} # [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}} # [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}} # [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}} # [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}} # [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}} # [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}} # [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}} # [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}} # [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}} # [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}} # [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}} # [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}} # [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}} # [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}} # [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}} # [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}} # [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}} # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}} # [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}} # [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}} # [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}} # [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}} # [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}} # [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}} # [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}} # [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}} # [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}} # [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}} # [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}} # [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}} # [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}} ==Motivation and emotion== # [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}} # [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}} [[Category:Motivation and emotion/Book]] 6lg7dk6vmkd77r3cfixb1cuu9844b7o 2815018 2815017 2026-06-10T09:48:32Z Jtneill 10242 /* Motivation and emotion */ + [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? 2815018 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}} # [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}} # [[/Attachment styles and relatedness motivation/]] - What is the relationship between attachment styles and pursuit of social connection? {{ME-By|User Name}} # [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}} # [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}} # [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}} # [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}} # [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}} # [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}} # [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}} # [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}} # [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}} # [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}} # [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}} # [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}} # [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}} # [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}} # [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}} # [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}} # [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}} # [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}} # [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}} # [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}} # [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}} # [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}} # [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}} # [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}} # [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}} # [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}} # [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}} # [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}} # [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}} # [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}} # [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}} # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}} # [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}} # [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}} # [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}} # [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}} # [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}} # [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}} # [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}} # [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}} # [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}} # [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}} # [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}} # [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}} # [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}} # [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}} # [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}} # [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}} # [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}} # [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}} # [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}} # [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}} # [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}} # [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}} # [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}} # [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}} # [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}} ==Emotion== # [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}} # [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}} # [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}} # [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}} # [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}} # [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}} # [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}} # [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}} # [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}} # [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}} # [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}} # [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}} # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}} # [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}} # [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}} # [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}} # [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}} # [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}} # [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}} # [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}} # [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}} # [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}} # [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}} # [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}} # [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}} # [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}} # [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}} # [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}} # [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}} # [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}} # [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}} # [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}} # [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}} # [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}} # [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}} # [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}} # [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}} # [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}} # [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}} # [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}} # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}} # [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}} # [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}} # [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}} # [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}} # [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}} # [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}} # [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}} # [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}} # [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}} # [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}} # [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}} # [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}} # [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}} ==Motivation and emotion== # [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}} # [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}} # [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}} [[Category:Motivation and emotion/Book]] j3lfqg9w6ezwk8qbhm0faeblpmfbkwh 2815020 2815018 2026-06-10T10:07:12Z Jtneill 10242 /* Emotion */ + [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? 2815020 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}} # [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}} # [[/Attachment styles and relatedness motivation/]] - What is the relationship between attachment styles and pursuit of social connection? {{ME-By|User Name}} # [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}} # [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}} # [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}} # [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}} # [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}} # [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}} # [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}} # [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}} # [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}} # [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}} # [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}} # [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}} # [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}} # [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}} # [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}} # [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}} # [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}} # [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}} # [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}} # [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}} # [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}} # [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}} # [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}} # [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}} # [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}} # [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}} # [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}} # [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}} # [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}} # [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}} # [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}} # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}} # [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}} # [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}} # [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}} # [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}} # [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}} # [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}} # [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}} # [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}} # [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}} # [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}} # [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}} # [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}} # [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}} # [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}} # [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}} # [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}} # [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}} # [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}} # [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}} # [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}} # [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}} # [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}} # [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}} # [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}} # [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}} ==Emotion== # [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}} # [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}} # [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}} # [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}} # [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}} # [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}} # [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}} # [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}} # [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}} # [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}} # [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}} # [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}} # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}} # [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}} # [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}} # [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}} # [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}} # [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}} # [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}} # [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}} # [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}} # [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}} # [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}} # [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}} # [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}} # [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}} # [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}} # [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}} # [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}} # [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}} # [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}} # [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}} # [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}} # [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}} # [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}} # [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}} # [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}} # [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}} # [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}} # [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}} # [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}} # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}} # [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}} # [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}} # [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}} # [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}} # [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}} # [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}} # [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}} # [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}} # [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}} # [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}} # [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}} # [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}} # [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}} ==Motivation and emotion== # [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}} # [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}} # [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}} [[Category:Motivation and emotion/Book]] kjtj4fwxluec375v1mlfzyvn6794vxv 2815021 2815020 2026-06-10T10:10:30Z Jtneill 10242 /* Motivation */ + [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians 2815021 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}} # [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}} # [[/Attachment styles and relatedness motivation/]] - What is the relationship between attachment styles and pursuit of social connection? {{ME-By|User Name}} # [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}} # [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}} # [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}} # [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}} # [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}} # [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}} # [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}} # [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}} # [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}} # [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}} # [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}} # [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}} # [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}} # [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}} # [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}} # [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}} # [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}} # [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}} # [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}} # [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}} # [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}} # [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}} # [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}} # [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}} # [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}} # [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}} # [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}} # [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}} # [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}} # [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}} # [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}} # [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}} # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}} # [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}} # [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}} # [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}} # [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}} # [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}} # [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}} # [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}} # [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}} # [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}} # [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}} # [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}} # [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}} # [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}} # [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}} # [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}} # [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}} # [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}} # [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}} # [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}} # [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}} # [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}} # [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}} # [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}} # [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}} # [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}} ==Emotion== # [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}} # [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}} # [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}} # [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}} # [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}} # [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}} # [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}} # [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}} # [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}} # [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}} # [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}} # [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}} # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}} # [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}} # [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}} # [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}} # [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}} # [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}} # [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}} # [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}} # [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}} # [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}} # [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}} # [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}} # [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}} # [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}} # [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}} # [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}} # [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}} # [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}} # [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}} # [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}} # [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}} # [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}} # [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}} # [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}} # [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}} # [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}} # [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}} # [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}} # [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}} # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}} # [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}} # [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}} # [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}} # [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}} # [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}} # [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}} # [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}} # [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}} # [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}} # [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}} # [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}} # [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}} # [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}} ==Motivation and emotion== # [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}} # [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}} # [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}} [[Category:Motivation and emotion/Book]] 071m099kijf7s923p7a0th7fvunwetc 2815022 2815021 2026-06-10T10:15:57Z Jtneill 10242 /* Emotion */ [[/Emotional effects of incaraceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians? 2815022 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}} # [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}} # [[/Attachment styles and relatedness motivation/]] - What is the relationship between attachment styles and pursuit of social connection? {{ME-By|User Name}} # [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}} # [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}} # [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}} # [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}} # [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}} # [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}} # [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}} # [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}} # [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}} # [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}} # [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}} # [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}} # [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}} # [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}} # [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}} # [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}} # [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}} # [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}} # [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}} # [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}} # [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}} # [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}} # [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}} # [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}} # [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}} # [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}} # [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}} # [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}} # [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}} # [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}} # [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}} # [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}} # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}} # [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}} # [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}} # [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}} # [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}} # [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}} # [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}} # [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}} # [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}} # [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}} # [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}} # [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}} # [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}} # [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}} # [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}} # [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}} # [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}} # [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}} # [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}} # [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}} # [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}} # [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}} # [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}} # [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}} # [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}} # [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}} ==Emotion== # [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}} # [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}} # [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}} # [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}} # [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}} # [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}} # [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}} # [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}} # [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}} # [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}} # [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}} # [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}} # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}} # [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}} # [[/Emotional effects of incaraceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}} # [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}} # [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}} # [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}} # [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}} # [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}} # [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}} # [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}} # [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}} # [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}} # [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}} # [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}} # [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}} # [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}} # [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}} # [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}} # [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}} # [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}} # [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}} # [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}} # [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}} # [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}} # [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}} # [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}} # [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}} # [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}} # [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}} # [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}} # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}} # [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}} # [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}} # [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}} # [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}} # [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}} # [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}} # [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}} # [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}} # [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}} # [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}} # [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}} # [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}} # [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}} ==Motivation and emotion== # [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}} # [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}} # [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}} [[Category:Motivation and emotion/Book]] q941n1kpia5jaards40bsunx089uuy3 2815023 2815022 2026-06-10T10:16:23Z Jtneill 10242 /* Emotion */ 2815023 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}} # [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}} # [[/Attachment styles and relatedness motivation/]] - What is the relationship between attachment styles and pursuit of social connection? {{ME-By|User Name}} # [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}} # [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}} # [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}} # [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}} # [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}} # [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}} # [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}} # [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}} # [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}} # [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}} # [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}} # [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}} # [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}} # [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}} # [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}} # [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}} # [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}} # [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}} # [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}} # [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}} # [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}} # [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}} # [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}} # [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}} # [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}} # [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}} # [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}} # [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}} # [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}} # [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}} # [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}} # [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}} # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}} # [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}} # [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}} # [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}} # [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}} # [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}} # [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}} # [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}} # [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}} # [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}} # [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}} # [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}} # [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}} # [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}} # [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}} # [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}} # [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}} # [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}} # [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}} # [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}} # [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}} # [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}} # [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}} # [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}} # [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}} # [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}} ==Emotion== # [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}} # [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}} # [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}} # [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}} # [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}} # [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}} # [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}} # [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}} # [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}} # [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}} # [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}} # [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}} # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}} # [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}} # [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}} # [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}} # [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}} # [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}} # [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}} # [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}} # [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}} # [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}} # [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}} # [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}} # [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}} # [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}} # [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}} # [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}} # [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}} # [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}} # [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}} # [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}} # [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}} # [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}} # [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}} # [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}} # [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}} # [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}} # [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}} # [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}} # [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}} # [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}} # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}} # [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}} # [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}} # [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}} # [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}} # [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}} # [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}} # [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}} # [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}} # [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}} # [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}} # [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}} # [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}} # [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}} ==Motivation and emotion== # [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}} # [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}} # [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}} [[Category:Motivation and emotion/Book]] b6ec2qa8g0sq006ncuj4gbtry3hl1ko 2815024 2815023 2026-06-10T10:17:43Z Jtneill 10242 /* Motivation */ [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians? 2815024 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}} # [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}} # [[/Attachment styles and relatedness motivation/]] - What is the relationship between attachment styles and pursuit of social connection? {{ME-By|User Name}} # [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}} # [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}} # [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}} # [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}} # [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}} # [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}} # [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}} # [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}} # [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}} # [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}} # [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}} # [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}} # [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}} # [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}} # [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}} # [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}} # [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}} # [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}} # [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}} # [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}} # [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}} # [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}} # [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}} # [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}} # [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}} # [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}} # [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}} # [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}} # [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}} # [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}} # [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}} # [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}} # [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}} # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}} # [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}} # [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}} # [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}} # [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}} # [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}} # [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}} # [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}} # [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}} # [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}} # [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}} # [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}} # [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}} # [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}} # [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}} # [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}} # [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}} # [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}} # [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}} # [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}} # [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}} # [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}} # [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}} # [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}} # [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}} # [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}} ==Emotion== # [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}} # [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}} # [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}} # [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}} # [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}} # [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}} # [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}} # [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}} # [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}} # [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}} # [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}} # [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}} # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}} # [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}} # [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}} # [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}} # [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}} # [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}} # [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}} # [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}} # [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}} # [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}} # [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}} # [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}} # [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}} # [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}} # [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}} # [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}} # [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}} # [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}} # [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}} # [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}} # [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}} # [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}} # [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}} # [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}} # [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}} # [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}} # [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}} # [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}} # [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}} # [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}} # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}} # [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}} # [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}} # [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}} # [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}} # [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}} # [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}} # [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}} # [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}} # [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}} # [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}} # [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}} # [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}} # [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}} ==Motivation and emotion== # [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}} # [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}} # [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}} [[Category:Motivation and emotion/Book]] iwjpl8c6z4r6qpeo78bj9vehik5wmru 2815025 2815024 2026-06-10T10:24:49Z Jtneill 10242 /* Motivation and emotion */ [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? 2815025 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}} # [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}} # [[/Attachment styles and relatedness motivation/]] - What is the relationship between attachment styles and pursuit of social connection? {{ME-By|User Name}} # [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}} # [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}} # [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}} # [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}} # [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}} # [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}} # [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}} # [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}} # [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}} # [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}} # [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}} # [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}} # [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}} # [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}} # [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}} # [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}} # [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}} # [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}} # [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}} # [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}} # [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}} # [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}} # [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}} # [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}} # [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}} # [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}} # [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}} # [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}} # [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}} # [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}} # [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}} # [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}} # [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}} # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}} # [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}} # [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}} # [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}} # [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}} # [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}} # [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}} # [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}} # [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}} # [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}} # [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}} # [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}} # [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}} # [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}} # [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}} # [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}} # [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}} # [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}} # [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}} # [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}} # [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}} # [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}} # [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}} # [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}} # [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}} # [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}} ==Emotion== # [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}} # [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}} # [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}} # [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}} # [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}} # [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}} # [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}} # [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}} # [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}} # [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}} # [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}} # [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}} # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}} # [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}} # [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}} # [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}} # [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}} # [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}} # [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}} # [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}} # [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}} # [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}} # [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}} # [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}} # [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}} # [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}} # [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}} # [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}} # [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}} # [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}} # [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}} # [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}} # [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}} # [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}} # [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}} # [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}} # [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}} # [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}} # [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}} # [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}} # [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}} # [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}} # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}} # [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}} # [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}} # [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}} # [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}} # [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}} # [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}} # [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}} # [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}} # [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}} # [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}} # [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}} # [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}} # [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}} ==Motivation and emotion== # [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}} # [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}} # [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}} # [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}} [[Category:Motivation and emotion/Book]] ja4wrf0y8v52p7f1bxdq4kutdej2kxp 2815027 2815025 2026-06-10T10:45:20Z Jtneill 10242 /* Motivation and emotion */ + [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? 2815027 wikitext text/x-wiki {{/Banner}} ==Motivation== # [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}} # [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}} # [[/Attachment styles and relatedness motivation/]] - What is the relationship between attachment styles and pursuit of social connection? {{ME-By|User Name}} # [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}} # [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}} # [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}} # [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}} # [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}} # [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}} # [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}} # [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}} # [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}} # [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}} # [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}} # [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}} # [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}} # [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}} # [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}} # [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}} # [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}} # [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}} # [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}} # [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}} # [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}} # [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}} # [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}} # [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}} # [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}} # [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}} # [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}} # [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}} # [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}} # [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}} # [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}} # [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}} # [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}} # [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}} # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}} # [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}} # [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}} # [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}} # [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}} # [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}} # [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}} # [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}} # [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}} # [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}} # [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}} # [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}} # [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}} # [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}} # [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}} # [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}} # [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}} # [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}} # [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}} # [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}} # [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}} # [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}} # [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}} # [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}} # [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}} # [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}} # [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}} # [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}} ==Emotion== # [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}} # [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}} # [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}} # [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}} # [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}} # [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}} # [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}} # [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}} # [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}} # [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}} # [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}} # [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}} # [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}} # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}} # [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}} # [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}} # [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}} # [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}} # [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}} # [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}} # [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}} # [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}} # [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}} # [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}} # [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}} # [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}} # [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}} # [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}} # [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}} # [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}} # [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}} # [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}} # [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}} # [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}} # [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}} # [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}} # [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}} # [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}} # [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}} # [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}} # [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}} # [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}} # [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}} # [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}} # [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}} # [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}} # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}} # [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}} # [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}} # [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}} # [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}} # [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}} # [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}} # [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}} # [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}} # [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}} # [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}} # [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}} # [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}} # [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}} # [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}} # [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}} ==Motivation and emotion== # [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}} # [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}} # [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}} # [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}} # [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}} # [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}} # [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}} [[Category:Motivation and emotion/Book]] nmvez0j06o9zam5l7co4pc5c3ttol1n 0 324264 2814966 2759822 2026-06-10T03:45:37Z Dian (WMID) 2933238 augustin > agustin 2814966 wikitext text/x-wiki {{Article info | first1 = Dian | last1 = Agustin | orcid1 = | affiliation1 = Wikimedia Indonesia | submitted= 2025-06-24 | correspondence1 = {{nospam|dian.agustin|wikimedia.or.id}} | journal = WikiJournal of Humanities | w1 = | license = | abstract = }} '''Abstract''' Free online courses are essential for everyone who wants to learn new digital skills and knowledge, including how Wikipedia works and how to edit Wikipedia articles, since the core value of free online courses lies in its ability to provide accessibility to educational resources regardless of geographic location, social and economic status, or other traditional barriers. In our case, Online Courses can also reach wider participants from various regions in Indonesia. Since the popularity of Online Courses (WikiLatih Daring) in Indonesia has grown significantly, we have implemented several enhancements to maintain participant attention and engagement in online environments by producing shorter videos (2-6 minutes) and improving platform compatibility for seamless mobile and computer access. We have also collaborated with Learnext to offer free and self-paced online courses that can be enrolled and accessed anytime, anywhere, by everyone. Currently, we are focusing on improving content quality and learning experiences on the LMS and designing advanced training programs (WikiLatih Mahir) for participants. We also have plans to extend this initiative to other Wikimedia projects and beyond. olrpd3ekhnztnn8uy4jghjszvvqkyjl 102 324936 2814857 2807099 2026-06-09T13:20:40Z Projet PEP 3002502 2814857 wikitext text/x-wiki <!-- BANNER ACROSS TOP OF PAGE --> <!---------------------- TABS -------------------------> {{Portal:Wikilang/start tab}} Welcome to the portal "Plurilingual education". It is a collection of free resources dedicated to plurilingual education to be used for pre-service and in-service training of language teachers. It has been created by the European project PEP, which is co-funded by the European Commission within the Erasmus+ programme (Promoting plurilingual education, 2023-1-FR01-KA220-HED-000160820). {{end tab}} <!---------------------- WELCOME -------------------------> {| cellspacing="0" cellpadding="0" | colspan="2" | |- style="vertical-align: top;" | style="padding-right: 1.2em; width: 50%;" | <!---------------------- FEATURED CONTENT -----------------------> {{Frame alt | color = 006699 | title = Featured resources | content = Following "lessons" are available. More are coming soon! * [[Awakening to languages]] * [[CLIL (Content and Language Integrated Learning)]] * [[Decolonial perspective in plurilingual education]] * [[Plurilingual education and digital technologies|Digital technologies in plurilingual 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 & parental involvement]] * [[Multilingual awareness - Language awareness - Metacompetencies]] * [[Multulingual turn]] * [[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]] * [[Pedagogy of variation]] * [[Plurilingual and inter/transcultural competence]] * [[Plurilingualism and plurilingual education in the past]] * [[Telecollaboration and plurilingualism]] * [[Tertiary language teaching]] * [[Terminology and plurilingual education]] * [[Teachers’ beliefs and plurilingualism]] * [[Theories and models of plurilingualism]] * [[Translanguaging]] }} <!------------------------ LANGUAGES ---------------------------> {{Frame alt | color = 006699 | title = Library | content = Useful ressources to read * 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 ] }} | style="padding-left: 1.2em; width: 50%;" | <!-------------------------- NEWS ---------------------------> {{Frame alt | color = 990000 | title = News | content = Selected worldwide news about plurilingual education: * '''[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). }}<!------------------------ EXTERNAL RESOURCES ------------------------->{{Frame alt | title = External ressources | content = Projects and materials to "teach" plurilingual education *[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] }}<!------------------------ OTHER --------------------------->{{Frame alt | color = 339966 | title = Other resources in the Wikiversity | content = '''Learning Groups''' * [[Portal:Foreign Language Learning|Foreign Language Learning]] * [[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] }} [[Category:Wikilang|*]] [[Category:Foreign Language Learning|*]] [[fr:Faculté:Wikilangues]] <!-- {{Frame alt | color = 990000 | title = Click on a continent | content = <div>{{Wikilang map}}</div><br>Click on a continent to get to a portal of languages of this continent. }} |}--> |} nytwbggywa1hrx4d9vlhl0dzrxfzyba 2814877 2814857 2026-06-09T14:35:15Z Projet PEP 3002502 2814877 wikitext text/x-wiki <!-- BANNER ACROSS TOP OF PAGE --> <!---------------------- TABS -------------------------> {{Portal:Wikilang/start tab}} Welcome to the portal "Plurilingual education". It is a collection of free resources dedicated to plurilingual education to be used for pre-service and in-service training of language teachers. It has been created by the European project PEP, which is co-funded by the European Commission within the Erasmus+ programme (Promoting plurilingual education, 2023-1-FR01-KA220-HED-000160820). {{end tab}} <!---------------------- WELCOME -------------------------> {| cellspacing="0" cellpadding="0" | colspan="2" | |- style="vertical-align: top;" | style="padding-right: 1.2em; width: 50%;" | <!---------------------- FEATURED CONTENT -----------------------> {{Frame alt | color = 006699 | title = Featured resources | content = Following "lessons" are available. More are coming soon! * [[Awakening to languages]] * [[CLIL (Content and Language Integrated Learning)]] * [[Decolonial perspective in plurilingual education]] * [[Plurilingual education and digital technologies|Digital technologies in plurilingual 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 & 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]] * [[Pedagogy of variation]] * [[Plurilingual and inter/transcultural competence]] * [[Plurilingualism and plurilingual education in the past]] * [[Telecollaboration and plurilingualism]] * [[Tertiary language teaching]] * [[Terminology and plurilingual education]] * [[Teachers’ beliefs and plurilingualism]] * [[Theories and models of plurilingualism]] * [[Translanguaging]] }} <!------------------------ LANGUAGES ---------------------------> {{Frame alt | color = 006699 | title = Library | content = Useful ressources to read * 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 ] }} | style="padding-left: 1.2em; width: 50%;" | <!-------------------------- NEWS ---------------------------> {{Frame alt | color = 990000 | title = News | content = Selected worldwide news about plurilingual education: * '''[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). }}<!------------------------ EXTERNAL RESOURCES ------------------------->{{Frame alt | title = External ressources | content = Projects and materials to "teach" plurilingual education *[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] }}<!------------------------ OTHER --------------------------->{{Frame alt | color = 339966 | title = Other resources in the Wikiversity | content = '''Learning Groups''' * [[Portal:Foreign Language Learning|Foreign Language Learning]] * [[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] }} [[Category:Wikilang|*]] [[Category:Foreign Language Learning|*]] [[fr:Faculté:Wikilangues]] <!-- {{Frame alt | color = 990000 | title = Click on a continent | content = <div>{{Wikilang map}}</div><br>Click on a continent to get to a portal of languages of this continent. }} |}--> |} evxtdh6dwgpb0aebhvc8q320scvfqdz 2 326765 2814910 2814714 2026-06-09T19:43:20Z Dc.samizdat 2856930 /* Finally the 120-cell */ 2814910 wikitext text/x-wiki = 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}} <blockquote>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.</blockquote> == 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="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "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]]."}} 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="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{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 5² 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & 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 <math>\phi^6</math> 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="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == Complementary chord pairs == The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 kinds of vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ⁻²√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1₀ − 16₀|ps=, but 14₀ and 16₀ are congruent opposing sections and 15₀ opposes itself; there are 29 non-point sections, denoted 1₀ − 29₀, in 15 opposing pairs.}} |- ! colspan="4" |Short chord ! colspan="2" |Great circle polygons !Rotation ! colspan="4" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° |{{radic|0}} |{{radic|0}} | rowspan="3" | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | |180° |{{radic|4}} |{{radic|4}} | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |0 |0 |{{radic|4}} |2 |2 |- style="background: palegreen;" | |0 |0 |<small><math>0\times\zeta</math></small> |2 |7.405~ |<small><math>2\phi^2\sqrt{2}\times\zeta</math></small> |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° |{{radic|0.𝜀}} |<small><math>\sqrt{1/2\phi^4}</math></small> | rowspan="3" |[[File:Irregular great hexagons of the 120-cell.png|100px]] | rowspan="3" |400 irregular great hexagons<br> (600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}} |164.5~° |{{radic|3.93~}} |<small><math>\sqrt{3\phi^2/2}</math></small> | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |1.982~ |<small><math>1 / \phi^2\sqrt{2}</math></small> |{{radic|3.927~}} |1.982~ |<small><math>\phi\sqrt{1.5}</math></small> |- style="background: palegreen;" | |0.270~ |1 |<small><math>1\times\zeta</math></small> |1.982~ |7.337~ |<small><math>\phi^3\sqrt{3}\times\zeta</math></small> |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° |{{radic|0.19~}} |<small><math>\sqrt{1/2\phi^2}</math></small> | rowspan="3" |[[File:25.2° × 154.8° chords great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:Triacontagon#Triacontagram|{30/13}]]<br>#13 |154.8~° |{{radic|3.81~}} | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |0.437~ |<small><math>1 / \phi\sqrt{2}</math></small> |{{radic|3.809~}} |1.952~ | |- style="background: gainsboro;" | |0.437~ |1.618~ |<small><math>\phi\times\zeta</math></small> |1.952~ |7.226~ |<small><math>\text{‡}\times\zeta</math></small> {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=<br>‡ For simplicity we omit the value of <math>a</math> whenever it is not mononomial in <math>\chi</math>, <math>\psi</math> and <math>\phi</math>.}} |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° |{{radic|0.𝚫}} |<small><math>\sqrt{1/\phi^2}</math></small> | rowspan="3" |[[File:Great decagon rectangle.png|100px]] | rowspan="3" |720 great decagons<br>(3600 great rectangles)<br>in 720 <big>𝜙</big> planes | rowspan="3" |5𝝅<br>[[600-cell#Decagons and pentadecagrams|{15/2}]]<br>#5 |144° |{{radic|3.𝚽}} |<small><math>\sqrt{2+\phi}</math></small> | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |0.618~ |<small><math>1 / \phi</math></small> |{{radic|3.618~}} |1.902~ |<small><math>1+1/{\phi^2}</math></small> |- style="background: yellow;" | |0.618~ |2.288~ |<small><math>\phi\sqrt{2}\times\zeta</math></small> |1.902~ |7.0425 |<small><math>\sqrt{2\phi^5\sqrt{5}}\times\zeta</math></small> |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° |{{radic|0.5}} |<small><math>\sqrt{1/2}</math></small> | rowspan="3" |[[File:√0.5 × √3.5 great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |138.6~° |{{radic|3.5}} |<small><math>\sqrt{7/2}</math></small> | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |0.707~ |<small><math>\sqrt{2}/2</math></small> |{{radic|3.5}} |1.871~ | |- style="background: gainsboro;" | |0.707~ |2.618~ |<small><math>\phi^2\times\zeta</math></small> |1.871~ |6.927~ |<small><math>\phi^2\sqrt{7}\times\zeta</math></small> |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° |{{radic|0.57~}} |<small><math>\sqrt{3/{2\phi^2}}</math></small> | rowspan="3" |[[File:Irregular great dodecagon.png|100px]] | rowspan="3" |200 irregular great dodecagons<br>(600 great rectangles)<br>in 200 △ planes | rowspan="3" | |135.5~° |{{radic|3.43~}} |<small><math>\sqrt{\phi^4/2}</math></small> | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |0.757~ |<small><math>\sqrt{3} / \phi\sqrt{2}</math></small> |{{radic|3.427~}} |1.851~ |<small><math>\phi^2 / \sqrt{2}</math></small> |- style="background: palegreen;" | |0.757~ |2.803~ |<small><math>\phi\sqrt{3}\times\zeta</math></small> |1.851~ |6.854~ |<small><math>\phi^4\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° |{{radic|0.69~}} |<small><math>\sqrt{\sqrt{5}/{2\phi}}</math></small> | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |130.9~° |{{radic|3.31~}} |<small><math>\sqrt{4 - \sqrt{5}/{2\phi}}</math></small> | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |0.831~ | |{{radic|3.309~}} |1.819~ | |- style="background: gainsboro;" | |0.831~ |3.078~ |<small><math>\sqrt{\phi^3\sqrt{5}}\times\zeta</math></small> |1.819~ |6.735~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° |{{radic|0.88~}} |<small><math>\sqrt{\psi/{2\phi}}</math></small> | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |124° |{{radic|3.12~}} |<small><math>\sqrt{4 - \psi/{2\phi}}</math></small> | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |0.939~ | |{{radic|3.118~}} |1.766~ | |- style="background: gainsboro;" | |0.939~ |3.477~ |<small><math>\sqrt{\psi\phi^3}\times\zeta</math></small> |1.766~ |6.538~ |<small><math>\sqrt{\chi\phi^5}\times\zeta</math></small>{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:<br> {{indent|4}}<math>11/\chi = \psi</math> <br> {{indent|4}}<math>\chi=(3\sqrt{5}+1)/2 \approx 3.854~</math> {{indent|4}}<math>\psi=(3\sqrt{5}-1)/2 \approx 2.854~</math>}} |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° |{{radic|1}} |<small><math>\sqrt{1}</math></small> | rowspan="3" |[[File:Great hexagon.png|100px]] | rowspan="3" |400 regular [[600-cell#Hexagons|great hexagons]]<br> (1200 great rectangles)<br>in 200 △ planes | rowspan="3" |4𝝅<br>[[600-cell#Hexagons and hexagrams|2{10/3}]]<br>#4 |120° |{{radic|3}} |<small><math>\sqrt{3}</math></small> | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |1 | |{{radic|3}} |1.732~ | |- style="background: palegreen;" | |1 |3.702~ |<small><math>\phi^2\sqrt{2}\times\zeta</math></small> |1.732~ |6.413~ |<small><math>\phi^2\sqrt{6}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° |{{radic|1.19~}} |<small><math>\sqrt{\chi/2\phi}</math></small> | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |113.9~° |{{radic|2.81~}} |<small><math>\sqrt{4 - \chi/2\phi}</math></small> | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |1.091~ | |{{radic|2.809~}} |1.676~ | |- style="background: gainsboro;" | |1.091~ |4.041~ |<small><math>\sqrt{\chi/\phi^3}\times\zeta</math></small> |1.676~ |6.205~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° |{{radic|1.31~}} |<small><math>\sqrt{\phi^2/2}</math></small> | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |110.2~° |{{radic|2.69~}} |<small><math>\sqrt{4 - \phi^2/2}</math></small> | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |1.144~ |<small><math>\phi/\sqrt{2}</math></small> |{{radic|2.691~}} |1.640~ | |- style="background: gainsboro;" | |1.144~ |4.236~ |<small><math>\phi^3\times\zeta</math></small> |1.640~ |6.074~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° |{{radic|1.618~}} |<small><math>\sqrt{3-\phi}</math></small> | rowspan="3" |[[File:Great pentagons rectangle.png|100px]] | rowspan="3" |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]<br>(3600 great rectangles)<br> in 720 <big>𝜙</big> planes | rowspan="3" |4𝝅<br>[[600-cell#Squares and octagrams|{24/5}]]<br>#9 |108° |{{radic|2.𝚽}} |<small><math>\sqrt{\phi^2}</math></small> | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |1.176~ |<small><math>\sqrt{\sqrt{5}/\phi}</math></small> |{{radic|2.618~}} |1.618~ |<small><math>\phi</math></small> |- style="background: yellow;" | |1.176~ |4.353~ |<small><math>\sqrt{2\phi^3\sqrt{5}}\times\zeta</math></small> |1.618~ |5.991~ |<small><math>\phi^3\sqrt{2}\times\zeta</math></small> |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° |{{radic|1.5}} |<small><math>\sqrt{3/2}</math></small> | rowspan="3" |[[File:Great 5-cell digons rectangle.png|100px]] | rowspan="3" |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]<br>(600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Pentagram|{5/2}]]<br>#8 |104.5~° |{{radic|2.5}} |<small><math>\sqrt{5/2}</math></small> | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |1.224~ | |{{radic|2.5}} |1.581~ | |- style="background: palegreen;" | |1.224~ |4.535~ |<small><math>\phi^2\sqrt{3}\times\zeta</math></small> |1.581~ |5.854~ |<small><math>\sqrt{5\phi^4}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° |{{radic|1.69~}} |<small><math>\sqrt{\tfrac{1}{4}(9-\sqrt{5})}</math></small> | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |98.9~° |{{radic|2.31~}} | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |1.300~ |<small><math>\tfrac{1}{2}\sqrt{9-\sqrt{5}}</math></small> |{{radic|2.309~}} |1.520~ | |- style="background: gainsboro;" | |1.300~ |4.815~ |<small><math>\text{‡}\times\zeta</math></small> |1.520~ |5.626~ |<small><math>\sqrt{\psi\phi^5}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° |{{radic|0.81~}} |<small><math>\sqrt{\tfrac{2\phi\sqrt{5}}{4}}</math></small> | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |95.5~° |{{radic|2.19~}} |<small><math>\sqrt{\tfrac{11-\sqrt{5}}{4}}</math></small> | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |1.345~ | |{{radic|2.191~}} |1.480~ | |- style="background: gainsboro;" | |1.345~ |4.980~ |<small><math>\sqrt{\phi^5\sqrt{5}}\times\zeta</math></small> |1.480~ |5.480~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |[[File:Great square rectangle.png|100px]] | rowspan="3" |4050 [[600-cell#Squares|great squares]]<br> in 4050 <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:30-gon#Triacontagram|{30/7}]]<br>#7 |90° |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |1.414~ | |{{radic|2}} |1.414~ | |- style="background: seashell;" | |1.414~ |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |1.414~ |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |} == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>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</math> The chord ratio <math>r_3=\sqrt{2}+1</math> 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: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. 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: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> 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: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ 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]] <small><math>\{3,3,4\}</math></small>. 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 <math>\sqrt{2}</math> 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: :<math>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</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> 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 <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, 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 <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> 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 <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> 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 <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> 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° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> 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 <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> 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 <small><math>\{4,3,3\}</math></small> 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]] <small><math>\{4,3,3\}</math></small>. 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 <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. 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: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> 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 <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> 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: :<math>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}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. The <math>r_1</math> and <math>r_5</math> 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 <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ 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 <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, 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. [[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]] The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> 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.]] [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of <math>r_7</math> edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference <math>6\pi</math> over <math>r_7</math> chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges. [[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of <math>r_9</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges. [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of <math>r_4</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_4</math> chords. Successive <math>r_4</math> chords are edges of different 24-cells. The rotational curve over each <math>r_4</math> chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges. In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 <math>\sqrt{1}</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == The 5-point (5-cell) 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. 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 <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> 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 & 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 & 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}} j4xaa4j4n0pjv1w3kufu9py4ivq8dzf 2814911 2814910 2026-06-09T19:46:31Z Dc.samizdat 2856930 /* Finally the 120-cell */ 2814911 wikitext text/x-wiki = 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}} <blockquote>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.</blockquote> == 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="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "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]]."}} 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="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{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 5² 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & 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 <math>\phi^6</math> 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="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == Complementary chord pairs == The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 kinds of vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ⁻²√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1₀ − 16₀|ps=, but 14₀ and 16₀ are congruent opposing sections and 15₀ opposes itself; there are 29 non-point sections, denoted 1₀ − 29₀, in 15 opposing pairs.}} |- ! colspan="4" |Short chord ! colspan="2" |Great circle polygons !Rotation ! colspan="4" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° |{{radic|0}} |{{radic|0}} | rowspan="3" | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | |180° |{{radic|4}} |{{radic|4}} | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |0 |0 |{{radic|4}} |2 |2 |- style="background: palegreen;" | |0 |0 |<small><math>0\times\zeta</math></small> |2 |7.405~ |<small><math>2\phi^2\sqrt{2}\times\zeta</math></small> |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° |{{radic|0.𝜀}} |<small><math>\sqrt{1/2\phi^4}</math></small> | rowspan="3" |[[File:Irregular great hexagons of the 120-cell.png|100px]] | rowspan="3" |400 irregular great hexagons<br> (600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}} |164.5~° |{{radic|3.93~}} |<small><math>\sqrt{3\phi^2/2}</math></small> | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |1.982~ |<small><math>1 / \phi^2\sqrt{2}</math></small> |{{radic|3.927~}} |1.982~ |<small><math>\phi\sqrt{1.5}</math></small> |- style="background: palegreen;" | |0.270~ |1 |<small><math>1\times\zeta</math></small> |1.982~ |7.337~ |<small><math>\phi^3\sqrt{3}\times\zeta</math></small> |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° |{{radic|0.19~}} |<small><math>\sqrt{1/2\phi^2}</math></small> | rowspan="3" |[[File:25.2° × 154.8° chords great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:Triacontagon#Triacontagram|{30/13}]]<br>#13 |154.8~° |{{radic|3.81~}} | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |0.437~ |<small><math>1 / \phi\sqrt{2}</math></small> |{{radic|3.809~}} |1.952~ | |- style="background: gainsboro;" | |0.437~ |1.618~ |<small><math>\phi\times\zeta</math></small> |1.952~ |7.226~ |<small><math>\text{‡}\times\zeta</math></small> {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=<br>‡ For simplicity we omit the value of <math>a</math> whenever it is not mononomial in <math>\chi</math>, <math>\psi</math> and <math>\phi</math>.}} |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° |{{radic|0.𝚫}} |<small><math>\sqrt{1/\phi^2}</math></small> | rowspan="3" |[[File:Great decagon rectangle.png|100px]] | rowspan="3" |720 great decagons<br>(3600 great rectangles)<br>in 720 <big>𝜙</big> planes | rowspan="3" |5𝝅<br>[[600-cell#Decagons and pentadecagrams|{15/2}]]<br>#5 |144° |{{radic|3.𝚽}} |<small><math>\sqrt{2+\phi}</math></small> | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |0.618~ |<small><math>1 / \phi</math></small> |{{radic|3.618~}} |1.902~ |<small><math>1+1/{\phi^2}</math></small> |- style="background: yellow;" | |0.618~ |2.288~ |<small><math>\phi\sqrt{2}\times\zeta</math></small> |1.902~ |7.0425 |<small><math>\sqrt{2\phi^5\sqrt{5}}\times\zeta</math></small> |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° |{{radic|0.5}} |<small><math>\sqrt{1/2}</math></small> | rowspan="3" |[[File:√0.5 × √3.5 great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |138.6~° |{{radic|3.5}} |<small><math>\sqrt{7/2}</math></small> | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |0.707~ |<small><math>\sqrt{2}/2</math></small> |{{radic|3.5}} |1.871~ | |- style="background: gainsboro;" | |0.707~ |2.618~ |<small><math>\phi^2\times\zeta</math></small> |1.871~ |6.927~ |<small><math>\phi^2\sqrt{7}\times\zeta</math></small> |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° |{{radic|0.57~}} |<small><math>\sqrt{3/{2\phi^2}}</math></small> | rowspan="3" |[[File:Irregular great dodecagon.png|100px]] | rowspan="3" |200 irregular great dodecagons<br>(600 great rectangles)<br>in 200 △ planes | rowspan="3" | |135.5~° |{{radic|3.43~}} |<small><math>\sqrt{\phi^4/2}</math></small> | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |0.757~ |<small><math>\sqrt{3} / \phi\sqrt{2}</math></small> |{{radic|3.427~}} |1.851~ |<small><math>\phi^2 / \sqrt{2}</math></small> |- style="background: palegreen;" | |0.757~ |2.803~ |<small><math>\phi\sqrt{3}\times\zeta</math></small> |1.851~ |6.854~ |<small><math>\phi^4\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° |{{radic|0.69~}} |<small><math>\sqrt{\sqrt{5}/{2\phi}}</math></small> | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |130.9~° |{{radic|3.31~}} |<small><math>\sqrt{4 - \sqrt{5}/{2\phi}}</math></small> | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |0.831~ | |{{radic|3.309~}} |1.819~ | |- style="background: gainsboro;" | |0.831~ |3.078~ |<small><math>\sqrt{\phi^3\sqrt{5}}\times\zeta</math></small> |1.819~ |6.735~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° |{{radic|0.88~}} |<small><math>\sqrt{\psi/{2\phi}}</math></small> | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |124° |{{radic|3.12~}} |<small><math>\sqrt{4 - \psi/{2\phi}}</math></small> | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |0.939~ | |{{radic|3.118~}} |1.766~ | |- style="background: gainsboro;" | |0.939~ |3.477~ |<small><math>\sqrt{\psi\phi^3}\times\zeta</math></small> |1.766~ |6.538~ |<small><math>\sqrt{\chi\phi^5}\times\zeta</math></small>{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:<br> {{indent|4}}<math>11/\chi = \psi</math> <br> {{indent|4}}<math>\chi=(3\sqrt{5}+1)/2 \approx 3.854~</math> {{indent|4}}<math>\psi=(3\sqrt{5}-1)/2 \approx 2.854~</math>}} |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° |{{radic|1}} |<small><math>\sqrt{1}</math></small> | rowspan="3" |[[File:Great hexagon.png|100px]] | rowspan="3" |400 regular [[600-cell#Hexagons|great hexagons]]<br> (1200 great rectangles)<br>in 200 △ planes | rowspan="3" |4𝝅<br>[[600-cell#Hexagons and hexagrams|2{10/3}]]<br>#4 |120° |{{radic|3}} |<small><math>\sqrt{3}</math></small> | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |1 | |{{radic|3}} |1.732~ | |- style="background: palegreen;" | |1 |3.702~ |<small><math>\phi^2\sqrt{2}\times\zeta</math></small> |1.732~ |6.413~ |<small><math>\phi^2\sqrt{6}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° |{{radic|1.19~}} |<small><math>\sqrt{\chi/2\phi}</math></small> | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |113.9~° |{{radic|2.81~}} |<small><math>\sqrt{4 - \chi/2\phi}</math></small> | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |1.091~ | |{{radic|2.809~}} |1.676~ | |- style="background: gainsboro;" | |1.091~ |4.041~ |<small><math>\sqrt{\chi/\phi^3}\times\zeta</math></small> |1.676~ |6.205~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° |{{radic|1.31~}} |<small><math>\sqrt{\phi^2/2}</math></small> | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |110.2~° |{{radic|2.69~}} |<small><math>\sqrt{4 - \phi^2/2}</math></small> | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |1.144~ |<small><math>\phi/\sqrt{2}</math></small> |{{radic|2.691~}} |1.640~ | |- style="background: gainsboro;" | |1.144~ |4.236~ |<small><math>\phi^3\times\zeta</math></small> |1.640~ |6.074~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° |{{radic|1.618~}} |<small><math>\sqrt{3-\phi}</math></small> | rowspan="3" |[[File:Great pentagons rectangle.png|100px]] | rowspan="3" |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]<br>(3600 great rectangles)<br> in 720 <big>𝜙</big> planes | rowspan="3" |4𝝅<br>[[600-cell#Squares and octagrams|{24/5}]]<br>#9 |108° |{{radic|2.𝚽}} |<small><math>\sqrt{\phi^2}</math></small> | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |1.176~ |<small><math>\sqrt{\sqrt{5}/\phi}</math></small> |{{radic|2.618~}} |1.618~ |<small><math>\phi</math></small> |- style="background: yellow;" | |1.176~ |4.353~ |<small><math>\sqrt{2\phi^3\sqrt{5}}\times\zeta</math></small> |1.618~ |5.991~ |<small><math>\phi^3\sqrt{2}\times\zeta</math></small> |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° |{{radic|1.5}} |<small><math>\sqrt{3/2}</math></small> | rowspan="3" |[[File:Great 5-cell digons rectangle.png|100px]] | rowspan="3" |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]<br>(600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Pentagram|{5/2}]]<br>#8 |104.5~° |{{radic|2.5}} |<small><math>\sqrt{5/2}</math></small> | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |1.224~ | |{{radic|2.5}} |1.581~ | |- style="background: palegreen;" | |1.224~ |4.535~ |<small><math>\phi^2\sqrt{3}\times\zeta</math></small> |1.581~ |5.854~ |<small><math>\sqrt{5\phi^4}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° |{{radic|1.69~}} |<small><math>\sqrt{\tfrac{1}{4}(9-\sqrt{5})}</math></small> | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |98.9~° |{{radic|2.31~}} | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |1.300~ |<small><math>\tfrac{1}{2}\sqrt{9-\sqrt{5}}</math></small> |{{radic|2.309~}} |1.520~ | |- style="background: gainsboro;" | |1.300~ |4.815~ |<small><math>\text{‡}\times\zeta</math></small> |1.520~ |5.626~ |<small><math>\sqrt{\psi\phi^5}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° |{{radic|0.81~}} |<small><math>\sqrt{\tfrac{2\phi\sqrt{5}}{4}}</math></small> | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |95.5~° |{{radic|2.19~}} |<small><math>\sqrt{\tfrac{11-\sqrt{5}}{4}}</math></small> | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |1.345~ | |{{radic|2.191~}} |1.480~ | |- style="background: gainsboro;" | |1.345~ |4.980~ |<small><math>\sqrt{\phi^5\sqrt{5}}\times\zeta</math></small> |1.480~ |5.480~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |[[File:Great square rectangle.png|100px]] | rowspan="3" |4050 [[600-cell#Squares|great squares]]<br> in 4050 <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:30-gon#Triacontagram|{30/7}]]<br>#7 |90° |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |1.414~ | |{{radic|2}} |1.414~ | |- style="background: seashell;" | |1.414~ |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |1.414~ |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |} == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>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</math> The chord ratio <math>r_3=\sqrt{2}+1</math> 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: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. 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: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> 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: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ 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]] <small><math>\{3,3,4\}</math></small>. 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 <math>\sqrt{2}</math> 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: :<math>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</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> 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 <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, 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 <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> 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 <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> 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 <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> 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° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> 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 <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> 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 <small><math>\{4,3,3\}</math></small> 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]] <small><math>\{4,3,3\}</math></small>. 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 <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. 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: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> 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 <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> 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: :<math>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}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. The <math>r_1</math> and <math>r_5</math> 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 <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ 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 <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, 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. [[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]] The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> 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.]] [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of <math>r_7</math> edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference <math>6\pi</math> over <math>r_7</math> chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges. [[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of <math>r_9</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges. [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of <math>r_4</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_4</math> chords. Successive <math>r_4</math> chords are edges of different 24-cells. The rotational curve over each <math>r_4</math> chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges. In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 <math>\sqrt{1}</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == The 5-point (5-cell) 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell's Petrie {30}-gon has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> 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 & 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 & 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}} spwq1vg7ovn0nloyw9z7fyg547i0k9f 2814913 2814911 2026-06-09T19:48:47Z Dc.samizdat 2856930 /* The 5-point (5-cell) 4-simplex */ 2814913 wikitext text/x-wiki = 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}} <blockquote>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.</blockquote> == 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="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "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]]."}} 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="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{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 5² 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & 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 <math>\phi^6</math> 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="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == Complementary chord pairs == The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 kinds of vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ⁻²√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1₀ − 16₀|ps=, but 14₀ and 16₀ are congruent opposing sections and 15₀ opposes itself; there are 29 non-point sections, denoted 1₀ − 29₀, in 15 opposing pairs.}} |- ! colspan="4" |Short chord ! colspan="2" |Great circle polygons !Rotation ! colspan="4" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° |{{radic|0}} |{{radic|0}} | rowspan="3" | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | |180° |{{radic|4}} |{{radic|4}} | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |0 |0 |{{radic|4}} |2 |2 |- style="background: palegreen;" | |0 |0 |<small><math>0\times\zeta</math></small> |2 |7.405~ |<small><math>2\phi^2\sqrt{2}\times\zeta</math></small> |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° |{{radic|0.𝜀}} |<small><math>\sqrt{1/2\phi^4}</math></small> | rowspan="3" |[[File:Irregular great hexagons of the 120-cell.png|100px]] | rowspan="3" |400 irregular great hexagons<br> (600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}} |164.5~° |{{radic|3.93~}} |<small><math>\sqrt{3\phi^2/2}</math></small> | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |1.982~ |<small><math>1 / \phi^2\sqrt{2}</math></small> |{{radic|3.927~}} |1.982~ |<small><math>\phi\sqrt{1.5}</math></small> |- style="background: palegreen;" | |0.270~ |1 |<small><math>1\times\zeta</math></small> |1.982~ |7.337~ |<small><math>\phi^3\sqrt{3}\times\zeta</math></small> |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° |{{radic|0.19~}} |<small><math>\sqrt{1/2\phi^2}</math></small> | rowspan="3" |[[File:25.2° × 154.8° chords great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:Triacontagon#Triacontagram|{30/13}]]<br>#13 |154.8~° |{{radic|3.81~}} | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |0.437~ |<small><math>1 / \phi\sqrt{2}</math></small> |{{radic|3.809~}} |1.952~ | |- style="background: gainsboro;" | |0.437~ |1.618~ |<small><math>\phi\times\zeta</math></small> |1.952~ |7.226~ |<small><math>\text{‡}\times\zeta</math></small> {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=<br>‡ For simplicity we omit the value of <math>a</math> whenever it is not mononomial in <math>\chi</math>, <math>\psi</math> and <math>\phi</math>.}} |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° |{{radic|0.𝚫}} |<small><math>\sqrt{1/\phi^2}</math></small> | rowspan="3" |[[File:Great decagon rectangle.png|100px]] | rowspan="3" |720 great decagons<br>(3600 great rectangles)<br>in 720 <big>𝜙</big> planes | rowspan="3" |5𝝅<br>[[600-cell#Decagons and pentadecagrams|{15/2}]]<br>#5 |144° |{{radic|3.𝚽}} |<small><math>\sqrt{2+\phi}</math></small> | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |0.618~ |<small><math>1 / \phi</math></small> |{{radic|3.618~}} |1.902~ |<small><math>1+1/{\phi^2}</math></small> |- style="background: yellow;" | |0.618~ |2.288~ |<small><math>\phi\sqrt{2}\times\zeta</math></small> |1.902~ |7.0425 |<small><math>\sqrt{2\phi^5\sqrt{5}}\times\zeta</math></small> |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° |{{radic|0.5}} |<small><math>\sqrt{1/2}</math></small> | rowspan="3" |[[File:√0.5 × √3.5 great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |138.6~° |{{radic|3.5}} |<small><math>\sqrt{7/2}</math></small> | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |0.707~ |<small><math>\sqrt{2}/2</math></small> |{{radic|3.5}} |1.871~ | |- style="background: gainsboro;" | |0.707~ |2.618~ |<small><math>\phi^2\times\zeta</math></small> |1.871~ |6.927~ |<small><math>\phi^2\sqrt{7}\times\zeta</math></small> |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° |{{radic|0.57~}} |<small><math>\sqrt{3/{2\phi^2}}</math></small> | rowspan="3" |[[File:Irregular great dodecagon.png|100px]] | rowspan="3" |200 irregular great dodecagons<br>(600 great rectangles)<br>in 200 △ planes | rowspan="3" | |135.5~° |{{radic|3.43~}} |<small><math>\sqrt{\phi^4/2}</math></small> | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |0.757~ |<small><math>\sqrt{3} / \phi\sqrt{2}</math></small> |{{radic|3.427~}} |1.851~ |<small><math>\phi^2 / \sqrt{2}</math></small> |- style="background: palegreen;" | |0.757~ |2.803~ |<small><math>\phi\sqrt{3}\times\zeta</math></small> |1.851~ |6.854~ |<small><math>\phi^4\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° |{{radic|0.69~}} |<small><math>\sqrt{\sqrt{5}/{2\phi}}</math></small> | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |130.9~° |{{radic|3.31~}} |<small><math>\sqrt{4 - \sqrt{5}/{2\phi}}</math></small> | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |0.831~ | |{{radic|3.309~}} |1.819~ | |- style="background: gainsboro;" | |0.831~ |3.078~ |<small><math>\sqrt{\phi^3\sqrt{5}}\times\zeta</math></small> |1.819~ |6.735~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° |{{radic|0.88~}} |<small><math>\sqrt{\psi/{2\phi}}</math></small> | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |124° |{{radic|3.12~}} |<small><math>\sqrt{4 - \psi/{2\phi}}</math></small> | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |0.939~ | |{{radic|3.118~}} |1.766~ | |- style="background: gainsboro;" | |0.939~ |3.477~ |<small><math>\sqrt{\psi\phi^3}\times\zeta</math></small> |1.766~ |6.538~ |<small><math>\sqrt{\chi\phi^5}\times\zeta</math></small>{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:<br> {{indent|4}}<math>11/\chi = \psi</math> <br> {{indent|4}}<math>\chi=(3\sqrt{5}+1)/2 \approx 3.854~</math> {{indent|4}}<math>\psi=(3\sqrt{5}-1)/2 \approx 2.854~</math>}} |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° |{{radic|1}} |<small><math>\sqrt{1}</math></small> | rowspan="3" |[[File:Great hexagon.png|100px]] | rowspan="3" |400 regular [[600-cell#Hexagons|great hexagons]]<br> (1200 great rectangles)<br>in 200 △ planes | rowspan="3" |4𝝅<br>[[600-cell#Hexagons and hexagrams|2{10/3}]]<br>#4 |120° |{{radic|3}} |<small><math>\sqrt{3}</math></small> | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |1 | |{{radic|3}} |1.732~ | |- style="background: palegreen;" | |1 |3.702~ |<small><math>\phi^2\sqrt{2}\times\zeta</math></small> |1.732~ |6.413~ |<small><math>\phi^2\sqrt{6}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° |{{radic|1.19~}} |<small><math>\sqrt{\chi/2\phi}</math></small> | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |113.9~° |{{radic|2.81~}} |<small><math>\sqrt{4 - \chi/2\phi}</math></small> | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |1.091~ | |{{radic|2.809~}} |1.676~ | |- style="background: gainsboro;" | |1.091~ |4.041~ |<small><math>\sqrt{\chi/\phi^3}\times\zeta</math></small> |1.676~ |6.205~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° |{{radic|1.31~}} |<small><math>\sqrt{\phi^2/2}</math></small> | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |110.2~° |{{radic|2.69~}} |<small><math>\sqrt{4 - \phi^2/2}</math></small> | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |1.144~ |<small><math>\phi/\sqrt{2}</math></small> |{{radic|2.691~}} |1.640~ | |- style="background: gainsboro;" | |1.144~ |4.236~ |<small><math>\phi^3\times\zeta</math></small> |1.640~ |6.074~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° |{{radic|1.618~}} |<small><math>\sqrt{3-\phi}</math></small> | rowspan="3" |[[File:Great pentagons rectangle.png|100px]] | rowspan="3" |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]<br>(3600 great rectangles)<br> in 720 <big>𝜙</big> planes | rowspan="3" |4𝝅<br>[[600-cell#Squares and octagrams|{24/5}]]<br>#9 |108° |{{radic|2.𝚽}} |<small><math>\sqrt{\phi^2}</math></small> | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |1.176~ |<small><math>\sqrt{\sqrt{5}/\phi}</math></small> |{{radic|2.618~}} |1.618~ |<small><math>\phi</math></small> |- style="background: yellow;" | |1.176~ |4.353~ |<small><math>\sqrt{2\phi^3\sqrt{5}}\times\zeta</math></small> |1.618~ |5.991~ |<small><math>\phi^3\sqrt{2}\times\zeta</math></small> |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° |{{radic|1.5}} |<small><math>\sqrt{3/2}</math></small> | rowspan="3" |[[File:Great 5-cell digons rectangle.png|100px]] | rowspan="3" |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]<br>(600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Pentagram|{5/2}]]<br>#8 |104.5~° |{{radic|2.5}} |<small><math>\sqrt{5/2}</math></small> | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |1.224~ | |{{radic|2.5}} |1.581~ | |- style="background: palegreen;" | |1.224~ |4.535~ |<small><math>\phi^2\sqrt{3}\times\zeta</math></small> |1.581~ |5.854~ |<small><math>\sqrt{5\phi^4}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° |{{radic|1.69~}} |<small><math>\sqrt{\tfrac{1}{4}(9-\sqrt{5})}</math></small> | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |98.9~° |{{radic|2.31~}} | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |1.300~ |<small><math>\tfrac{1}{2}\sqrt{9-\sqrt{5}}</math></small> |{{radic|2.309~}} |1.520~ | |- style="background: gainsboro;" | |1.300~ |4.815~ |<small><math>\text{‡}\times\zeta</math></small> |1.520~ |5.626~ |<small><math>\sqrt{\psi\phi^5}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° |{{radic|0.81~}} |<small><math>\sqrt{\tfrac{2\phi\sqrt{5}}{4}}</math></small> | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |95.5~° |{{radic|2.19~}} |<small><math>\sqrt{\tfrac{11-\sqrt{5}}{4}}</math></small> | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |1.345~ | |{{radic|2.191~}} |1.480~ | |- style="background: gainsboro;" | |1.345~ |4.980~ |<small><math>\sqrt{\phi^5\sqrt{5}}\times\zeta</math></small> |1.480~ |5.480~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |[[File:Great square rectangle.png|100px]] | rowspan="3" |4050 [[600-cell#Squares|great squares]]<br> in 4050 <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:30-gon#Triacontagram|{30/7}]]<br>#7 |90° |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |1.414~ | |{{radic|2}} |1.414~ | |- style="background: seashell;" | |1.414~ |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |1.414~ |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |} == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>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</math> The chord ratio <math>r_3=\sqrt{2}+1</math> 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: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. 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: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> 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: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ 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]] <small><math>\{3,3,4\}</math></small>. 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 <math>\sqrt{2}</math> 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: :<math>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</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> 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 <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, 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 <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> 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 <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> 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 <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> 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° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> 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 <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> 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 <small><math>\{4,3,3\}</math></small> 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]] <small><math>\{4,3,3\}</math></small>. 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 <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. 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: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> 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 <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> 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: :<math>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}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. The <math>r_1</math> and <math>r_5</math> 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 <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ 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 <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, 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. [[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]] The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> 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.]] [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of <math>r_7</math> edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference <math>6\pi</math> over <math>r_7</math> chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges. [[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of <math>r_9</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges. [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of <math>r_4</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_4</math> chords. Successive <math>r_4</math> chords are edges of different 24-cells. The rotational curve over each <math>r_4</math> chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges. In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 <math>\sqrt{1}</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == The 5-point (5-cell) 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the planar {30)-gon and the 600-cell. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> 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 & 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 & 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}} dut4beq9r5pmjsooaihhtlj813wpax0 2814915 2814913 2026-06-09T19:50:16Z Dc.samizdat 2856930 /* The 5-point (5-cell) 4-simplex */ 2814915 wikitext text/x-wiki = 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}} <blockquote>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.</blockquote> == 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="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "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]]."}} 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="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{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 5² 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & 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 <math>\phi^6</math> 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="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == Complementary chord pairs == The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 kinds of vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ⁻²√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1₀ − 16₀|ps=, but 14₀ and 16₀ are congruent opposing sections and 15₀ opposes itself; there are 29 non-point sections, denoted 1₀ − 29₀, in 15 opposing pairs.}} |- ! colspan="4" |Short chord ! colspan="2" |Great circle polygons !Rotation ! colspan="4" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° |{{radic|0}} |{{radic|0}} | rowspan="3" | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | |180° |{{radic|4}} |{{radic|4}} | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |0 |0 |{{radic|4}} |2 |2 |- style="background: palegreen;" | |0 |0 |<small><math>0\times\zeta</math></small> |2 |7.405~ |<small><math>2\phi^2\sqrt{2}\times\zeta</math></small> |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° |{{radic|0.𝜀}} |<small><math>\sqrt{1/2\phi^4}</math></small> | rowspan="3" |[[File:Irregular great hexagons of the 120-cell.png|100px]] | rowspan="3" |400 irregular great hexagons<br> (600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}} |164.5~° |{{radic|3.93~}} |<small><math>\sqrt{3\phi^2/2}</math></small> | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |1.982~ |<small><math>1 / \phi^2\sqrt{2}</math></small> |{{radic|3.927~}} |1.982~ |<small><math>\phi\sqrt{1.5}</math></small> |- style="background: palegreen;" | |0.270~ |1 |<small><math>1\times\zeta</math></small> |1.982~ |7.337~ |<small><math>\phi^3\sqrt{3}\times\zeta</math></small> |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° |{{radic|0.19~}} |<small><math>\sqrt{1/2\phi^2}</math></small> | rowspan="3" |[[File:25.2° × 154.8° chords great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:Triacontagon#Triacontagram|{30/13}]]<br>#13 |154.8~° |{{radic|3.81~}} | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |0.437~ |<small><math>1 / \phi\sqrt{2}</math></small> |{{radic|3.809~}} |1.952~ | |- style="background: gainsboro;" | |0.437~ |1.618~ |<small><math>\phi\times\zeta</math></small> |1.952~ |7.226~ |<small><math>\text{‡}\times\zeta</math></small> {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=<br>‡ For simplicity we omit the value of <math>a</math> whenever it is not mononomial in <math>\chi</math>, <math>\psi</math> and <math>\phi</math>.}} |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° |{{radic|0.𝚫}} |<small><math>\sqrt{1/\phi^2}</math></small> | rowspan="3" |[[File:Great decagon rectangle.png|100px]] | rowspan="3" |720 great decagons<br>(3600 great rectangles)<br>in 720 <big>𝜙</big> planes | rowspan="3" |5𝝅<br>[[600-cell#Decagons and pentadecagrams|{15/2}]]<br>#5 |144° |{{radic|3.𝚽}} |<small><math>\sqrt{2+\phi}</math></small> | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |0.618~ |<small><math>1 / \phi</math></small> |{{radic|3.618~}} |1.902~ |<small><math>1+1/{\phi^2}</math></small> |- style="background: yellow;" | |0.618~ |2.288~ |<small><math>\phi\sqrt{2}\times\zeta</math></small> |1.902~ |7.0425 |<small><math>\sqrt{2\phi^5\sqrt{5}}\times\zeta</math></small> |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° |{{radic|0.5}} |<small><math>\sqrt{1/2}</math></small> | rowspan="3" |[[File:√0.5 × √3.5 great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |138.6~° |{{radic|3.5}} |<small><math>\sqrt{7/2}</math></small> | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |0.707~ |<small><math>\sqrt{2}/2</math></small> |{{radic|3.5}} |1.871~ | |- style="background: gainsboro;" | |0.707~ |2.618~ |<small><math>\phi^2\times\zeta</math></small> |1.871~ |6.927~ |<small><math>\phi^2\sqrt{7}\times\zeta</math></small> |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° |{{radic|0.57~}} |<small><math>\sqrt{3/{2\phi^2}}</math></small> | rowspan="3" |[[File:Irregular great dodecagon.png|100px]] | rowspan="3" |200 irregular great dodecagons<br>(600 great rectangles)<br>in 200 △ planes | rowspan="3" | |135.5~° |{{radic|3.43~}} |<small><math>\sqrt{\phi^4/2}</math></small> | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |0.757~ |<small><math>\sqrt{3} / \phi\sqrt{2}</math></small> |{{radic|3.427~}} |1.851~ |<small><math>\phi^2 / \sqrt{2}</math></small> |- style="background: palegreen;" | |0.757~ |2.803~ |<small><math>\phi\sqrt{3}\times\zeta</math></small> |1.851~ |6.854~ |<small><math>\phi^4\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° |{{radic|0.69~}} |<small><math>\sqrt{\sqrt{5}/{2\phi}}</math></small> | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |130.9~° |{{radic|3.31~}} |<small><math>\sqrt{4 - \sqrt{5}/{2\phi}}</math></small> | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |0.831~ | |{{radic|3.309~}} |1.819~ | |- style="background: gainsboro;" | |0.831~ |3.078~ |<small><math>\sqrt{\phi^3\sqrt{5}}\times\zeta</math></small> |1.819~ |6.735~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° |{{radic|0.88~}} |<small><math>\sqrt{\psi/{2\phi}}</math></small> | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |124° |{{radic|3.12~}} |<small><math>\sqrt{4 - \psi/{2\phi}}</math></small> | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |0.939~ | |{{radic|3.118~}} |1.766~ | |- style="background: gainsboro;" | |0.939~ |3.477~ |<small><math>\sqrt{\psi\phi^3}\times\zeta</math></small> |1.766~ |6.538~ |<small><math>\sqrt{\chi\phi^5}\times\zeta</math></small>{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:<br> {{indent|4}}<math>11/\chi = \psi</math> <br> {{indent|4}}<math>\chi=(3\sqrt{5}+1)/2 \approx 3.854~</math> {{indent|4}}<math>\psi=(3\sqrt{5}-1)/2 \approx 2.854~</math>}} |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° |{{radic|1}} |<small><math>\sqrt{1}</math></small> | rowspan="3" |[[File:Great hexagon.png|100px]] | rowspan="3" |400 regular [[600-cell#Hexagons|great hexagons]]<br> (1200 great rectangles)<br>in 200 △ planes | rowspan="3" |4𝝅<br>[[600-cell#Hexagons and hexagrams|2{10/3}]]<br>#4 |120° |{{radic|3}} |<small><math>\sqrt{3}</math></small> | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |1 | |{{radic|3}} |1.732~ | |- style="background: palegreen;" | |1 |3.702~ |<small><math>\phi^2\sqrt{2}\times\zeta</math></small> |1.732~ |6.413~ |<small><math>\phi^2\sqrt{6}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° |{{radic|1.19~}} |<small><math>\sqrt{\chi/2\phi}</math></small> | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |113.9~° |{{radic|2.81~}} |<small><math>\sqrt{4 - \chi/2\phi}</math></small> | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |1.091~ | |{{radic|2.809~}} |1.676~ | |- style="background: gainsboro;" | |1.091~ |4.041~ |<small><math>\sqrt{\chi/\phi^3}\times\zeta</math></small> |1.676~ |6.205~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° |{{radic|1.31~}} |<small><math>\sqrt{\phi^2/2}</math></small> | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |110.2~° |{{radic|2.69~}} |<small><math>\sqrt{4 - \phi^2/2}</math></small> | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |1.144~ |<small><math>\phi/\sqrt{2}</math></small> |{{radic|2.691~}} |1.640~ | |- style="background: gainsboro;" | |1.144~ |4.236~ |<small><math>\phi^3\times\zeta</math></small> |1.640~ |6.074~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° |{{radic|1.618~}} |<small><math>\sqrt{3-\phi}</math></small> | rowspan="3" |[[File:Great pentagons rectangle.png|100px]] | rowspan="3" |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]<br>(3600 great rectangles)<br> in 720 <big>𝜙</big> planes | rowspan="3" |4𝝅<br>[[600-cell#Squares and octagrams|{24/5}]]<br>#9 |108° |{{radic|2.𝚽}} |<small><math>\sqrt{\phi^2}</math></small> | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |1.176~ |<small><math>\sqrt{\sqrt{5}/\phi}</math></small> |{{radic|2.618~}} |1.618~ |<small><math>\phi</math></small> |- style="background: yellow;" | |1.176~ |4.353~ |<small><math>\sqrt{2\phi^3\sqrt{5}}\times\zeta</math></small> |1.618~ |5.991~ |<small><math>\phi^3\sqrt{2}\times\zeta</math></small> |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° |{{radic|1.5}} |<small><math>\sqrt{3/2}</math></small> | rowspan="3" |[[File:Great 5-cell digons rectangle.png|100px]] | rowspan="3" |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]<br>(600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Pentagram|{5/2}]]<br>#8 |104.5~° |{{radic|2.5}} |<small><math>\sqrt{5/2}</math></small> | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |1.224~ | |{{radic|2.5}} |1.581~ | |- style="background: palegreen;" | |1.224~ |4.535~ |<small><math>\phi^2\sqrt{3}\times\zeta</math></small> |1.581~ |5.854~ |<small><math>\sqrt{5\phi^4}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° |{{radic|1.69~}} |<small><math>\sqrt{\tfrac{1}{4}(9-\sqrt{5})}</math></small> | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |98.9~° |{{radic|2.31~}} | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |1.300~ |<small><math>\tfrac{1}{2}\sqrt{9-\sqrt{5}}</math></small> |{{radic|2.309~}} |1.520~ | |- style="background: gainsboro;" | |1.300~ |4.815~ |<small><math>\text{‡}\times\zeta</math></small> |1.520~ |5.626~ |<small><math>\sqrt{\psi\phi^5}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° |{{radic|0.81~}} |<small><math>\sqrt{\tfrac{2\phi\sqrt{5}}{4}}</math></small> | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |95.5~° |{{radic|2.19~}} |<small><math>\sqrt{\tfrac{11-\sqrt{5}}{4}}</math></small> | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |1.345~ | |{{radic|2.191~}} |1.480~ | |- style="background: gainsboro;" | |1.345~ |4.980~ |<small><math>\sqrt{\phi^5\sqrt{5}}\times\zeta</math></small> |1.480~ |5.480~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |[[File:Great square rectangle.png|100px]] | rowspan="3" |4050 [[600-cell#Squares|great squares]]<br> in 4050 <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:30-gon#Triacontagram|{30/7}]]<br>#7 |90° |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |1.414~ | |{{radic|2}} |1.414~ | |- style="background: seashell;" | |1.414~ |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |1.414~ |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |} == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>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</math> The chord ratio <math>r_3=\sqrt{2}+1</math> 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: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. 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: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> 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: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ 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]] <small><math>\{3,3,4\}</math></small>. 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 <math>\sqrt{2}</math> 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: :<math>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</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> 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 <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, 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 <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> 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 <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> 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 <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> 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° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> 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 <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> 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 <small><math>\{4,3,3\}</math></small> 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]] <small><math>\{4,3,3\}</math></small>. 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 <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. 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: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> 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 <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> 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: :<math>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}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. The <math>r_1</math> and <math>r_5</math> 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 <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ 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 <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, 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. [[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]] The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> 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.]] [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of <math>r_7</math> edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference <math>6\pi</math> over <math>r_7</math> chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges. [[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of <math>r_9</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges. [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of <math>r_4</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_4</math> chords. Successive <math>r_4</math> chords are edges of different 24-cells. The rotational curve over each <math>r_4</math> chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges. In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 <math>\sqrt{1}</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == The 5-point (5-cell) 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> 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 & 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 & 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}} h7ggaoh60g79fzh8tdaqxdyrubxpxcj 2814956 2814915 2026-06-10T00:56:02Z Dc.samizdat 2856930 /* Complementary chord pairs */ 2814956 wikitext text/x-wiki = 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}} <blockquote>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.</blockquote> == 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="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "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]]."}} 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="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{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 5² 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & 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 <math>\phi^6</math> 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="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == Complementary chord pairs == The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 kinds of vertex-first polyhedral sections{{Sfn|Coxeter|1973|pp=300-301|loc=Table V:(v) Simplified sections of {5,3,3} (edge 2φ⁻²√2 [radius 4]) beginning with a vertex; Coxeter's table lists 16 non-point sections labelled 1₀ − 16₀|ps=, but 14₀ and 16₀ are congruent opposing sections and 15₀ opposes itself; there are 29 non-point sections, denoted 1₀ − 29₀, in 15 opposing pairs.}} |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° |{{radic|0}} |{{radic|0}} | rowspan="3" | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | |180° |{{radic|4}} |{{radic|4}} | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |0 |0 |{{radic|4}} |2 |2 |- style="background: palegreen;" | |0 |0 |<small><math>0\times\zeta</math></small> |2 |7.405~ |<small><math>2\phi^2\sqrt{2}\times\zeta</math></small> |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° |{{radic|0.𝜀}} |<small><math>\sqrt{1/2\phi^4}</math></small> | rowspan="3" |[[File:Irregular great hexagons of the 120-cell.png|100px]] | rowspan="3" |400 irregular great hexagons<br> (600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}} |164.5~° |{{radic|3.93~}} |<small><math>\sqrt{3\phi^2/2}</math></small> | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |1.982~ |<small><math>1 / \phi^2\sqrt{2}</math></small> |{{radic|3.927~}} |1.982~ |<small><math>\phi\sqrt{1.5}</math></small> |- style="background: palegreen;" | |0.270~ |1 |<small><math>1\times\zeta</math></small> |1.982~ |7.337~ |<small><math>\phi^3\sqrt{3}\times\zeta</math></small> |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° |{{radic|0.19~}} |<small><math>\sqrt{1/2\phi^2}</math></small> | rowspan="3" |[[File:25.2° × 154.8° chords great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:Triacontagon#Triacontagram|{30/13}]]<br>#13 |154.8~° |{{radic|3.81~}} | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |0.437~ |<small><math>1 / \phi\sqrt{2}</math></small> |{{radic|3.809~}} |1.952~ | |- style="background: gainsboro;" | |0.437~ |1.618~ |<small><math>\phi\times\zeta</math></small> |1.952~ |7.226~ |<small><math>\text{‡}\times\zeta</math></small> {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=<br>‡ For simplicity we omit the value of <math>a</math> whenever it is not mononomial in <math>\chi</math>, <math>\psi</math> and <math>\phi</math>.}} |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° |{{radic|0.𝚫}} |<small><math>\sqrt{1/\phi^2}</math></small> | rowspan="3" |[[File:Great decagon rectangle.png|100px]] | rowspan="3" |720 great decagons<br>(3600 great rectangles)<br>in 720 <big>𝜙</big> planes | rowspan="3" |5𝝅<br>[[600-cell#Decagons and pentadecagrams|{15/2}]]<br>#5 |144° |{{radic|3.𝚽}} |<small><math>\sqrt{2+\phi}</math></small> | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |0.618~ |<small><math>1 / \phi</math></small> |{{radic|3.618~}} |1.902~ |<small><math>1+1/{\phi^2}</math></small> |- style="background: yellow;" | |0.618~ |2.288~ |<small><math>\phi\sqrt{2}\times\zeta</math></small> |1.902~ |7.0425 |<small><math>\sqrt{2\phi^5\sqrt{5}}\times\zeta</math></small> |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° |{{radic|0.5}} |<small><math>\sqrt{1/2}</math></small> | rowspan="3" |[[File:√0.5 × √3.5 great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |138.6~° |{{radic|3.5}} |<small><math>\sqrt{7/2}</math></small> | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |0.707~ |<small><math>\sqrt{2}/2</math></small> |{{radic|3.5}} |1.871~ | |- style="background: gainsboro;" | |0.707~ |2.618~ |<small><math>\phi^2\times\zeta</math></small> |1.871~ |6.927~ |<small><math>\phi^2\sqrt{7}\times\zeta</math></small> |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° |{{radic|0.57~}} |<small><math>\sqrt{3/{2\phi^2}}</math></small> | rowspan="3" |[[File:Irregular great dodecagon.png|100px]] | rowspan="3" |200 irregular great dodecagons<br>(600 great rectangles)<br>in 200 △ planes | rowspan="3" | |135.5~° |{{radic|3.43~}} |<small><math>\sqrt{\phi^4/2}</math></small> | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |0.757~ |<small><math>\sqrt{3} / \phi\sqrt{2}</math></small> |{{radic|3.427~}} |1.851~ |<small><math>\phi^2 / \sqrt{2}</math></small> |- style="background: palegreen;" | |0.757~ |2.803~ |<small><math>\phi\sqrt{3}\times\zeta</math></small> |1.851~ |6.854~ |<small><math>\phi^4\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° |{{radic|0.69~}} |<small><math>\sqrt{\sqrt{5}/{2\phi}}</math></small> | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |130.9~° |{{radic|3.31~}} |<small><math>\sqrt{4 - \sqrt{5}/{2\phi}}</math></small> | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |0.831~ | |{{radic|3.309~}} |1.819~ | |- style="background: gainsboro;" | |0.831~ |3.078~ |<small><math>\sqrt{\phi^3\sqrt{5}}\times\zeta</math></small> |1.819~ |6.735~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° |{{radic|0.88~}} |<small><math>\sqrt{\psi/{2\phi}}</math></small> | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |124° |{{radic|3.12~}} |<small><math>\sqrt{4 - \psi/{2\phi}}</math></small> | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |0.939~ | |{{radic|3.118~}} |1.766~ | |- style="background: gainsboro;" | |0.939~ |3.477~ |<small><math>\sqrt{\psi\phi^3}\times\zeta</math></small> |1.766~ |6.538~ |<small><math>\sqrt{\chi\phi^5}\times\zeta</math></small>{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:<br> {{indent|4}}<math>11/\chi = \psi</math> <br> {{indent|4}}<math>\chi=(3\sqrt{5}+1)/2 \approx 3.854~</math> {{indent|4}}<math>\psi=(3\sqrt{5}-1)/2 \approx 2.854~</math>}} |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° |{{radic|1}} |<small><math>\sqrt{1}</math></small> | rowspan="3" |[[File:Great hexagon.png|100px]] | rowspan="3" |400 regular [[600-cell#Hexagons|great hexagons]]<br> (1200 great rectangles)<br>in 200 △ planes | rowspan="3" |4𝝅<br>[[600-cell#Hexagons and hexagrams|2{10/3}]]<br>#4 |120° |{{radic|3}} |<small><math>\sqrt{3}</math></small> | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |1 | |{{radic|3}} |1.732~ | |- style="background: palegreen;" | |1 |3.702~ |<small><math>\phi^2\sqrt{2}\times\zeta</math></small> |1.732~ |6.413~ |<small><math>\phi^2\sqrt{6}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° |{{radic|1.19~}} |<small><math>\sqrt{\chi/2\phi}</math></small> | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |113.9~° |{{radic|2.81~}} |<small><math>\sqrt{4 - \chi/2\phi}</math></small> | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |1.091~ | |{{radic|2.809~}} |1.676~ | |- style="background: gainsboro;" | |1.091~ |4.041~ |<small><math>\sqrt{\chi/\phi^3}\times\zeta</math></small> |1.676~ |6.205~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° |{{radic|1.31~}} |<small><math>\sqrt{\phi^2/2}</math></small> | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |110.2~° |{{radic|2.69~}} |<small><math>\sqrt{4 - \phi^2/2}</math></small> | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |1.144~ |<small><math>\phi/\sqrt{2}</math></small> |{{radic|2.691~}} |1.640~ | |- style="background: gainsboro;" | |1.144~ |4.236~ |<small><math>\phi^3\times\zeta</math></small> |1.640~ |6.074~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° |{{radic|1.618~}} |<small><math>\sqrt{3-\phi}</math></small> | rowspan="3" |[[File:Great pentagons rectangle.png|100px]] | rowspan="3" |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]<br>(3600 great rectangles)<br> in 720 <big>𝜙</big> planes | rowspan="3" |4𝝅<br>[[600-cell#Squares and octagrams|{24/5}]]<br>#9 |108° |{{radic|2.𝚽}} |<small><math>\sqrt{\phi^2}</math></small> | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |1.176~ |<small><math>\sqrt{\sqrt{5}/\phi}</math></small> |{{radic|2.618~}} |1.618~ |<small><math>\phi</math></small> |- style="background: yellow;" | |1.176~ |4.353~ |<small><math>\sqrt{2\phi^3\sqrt{5}}\times\zeta</math></small> |1.618~ |5.991~ |<small><math>\phi^3\sqrt{2}\times\zeta</math></small> |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° |{{radic|1.5}} |<small><math>\sqrt{3/2}</math></small> | rowspan="3" |[[File:Great 5-cell digons rectangle.png|100px]] | rowspan="3" |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]<br>(600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Pentagram|{5/2}]]<br>#8 |104.5~° |{{radic|2.5}} |<small><math>\sqrt{5/2}</math></small> | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |1.224~ | |{{radic|2.5}} |1.581~ | |- style="background: palegreen;" | |1.224~ |4.535~ |<small><math>\phi^2\sqrt{3}\times\zeta</math></small> |1.581~ |5.854~ |<small><math>\sqrt{5\phi^4}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° |{{radic|1.69~}} |<small><math>\sqrt{\tfrac{1}{4}(9-\sqrt{5})}</math></small> | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |98.9~° |{{radic|2.31~}} | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |1.300~ |<small><math>\tfrac{1}{2}\sqrt{9-\sqrt{5}}</math></small> |{{radic|2.309~}} |1.520~ | |- style="background: gainsboro;" | |1.300~ |4.815~ |<small><math>\text{‡}\times\zeta</math></small> |1.520~ |5.626~ |<small><math>\sqrt{\psi\phi^5}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° |{{radic|0.81~}} |<small><math>\sqrt{\tfrac{2\phi\sqrt{5}}{4}}</math></small> | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |95.5~° |{{radic|2.19~}} |<small><math>\sqrt{\tfrac{11-\sqrt{5}}{4}}</math></small> | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |1.345~ | |{{radic|2.191~}} |1.480~ | |- style="background: gainsboro;" | |1.345~ |4.980~ |<small><math>\sqrt{\phi^5\sqrt{5}}\times\zeta</math></small> |1.480~ |5.480~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |[[File:Great square rectangle.png|100px]] | rowspan="3" |4050 [[600-cell#Squares|great squares]]<br> in 4050 <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:30-gon#Triacontagram|{30/7}]]<br>#7 |90° |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |1.414~ | |{{radic|2}} |1.414~ | |- style="background: seashell;" | |1.414~ |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |1.414~ |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |} == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>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</math> The chord ratio <math>r_3=\sqrt{2}+1</math> 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: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. 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: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> 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: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ 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]] <small><math>\{3,3,4\}</math></small>. 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 <math>\sqrt{2}</math> 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: :<math>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</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> 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 <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, 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 <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> 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 <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> 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 <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> 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° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> 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 <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> 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 <small><math>\{4,3,3\}</math></small> 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]] <small><math>\{4,3,3\}</math></small>. 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 <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. 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: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> 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 <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> 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: :<math>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}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. The <math>r_1</math> and <math>r_5</math> 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 <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ 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 <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, 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. [[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]] The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> 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.]] [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of <math>r_7</math> edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference <math>6\pi</math> over <math>r_7</math> chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges. [[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of <math>r_9</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges. [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of <math>r_4</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_4</math> chords. Successive <math>r_4</math> chords are edges of different 24-cells. The rotational curve over each <math>r_4</math> chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges. In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 <math>\sqrt{1}</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == The 5-point (5-cell) 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> 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 & 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 & 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}} 0g7l5tj568eshcpb5xyuet0o6hjh3il 2814957 2814956 2026-06-10T00:58:36Z Dc.samizdat 2856930 /* Complementary chord pairs */ 2814957 wikitext text/x-wiki = 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}} <blockquote>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.</blockquote> == 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="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "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]]."}} 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="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{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 5² 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & 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 <math>\phi^6</math> 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="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == Complementary chord pairs == The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 kinds of polyhedral section |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° |{{radic|0}} |{{radic|0}} | rowspan="3" | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | |180° |{{radic|4}} |{{radic|4}} | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |0 |0 |{{radic|4}} |2 |2 |- style="background: palegreen;" | |0 |0 |<small><math>0\times\zeta</math></small> |2 |7.405~ |<small><math>2\phi^2\sqrt{2}\times\zeta</math></small> |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° |{{radic|0.𝜀}} |<small><math>\sqrt{1/2\phi^4}</math></small> | rowspan="3" |[[File:Irregular great hexagons of the 120-cell.png|100px]] | rowspan="3" |400 irregular great hexagons<br> (600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}} |164.5~° |{{radic|3.93~}} |<small><math>\sqrt{3\phi^2/2}</math></small> | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |1.982~ |<small><math>1 / \phi^2\sqrt{2}</math></small> |{{radic|3.927~}} |1.982~ |<small><math>\phi\sqrt{1.5}</math></small> |- style="background: palegreen;" | |0.270~ |1 |<small><math>1\times\zeta</math></small> |1.982~ |7.337~ |<small><math>\phi^3\sqrt{3}\times\zeta</math></small> |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° |{{radic|0.19~}} |<small><math>\sqrt{1/2\phi^2}</math></small> | rowspan="3" |[[File:25.2° × 154.8° chords great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:Triacontagon#Triacontagram|{30/13}]]<br>#13 |154.8~° |{{radic|3.81~}} | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |0.437~ |<small><math>1 / \phi\sqrt{2}</math></small> |{{radic|3.809~}} |1.952~ | |- style="background: gainsboro;" | |0.437~ |1.618~ |<small><math>\phi\times\zeta</math></small> |1.952~ |7.226~ |<small><math>\text{‡}\times\zeta</math></small> {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=<br>‡ For simplicity we omit the value of <math>a</math> whenever it is not mononomial in <math>\chi</math>, <math>\psi</math> and <math>\phi</math>.}} |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° |{{radic|0.𝚫}} |<small><math>\sqrt{1/\phi^2}</math></small> | rowspan="3" |[[File:Great decagon rectangle.png|100px]] | rowspan="3" |720 great decagons<br>(3600 great rectangles)<br>in 720 <big>𝜙</big> planes | rowspan="3" |5𝝅<br>[[600-cell#Decagons and pentadecagrams|{15/2}]]<br>#5 |144° |{{radic|3.𝚽}} |<small><math>\sqrt{2+\phi}</math></small> | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |0.618~ |<small><math>1 / \phi</math></small> |{{radic|3.618~}} |1.902~ |<small><math>1+1/{\phi^2}</math></small> |- style="background: yellow;" | |0.618~ |2.288~ |<small><math>\phi\sqrt{2}\times\zeta</math></small> |1.902~ |7.0425 |<small><math>\sqrt{2\phi^5\sqrt{5}}\times\zeta</math></small> |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° |{{radic|0.5}} |<small><math>\sqrt{1/2}</math></small> | rowspan="3" |[[File:√0.5 × √3.5 great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |138.6~° |{{radic|3.5}} |<small><math>\sqrt{7/2}</math></small> | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |0.707~ |<small><math>\sqrt{2}/2</math></small> |{{radic|3.5}} |1.871~ | |- style="background: gainsboro;" | |0.707~ |2.618~ |<small><math>\phi^2\times\zeta</math></small> |1.871~ |6.927~ |<small><math>\phi^2\sqrt{7}\times\zeta</math></small> |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° |{{radic|0.57~}} |<small><math>\sqrt{3/{2\phi^2}}</math></small> | rowspan="3" |[[File:Irregular great dodecagon.png|100px]] | rowspan="3" |200 irregular great dodecagons<br>(600 great rectangles)<br>in 200 △ planes | rowspan="3" | |135.5~° |{{radic|3.43~}} |<small><math>\sqrt{\phi^4/2}</math></small> | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |0.757~ |<small><math>\sqrt{3} / \phi\sqrt{2}</math></small> |{{radic|3.427~}} |1.851~ |<small><math>\phi^2 / \sqrt{2}</math></small> |- style="background: palegreen;" | |0.757~ |2.803~ |<small><math>\phi\sqrt{3}\times\zeta</math></small> |1.851~ |6.854~ |<small><math>\phi^4\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° |{{radic|0.69~}} |<small><math>\sqrt{\sqrt{5}/{2\phi}}</math></small> | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |130.9~° |{{radic|3.31~}} |<small><math>\sqrt{4 - \sqrt{5}/{2\phi}}</math></small> | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |0.831~ | |{{radic|3.309~}} |1.819~ | |- style="background: gainsboro;" | |0.831~ |3.078~ |<small><math>\sqrt{\phi^3\sqrt{5}}\times\zeta</math></small> |1.819~ |6.735~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° |{{radic|0.88~}} |<small><math>\sqrt{\psi/{2\phi}}</math></small> | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |124° |{{radic|3.12~}} |<small><math>\sqrt{4 - \psi/{2\phi}}</math></small> | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |0.939~ | |{{radic|3.118~}} |1.766~ | |- style="background: gainsboro;" | |0.939~ |3.477~ |<small><math>\sqrt{\psi\phi^3}\times\zeta</math></small> |1.766~ |6.538~ |<small><math>\sqrt{\chi\phi^5}\times\zeta</math></small>{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:<br> {{indent|4}}<math>11/\chi = \psi</math> <br> {{indent|4}}<math>\chi=(3\sqrt{5}+1)/2 \approx 3.854~</math> {{indent|4}}<math>\psi=(3\sqrt{5}-1)/2 \approx 2.854~</math>}} |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° |{{radic|1}} |<small><math>\sqrt{1}</math></small> | rowspan="3" |[[File:Great hexagon.png|100px]] | rowspan="3" |400 regular [[600-cell#Hexagons|great hexagons]]<br> (1200 great rectangles)<br>in 200 △ planes | rowspan="3" |4𝝅<br>[[600-cell#Hexagons and hexagrams|2{10/3}]]<br>#4 |120° |{{radic|3}} |<small><math>\sqrt{3}</math></small> | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |1 | |{{radic|3}} |1.732~ | |- style="background: palegreen;" | |1 |3.702~ |<small><math>\phi^2\sqrt{2}\times\zeta</math></small> |1.732~ |6.413~ |<small><math>\phi^2\sqrt{6}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° |{{radic|1.19~}} |<small><math>\sqrt{\chi/2\phi}</math></small> | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |113.9~° |{{radic|2.81~}} |<small><math>\sqrt{4 - \chi/2\phi}</math></small> | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |1.091~ | |{{radic|2.809~}} |1.676~ | |- style="background: gainsboro;" | |1.091~ |4.041~ |<small><math>\sqrt{\chi/\phi^3}\times\zeta</math></small> |1.676~ |6.205~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° |{{radic|1.31~}} |<small><math>\sqrt{\phi^2/2}</math></small> | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |110.2~° |{{radic|2.69~}} |<small><math>\sqrt{4 - \phi^2/2}</math></small> | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |1.144~ |<small><math>\phi/\sqrt{2}</math></small> |{{radic|2.691~}} |1.640~ | |- style="background: gainsboro;" | |1.144~ |4.236~ |<small><math>\phi^3\times\zeta</math></small> |1.640~ |6.074~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° |{{radic|1.618~}} |<small><math>\sqrt{3-\phi}</math></small> | rowspan="3" |[[File:Great pentagons rectangle.png|100px]] | rowspan="3" |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]<br>(3600 great rectangles)<br> in 720 <big>𝜙</big> planes | rowspan="3" |4𝝅<br>[[600-cell#Squares and octagrams|{24/5}]]<br>#9 |108° |{{radic|2.𝚽}} |<small><math>\sqrt{\phi^2}</math></small> | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |1.176~ |<small><math>\sqrt{\sqrt{5}/\phi}</math></small> |{{radic|2.618~}} |1.618~ |<small><math>\phi</math></small> |- style="background: yellow;" | |1.176~ |4.353~ |<small><math>\sqrt{2\phi^3\sqrt{5}}\times\zeta</math></small> |1.618~ |5.991~ |<small><math>\phi^3\sqrt{2}\times\zeta</math></small> |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° |{{radic|1.5}} |<small><math>\sqrt{3/2}</math></small> | rowspan="3" |[[File:Great 5-cell digons rectangle.png|100px]] | rowspan="3" |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]<br>(600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Pentagram|{5/2}]]<br>#8 |104.5~° |{{radic|2.5}} |<small><math>\sqrt{5/2}</math></small> | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |1.224~ | |{{radic|2.5}} |1.581~ | |- style="background: palegreen;" | |1.224~ |4.535~ |<small><math>\phi^2\sqrt{3}\times\zeta</math></small> |1.581~ |5.854~ |<small><math>\sqrt{5\phi^4}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° |{{radic|1.69~}} |<small><math>\sqrt{\tfrac{1}{4}(9-\sqrt{5})}</math></small> | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |98.9~° |{{radic|2.31~}} | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |1.300~ |<small><math>\tfrac{1}{2}\sqrt{9-\sqrt{5}}</math></small> |{{radic|2.309~}} |1.520~ | |- style="background: gainsboro;" | |1.300~ |4.815~ |<small><math>\text{‡}\times\zeta</math></small> |1.520~ |5.626~ |<small><math>\sqrt{\psi\phi^5}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° |{{radic|0.81~}} |<small><math>\sqrt{\tfrac{2\phi\sqrt{5}}{4}}</math></small> | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |95.5~° |{{radic|2.19~}} |<small><math>\sqrt{\tfrac{11-\sqrt{5}}{4}}</math></small> | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |1.345~ | |{{radic|2.191~}} |1.480~ | |- style="background: gainsboro;" | |1.345~ |4.980~ |<small><math>\sqrt{\phi^5\sqrt{5}}\times\zeta</math></small> |1.480~ |5.480~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |[[File:Great square rectangle.png|100px]] | rowspan="3" |4050 [[600-cell#Squares|great squares]]<br> in 4050 <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:30-gon#Triacontagram|{30/7}]]<br>#7 |90° |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |1.414~ | |{{radic|2}} |1.414~ | |- style="background: seashell;" | |1.414~ |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |1.414~ |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |} == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>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</math> The chord ratio <math>r_3=\sqrt{2}+1</math> 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: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. 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: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> 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: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ 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]] <small><math>\{3,3,4\}</math></small>. 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 <math>\sqrt{2}</math> 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: :<math>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</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> 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 <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, 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 <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> 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 <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> 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 <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> 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° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> 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 <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> 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 <small><math>\{4,3,3\}</math></small> 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]] <small><math>\{4,3,3\}</math></small>. 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 <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. 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: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> 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 <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> 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: :<math>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}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. The <math>r_1</math> and <math>r_5</math> 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 <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ 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 <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, 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. [[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]] The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> 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.]] [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of <math>r_7</math> edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference <math>6\pi</math> over <math>r_7</math> chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges. [[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of <math>r_9</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges. [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of <math>r_4</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_4</math> chords. Successive <math>r_4</math> chords are edges of different 24-cells. The rotational curve over each <math>r_4</math> chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges. In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 <math>\sqrt{1}</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == The 5-point (5-cell) 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> 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 & 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 & 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}} itup1oeev98d6icj8jn24medmkuth0s 2814958 2814957 2026-06-10T01:09:46Z Dc.samizdat 2856930 /* Complementary chord pairs */ 2814958 wikitext text/x-wiki = 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}} <blockquote>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.</blockquote> == 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="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "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]]."}} 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="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{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 5² 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & 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 <math>\phi^6</math> 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="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == Complementary chord pairs == The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 kinds of polyhedral section |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Irregular great hexagons of the 120-cell.png|100px]] | rowspan="3" |400 irregular great hexagons<br> (600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}} |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |1.982~ |<small><math>1 / \phi^2\sqrt{2}</math></small> |{{radic|3.927~}} |1.982~ |<small><math>\phi\sqrt{1.5}</math></small> |- style="background: palegreen;" | |0.270~ |1 |<small><math>1\times\zeta</math></small> |1.982~ |7.337~ |<small><math>\phi^3\sqrt{3}\times\zeta</math></small> |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° |{{radic|0.19~}} |<small><math>\sqrt{1/2\phi^2}</math></small> | rowspan="3" |[[File:25.2° × 154.8° chords great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:Triacontagon#Triacontagram|{30/13}]]<br>#13 |154.8~° |{{radic|3.81~}} | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |0.437~ |<small><math>1 / \phi\sqrt{2}</math></small> |{{radic|3.809~}} |1.952~ | |- style="background: gainsboro;" | |0.437~ |1.618~ |<small><math>\phi\times\zeta</math></small> |1.952~ |7.226~ |<small><math>\text{‡}\times\zeta</math></small> {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=<br>‡ For simplicity we omit the value of <math>a</math> whenever it is not mononomial in <math>\chi</math>, <math>\psi</math> and <math>\phi</math>.}} |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° |{{radic|0.𝚫}} |<small><math>\sqrt{1/\phi^2}</math></small> | rowspan="3" |[[File:Great decagon rectangle.png|100px]] | rowspan="3" |720 great decagons<br>(3600 great rectangles)<br>in 720 <big>𝜙</big> planes | rowspan="3" |5𝝅<br>[[600-cell#Decagons and pentadecagrams|{15/2}]]<br>#5 |144° |{{radic|3.𝚽}} |<small><math>\sqrt{2+\phi}</math></small> | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |0.618~ |<small><math>1 / \phi</math></small> |{{radic|3.618~}} |1.902~ |<small><math>1+1/{\phi^2}</math></small> |- style="background: yellow;" | |0.618~ |2.288~ |<small><math>\phi\sqrt{2}\times\zeta</math></small> |1.902~ |7.0425 |<small><math>\sqrt{2\phi^5\sqrt{5}}\times\zeta</math></small> |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° |{{radic|0.5}} |<small><math>\sqrt{1/2}</math></small> | rowspan="3" |[[File:√0.5 × √3.5 great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |138.6~° |{{radic|3.5}} |<small><math>\sqrt{7/2}</math></small> | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |0.707~ |<small><math>\sqrt{2}/2</math></small> |{{radic|3.5}} |1.871~ | |- style="background: gainsboro;" | |0.707~ |2.618~ |<small><math>\phi^2\times\zeta</math></small> |1.871~ |6.927~ |<small><math>\phi^2\sqrt{7}\times\zeta</math></small> |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° |{{radic|0.57~}} |<small><math>\sqrt{3/{2\phi^2}}</math></small> | rowspan="3" |[[File:Irregular great dodecagon.png|100px]] | rowspan="3" |200 irregular great dodecagons<br>(600 great rectangles)<br>in 200 △ planes | rowspan="3" | |135.5~° |{{radic|3.43~}} |<small><math>\sqrt{\phi^4/2}</math></small> | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |0.757~ |<small><math>\sqrt{3} / \phi\sqrt{2}</math></small> |{{radic|3.427~}} |1.851~ |<small><math>\phi^2 / \sqrt{2}</math></small> |- style="background: palegreen;" | |0.757~ |2.803~ |<small><math>\phi\sqrt{3}\times\zeta</math></small> |1.851~ |6.854~ |<small><math>\phi^4\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° |{{radic|0.69~}} |<small><math>\sqrt{\sqrt{5}/{2\phi}}</math></small> | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |130.9~° |{{radic|3.31~}} |<small><math>\sqrt{4 - \sqrt{5}/{2\phi}}</math></small> | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |0.831~ | |{{radic|3.309~}} |1.819~ | |- style="background: gainsboro;" | |0.831~ |3.078~ |<small><math>\sqrt{\phi^3\sqrt{5}}\times\zeta</math></small> |1.819~ |6.735~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° |{{radic|0.88~}} |<small><math>\sqrt{\psi/{2\phi}}</math></small> | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |124° |{{radic|3.12~}} |<small><math>\sqrt{4 - \psi/{2\phi}}</math></small> | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |0.939~ | |{{radic|3.118~}} |1.766~ | |- style="background: gainsboro;" | |0.939~ |3.477~ |<small><math>\sqrt{\psi\phi^3}\times\zeta</math></small> |1.766~ |6.538~ |<small><math>\sqrt{\chi\phi^5}\times\zeta</math></small>{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:<br> {{indent|4}}<math>11/\chi = \psi</math> <br> {{indent|4}}<math>\chi=(3\sqrt{5}+1)/2 \approx 3.854~</math> {{indent|4}}<math>\psi=(3\sqrt{5}-1)/2 \approx 2.854~</math>}} |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° |{{radic|1}} |<small><math>\sqrt{1}</math></small> | rowspan="3" |[[File:Great hexagon.png|100px]] | rowspan="3" |400 regular [[600-cell#Hexagons|great hexagons]]<br> (1200 great rectangles)<br>in 200 △ planes | rowspan="3" |4𝝅<br>[[600-cell#Hexagons and hexagrams|2{10/3}]]<br>#4 |120° |{{radic|3}} |<small><math>\sqrt{3}</math></small> | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |1 | |{{radic|3}} |1.732~ | |- style="background: palegreen;" | |1 |3.702~ |<small><math>\phi^2\sqrt{2}\times\zeta</math></small> |1.732~ |6.413~ |<small><math>\phi^2\sqrt{6}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° |{{radic|1.19~}} |<small><math>\sqrt{\chi/2\phi}</math></small> | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |113.9~° |{{radic|2.81~}} |<small><math>\sqrt{4 - \chi/2\phi}</math></small> | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |1.091~ | |{{radic|2.809~}} |1.676~ | |- style="background: gainsboro;" | |1.091~ |4.041~ |<small><math>\sqrt{\chi/\phi^3}\times\zeta</math></small> |1.676~ |6.205~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° |{{radic|1.31~}} |<small><math>\sqrt{\phi^2/2}</math></small> | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |110.2~° |{{radic|2.69~}} |<small><math>\sqrt{4 - \phi^2/2}</math></small> | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |1.144~ |<small><math>\phi/\sqrt{2}</math></small> |{{radic|2.691~}} |1.640~ | |- style="background: gainsboro;" | |1.144~ |4.236~ |<small><math>\phi^3\times\zeta</math></small> |1.640~ |6.074~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° |{{radic|1.618~}} |<small><math>\sqrt{3-\phi}</math></small> | rowspan="3" |[[File:Great pentagons rectangle.png|100px]] | rowspan="3" |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]<br>(3600 great rectangles)<br> in 720 <big>𝜙</big> planes | rowspan="3" |4𝝅<br>[[600-cell#Squares and octagrams|{24/5}]]<br>#9 |108° |{{radic|2.𝚽}} |<small><math>\sqrt{\phi^2}</math></small> | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |1.176~ |<small><math>\sqrt{\sqrt{5}/\phi}</math></small> |{{radic|2.618~}} |1.618~ |<small><math>\phi</math></small> |- style="background: yellow;" | |1.176~ |4.353~ |<small><math>\sqrt{2\phi^3\sqrt{5}}\times\zeta</math></small> |1.618~ |5.991~ |<small><math>\phi^3\sqrt{2}\times\zeta</math></small> |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° |{{radic|1.5}} |<small><math>\sqrt{3/2}</math></small> | rowspan="3" |[[File:Great 5-cell digons rectangle.png|100px]] | rowspan="3" |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]<br>(600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Pentagram|{5/2}]]<br>#8 |104.5~° |{{radic|2.5}} |<small><math>\sqrt{5/2}</math></small> | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |1.224~ | |{{radic|2.5}} |1.581~ | |- style="background: palegreen;" | |1.224~ |4.535~ |<small><math>\phi^2\sqrt{3}\times\zeta</math></small> |1.581~ |5.854~ |<small><math>\sqrt{5\phi^4}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° |{{radic|1.69~}} |<small><math>\sqrt{\tfrac{1}{4}(9-\sqrt{5})}</math></small> | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |98.9~° |{{radic|2.31~}} | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |1.300~ |<small><math>\tfrac{1}{2}\sqrt{9-\sqrt{5}}</math></small> |{{radic|2.309~}} |1.520~ | |- style="background: gainsboro;" | |1.300~ |4.815~ |<small><math>\text{‡}\times\zeta</math></small> |1.520~ |5.626~ |<small><math>\sqrt{\psi\phi^5}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° |{{radic|0.81~}} |<small><math>\sqrt{\tfrac{2\phi\sqrt{5}}{4}}</math></small> | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |95.5~° |{{radic|2.19~}} |<small><math>\sqrt{\tfrac{11-\sqrt{5}}{4}}</math></small> | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |1.345~ | |{{radic|2.191~}} |1.480~ | |- style="background: gainsboro;" | |1.345~ |4.980~ |<small><math>\sqrt{\phi^5\sqrt{5}}\times\zeta</math></small> |1.480~ |5.480~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |[[File:Great square rectangle.png|100px]] | rowspan="3" |4050 [[600-cell#Squares|great squares]]<br> in 4050 <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:30-gon#Triacontagram|{30/7}]]<br>#7 |90° |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |1.414~ | |{{radic|2}} |1.414~ | |- style="background: seashell;" | |1.414~ |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |1.414~ |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |} == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>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</math> The chord ratio <math>r_3=\sqrt{2}+1</math> 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: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. 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: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> 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: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ 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]] <small><math>\{3,3,4\}</math></small>. 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 <math>\sqrt{2}</math> 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: :<math>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</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> 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 <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, 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 <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> 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 <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> 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 <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> 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° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> 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 <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> 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 <small><math>\{4,3,3\}</math></small> 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]] <small><math>\{4,3,3\}</math></small>. 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 <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. 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: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> 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 <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> 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: :<math>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}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. The <math>r_1</math> and <math>r_5</math> 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 <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ 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 <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, 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. [[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]] The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> 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.]] [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of <math>r_7</math> edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference <math>6\pi</math> over <math>r_7</math> chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges. [[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of <math>r_9</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges. [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of <math>r_4</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_4</math> chords. Successive <math>r_4</math> chords are edges of different 24-cells. The rotational curve over each <math>r_4</math> chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges. In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 <math>\sqrt{1}</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == The 5-point (5-cell) 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> 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 & 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 & 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}} 70h90c769fvgw5chzo4s68cef74uuz6 2814959 2814958 2026-06-10T01:10:47Z Dc.samizdat 2856930 /* Complementary chord pairs */ 2814959 wikitext text/x-wiki = 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}} <blockquote>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.</blockquote> == 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="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "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]]."}} 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="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{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 5² 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & 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 <math>\phi^6</math> 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="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == Complementary chord pairs == The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 kinds of polyhedral section |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Irregular great hexagons of the 120-cell.png|100px]] | rowspan="3" |400 irregular great hexagons<br> (600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}} |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ |1 |<small><math>1\times\zeta</math></small> |1.982~ |7.337~ |<small><math>\phi^3\sqrt{3}\times\zeta</math></small> |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° |{{radic|0.19~}} |<small><math>\sqrt{1/2\phi^2}</math></small> | rowspan="3" |[[File:25.2° × 154.8° chords great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:Triacontagon#Triacontagram|{30/13}]]<br>#13 |154.8~° |{{radic|3.81~}} | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |0.437~ |<small><math>1 / \phi\sqrt{2}</math></small> |{{radic|3.809~}} |1.952~ | |- style="background: gainsboro;" | |0.437~ |1.618~ |<small><math>\phi\times\zeta</math></small> |1.952~ |7.226~ |<small><math>\text{‡}\times\zeta</math></small> {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=<br>‡ For simplicity we omit the value of <math>a</math> whenever it is not mononomial in <math>\chi</math>, <math>\psi</math> and <math>\phi</math>.}} |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° |{{radic|0.𝚫}} |<small><math>\sqrt{1/\phi^2}</math></small> | rowspan="3" |[[File:Great decagon rectangle.png|100px]] | rowspan="3" |720 great decagons<br>(3600 great rectangles)<br>in 720 <big>𝜙</big> planes | rowspan="3" |5𝝅<br>[[600-cell#Decagons and pentadecagrams|{15/2}]]<br>#5 |144° |{{radic|3.𝚽}} |<small><math>\sqrt{2+\phi}</math></small> | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |0.618~ |<small><math>1 / \phi</math></small> |{{radic|3.618~}} |1.902~ |<small><math>1+1/{\phi^2}</math></small> |- style="background: yellow;" | |0.618~ |2.288~ |<small><math>\phi\sqrt{2}\times\zeta</math></small> |1.902~ |7.0425 |<small><math>\sqrt{2\phi^5\sqrt{5}}\times\zeta</math></small> |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° |{{radic|0.5}} |<small><math>\sqrt{1/2}</math></small> | rowspan="3" |[[File:√0.5 × √3.5 great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |138.6~° |{{radic|3.5}} |<small><math>\sqrt{7/2}</math></small> | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |0.707~ |<small><math>\sqrt{2}/2</math></small> |{{radic|3.5}} |1.871~ | |- style="background: gainsboro;" | |0.707~ |2.618~ |<small><math>\phi^2\times\zeta</math></small> |1.871~ |6.927~ |<small><math>\phi^2\sqrt{7}\times\zeta</math></small> |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° |{{radic|0.57~}} |<small><math>\sqrt{3/{2\phi^2}}</math></small> | rowspan="3" |[[File:Irregular great dodecagon.png|100px]] | rowspan="3" |200 irregular great dodecagons<br>(600 great rectangles)<br>in 200 △ planes | rowspan="3" | |135.5~° |{{radic|3.43~}} |<small><math>\sqrt{\phi^4/2}</math></small> | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |0.757~ |<small><math>\sqrt{3} / \phi\sqrt{2}</math></small> |{{radic|3.427~}} |1.851~ |<small><math>\phi^2 / \sqrt{2}</math></small> |- style="background: palegreen;" | |0.757~ |2.803~ |<small><math>\phi\sqrt{3}\times\zeta</math></small> |1.851~ |6.854~ |<small><math>\phi^4\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° |{{radic|0.69~}} |<small><math>\sqrt{\sqrt{5}/{2\phi}}</math></small> | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |130.9~° |{{radic|3.31~}} |<small><math>\sqrt{4 - \sqrt{5}/{2\phi}}</math></small> | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |0.831~ | |{{radic|3.309~}} |1.819~ | |- style="background: gainsboro;" | |0.831~ |3.078~ |<small><math>\sqrt{\phi^3\sqrt{5}}\times\zeta</math></small> |1.819~ |6.735~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° |{{radic|0.88~}} |<small><math>\sqrt{\psi/{2\phi}}</math></small> | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |124° |{{radic|3.12~}} |<small><math>\sqrt{4 - \psi/{2\phi}}</math></small> | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |0.939~ | |{{radic|3.118~}} |1.766~ | |- style="background: gainsboro;" | |0.939~ |3.477~ |<small><math>\sqrt{\psi\phi^3}\times\zeta</math></small> |1.766~ |6.538~ |<small><math>\sqrt{\chi\phi^5}\times\zeta</math></small>{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:<br> {{indent|4}}<math>11/\chi = \psi</math> <br> {{indent|4}}<math>\chi=(3\sqrt{5}+1)/2 \approx 3.854~</math> {{indent|4}}<math>\psi=(3\sqrt{5}-1)/2 \approx 2.854~</math>}} |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° |{{radic|1}} |<small><math>\sqrt{1}</math></small> | rowspan="3" |[[File:Great hexagon.png|100px]] | rowspan="3" |400 regular [[600-cell#Hexagons|great hexagons]]<br> (1200 great rectangles)<br>in 200 △ planes | rowspan="3" |4𝝅<br>[[600-cell#Hexagons and hexagrams|2{10/3}]]<br>#4 |120° |{{radic|3}} |<small><math>\sqrt{3}</math></small> | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |1 | |{{radic|3}} |1.732~ | |- style="background: palegreen;" | |1 |3.702~ |<small><math>\phi^2\sqrt{2}\times\zeta</math></small> |1.732~ |6.413~ |<small><math>\phi^2\sqrt{6}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° |{{radic|1.19~}} |<small><math>\sqrt{\chi/2\phi}</math></small> | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |113.9~° |{{radic|2.81~}} |<small><math>\sqrt{4 - \chi/2\phi}</math></small> | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |1.091~ | |{{radic|2.809~}} |1.676~ | |- style="background: gainsboro;" | |1.091~ |4.041~ |<small><math>\sqrt{\chi/\phi^3}\times\zeta</math></small> |1.676~ |6.205~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° |{{radic|1.31~}} |<small><math>\sqrt{\phi^2/2}</math></small> | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |110.2~° |{{radic|2.69~}} |<small><math>\sqrt{4 - \phi^2/2}</math></small> | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |1.144~ |<small><math>\phi/\sqrt{2}</math></small> |{{radic|2.691~}} |1.640~ | |- style="background: gainsboro;" | |1.144~ |4.236~ |<small><math>\phi^3\times\zeta</math></small> |1.640~ |6.074~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° |{{radic|1.618~}} |<small><math>\sqrt{3-\phi}</math></small> | rowspan="3" |[[File:Great pentagons rectangle.png|100px]] | rowspan="3" |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]<br>(3600 great rectangles)<br> in 720 <big>𝜙</big> planes | rowspan="3" |4𝝅<br>[[600-cell#Squares and octagrams|{24/5}]]<br>#9 |108° |{{radic|2.𝚽}} |<small><math>\sqrt{\phi^2}</math></small> | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |1.176~ |<small><math>\sqrt{\sqrt{5}/\phi}</math></small> |{{radic|2.618~}} |1.618~ |<small><math>\phi</math></small> |- style="background: yellow;" | |1.176~ |4.353~ |<small><math>\sqrt{2\phi^3\sqrt{5}}\times\zeta</math></small> |1.618~ |5.991~ |<small><math>\phi^3\sqrt{2}\times\zeta</math></small> |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° |{{radic|1.5}} |<small><math>\sqrt{3/2}</math></small> | rowspan="3" |[[File:Great 5-cell digons rectangle.png|100px]] | rowspan="3" |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]<br>(600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Pentagram|{5/2}]]<br>#8 |104.5~° |{{radic|2.5}} |<small><math>\sqrt{5/2}</math></small> | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |1.224~ | |{{radic|2.5}} |1.581~ | |- style="background: palegreen;" | |1.224~ |4.535~ |<small><math>\phi^2\sqrt{3}\times\zeta</math></small> |1.581~ |5.854~ |<small><math>\sqrt{5\phi^4}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° |{{radic|1.69~}} |<small><math>\sqrt{\tfrac{1}{4}(9-\sqrt{5})}</math></small> | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |98.9~° |{{radic|2.31~}} | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |1.300~ |<small><math>\tfrac{1}{2}\sqrt{9-\sqrt{5}}</math></small> |{{radic|2.309~}} |1.520~ | |- style="background: gainsboro;" | |1.300~ |4.815~ |<small><math>\text{‡}\times\zeta</math></small> |1.520~ |5.626~ |<small><math>\sqrt{\psi\phi^5}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° |{{radic|0.81~}} |<small><math>\sqrt{\tfrac{2\phi\sqrt{5}}{4}}</math></small> | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |95.5~° |{{radic|2.19~}} |<small><math>\sqrt{\tfrac{11-\sqrt{5}}{4}}</math></small> | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |1.345~ | |{{radic|2.191~}} |1.480~ | |- style="background: gainsboro;" | |1.345~ |4.980~ |<small><math>\sqrt{\phi^5\sqrt{5}}\times\zeta</math></small> |1.480~ |5.480~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |[[File:Great square rectangle.png|100px]] | rowspan="3" |4050 [[600-cell#Squares|great squares]]<br> in 4050 <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:30-gon#Triacontagram|{30/7}]]<br>#7 |90° |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |1.414~ | |{{radic|2}} |1.414~ | |- style="background: seashell;" | |1.414~ |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |1.414~ |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |} == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>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</math> The chord ratio <math>r_3=\sqrt{2}+1</math> 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: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. 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: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> 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: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ 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]] <small><math>\{3,3,4\}</math></small>. 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 <math>\sqrt{2}</math> 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: :<math>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</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> 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 <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, 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 <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> 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 <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> 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 <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> 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° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> 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 <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> 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 <small><math>\{4,3,3\}</math></small> 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]] <small><math>\{4,3,3\}</math></small>. 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 <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. 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: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> 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 <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> 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: :<math>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}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. The <math>r_1</math> and <math>r_5</math> 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 <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ 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 <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, 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. [[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]] The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> 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.]] [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of <math>r_7</math> edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference <math>6\pi</math> over <math>r_7</math> chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges. [[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of <math>r_9</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges. [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of <math>r_4</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_4</math> chords. Successive <math>r_4</math> chords are edges of different 24-cells. The rotational curve over each <math>r_4</math> chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges. In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 <math>\sqrt{1}</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == The 5-point (5-cell) 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> 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 & 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 & 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}} hwn6ob2ov3ra8rj77uuvy8w7lleihg3 2814960 2814959 2026-06-10T01:11:44Z Dc.samizdat 2856930 /* Complementary chord pairs */ 2814960 wikitext text/x-wiki = 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}} <blockquote>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.</blockquote> == 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="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "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]]."}} 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="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{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 5² 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & 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 <math>\phi^6</math> 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="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == Complementary chord pairs == The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 kinds of polyhedral section |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Irregular great hexagons of the 120-cell.png|100px]] | rowspan="3" |400 irregular great hexagons<br> (600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}} |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:25.2° × 154.8° chords great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:Triacontagon#Triacontagram|{30/13}]]<br>#13 |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |0.437~ |<small><math>1 / \phi\sqrt{2}</math></small> |{{radic|3.809~}} |1.952~ | |- style="background: gainsboro;" | |0.437~ |1.618~ |<small><math>\phi\times\zeta</math></small> |1.952~ |7.226~ |<small><math>\text{‡}\times\zeta</math></small> {{Sfn|Coxeter|1973|pp=300-301|loc=footnote:|ps=<br>‡ For simplicity we omit the value of <math>a</math> whenever it is not mononomial in <math>\chi</math>, <math>\psi</math> and <math>\phi</math>.}} |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° |{{radic|0.𝚫}} |<small><math>\sqrt{1/\phi^2}</math></small> | rowspan="3" |[[File:Great decagon rectangle.png|100px]] | rowspan="3" |720 great decagons<br>(3600 great rectangles)<br>in 720 <big>𝜙</big> planes | rowspan="3" |5𝝅<br>[[600-cell#Decagons and pentadecagrams|{15/2}]]<br>#5 |144° |{{radic|3.𝚽}} |<small><math>\sqrt{2+\phi}</math></small> | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |0.618~ |<small><math>1 / \phi</math></small> |{{radic|3.618~}} |1.902~ |<small><math>1+1/{\phi^2}</math></small> |- style="background: yellow;" | |0.618~ |2.288~ |<small><math>\phi\sqrt{2}\times\zeta</math></small> |1.902~ |7.0425 |<small><math>\sqrt{2\phi^5\sqrt{5}}\times\zeta</math></small> |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° |{{radic|0.5}} |<small><math>\sqrt{1/2}</math></small> | rowspan="3" |[[File:√0.5 × √3.5 great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |138.6~° |{{radic|3.5}} |<small><math>\sqrt{7/2}</math></small> | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |0.707~ |<small><math>\sqrt{2}/2</math></small> |{{radic|3.5}} |1.871~ | |- style="background: gainsboro;" | |0.707~ |2.618~ |<small><math>\phi^2\times\zeta</math></small> |1.871~ |6.927~ |<small><math>\phi^2\sqrt{7}\times\zeta</math></small> |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° |{{radic|0.57~}} |<small><math>\sqrt{3/{2\phi^2}}</math></small> | rowspan="3" |[[File:Irregular great dodecagon.png|100px]] | rowspan="3" |200 irregular great dodecagons<br>(600 great rectangles)<br>in 200 △ planes | rowspan="3" | |135.5~° |{{radic|3.43~}} |<small><math>\sqrt{\phi^4/2}</math></small> | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |0.757~ |<small><math>\sqrt{3} / \phi\sqrt{2}</math></small> |{{radic|3.427~}} |1.851~ |<small><math>\phi^2 / \sqrt{2}</math></small> |- style="background: palegreen;" | |0.757~ |2.803~ |<small><math>\phi\sqrt{3}\times\zeta</math></small> |1.851~ |6.854~ |<small><math>\phi^4\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° |{{radic|0.69~}} |<small><math>\sqrt{\sqrt{5}/{2\phi}}</math></small> | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |130.9~° |{{radic|3.31~}} |<small><math>\sqrt{4 - \sqrt{5}/{2\phi}}</math></small> | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |0.831~ | |{{radic|3.309~}} |1.819~ | |- style="background: gainsboro;" | |0.831~ |3.078~ |<small><math>\sqrt{\phi^3\sqrt{5}}\times\zeta</math></small> |1.819~ |6.735~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° |{{radic|0.88~}} |<small><math>\sqrt{\psi/{2\phi}}</math></small> | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |124° |{{radic|3.12~}} |<small><math>\sqrt{4 - \psi/{2\phi}}</math></small> | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |0.939~ | |{{radic|3.118~}} |1.766~ | |- style="background: gainsboro;" | |0.939~ |3.477~ |<small><math>\sqrt{\psi\phi^3}\times\zeta</math></small> |1.766~ |6.538~ |<small><math>\sqrt{\chi\phi^5}\times\zeta</math></small>{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:<br> {{indent|4}}<math>11/\chi = \psi</math> <br> {{indent|4}}<math>\chi=(3\sqrt{5}+1)/2 \approx 3.854~</math> {{indent|4}}<math>\psi=(3\sqrt{5}-1)/2 \approx 2.854~</math>}} |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° |{{radic|1}} |<small><math>\sqrt{1}</math></small> | rowspan="3" |[[File:Great hexagon.png|100px]] | rowspan="3" |400 regular [[600-cell#Hexagons|great hexagons]]<br> (1200 great rectangles)<br>in 200 △ planes | rowspan="3" |4𝝅<br>[[600-cell#Hexagons and hexagrams|2{10/3}]]<br>#4 |120° |{{radic|3}} |<small><math>\sqrt{3}</math></small> | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |1 | |{{radic|3}} |1.732~ | |- style="background: palegreen;" | |1 |3.702~ |<small><math>\phi^2\sqrt{2}\times\zeta</math></small> |1.732~ |6.413~ |<small><math>\phi^2\sqrt{6}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° |{{radic|1.19~}} |<small><math>\sqrt{\chi/2\phi}</math></small> | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |113.9~° |{{radic|2.81~}} |<small><math>\sqrt{4 - \chi/2\phi}</math></small> | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |1.091~ | |{{radic|2.809~}} |1.676~ | |- style="background: gainsboro;" | |1.091~ |4.041~ |<small><math>\sqrt{\chi/\phi^3}\times\zeta</math></small> |1.676~ |6.205~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° |{{radic|1.31~}} |<small><math>\sqrt{\phi^2/2}</math></small> | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |110.2~° |{{radic|2.69~}} |<small><math>\sqrt{4 - \phi^2/2}</math></small> | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |1.144~ |<small><math>\phi/\sqrt{2}</math></small> |{{radic|2.691~}} |1.640~ | |- style="background: gainsboro;" | |1.144~ |4.236~ |<small><math>\phi^3\times\zeta</math></small> |1.640~ |6.074~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° |{{radic|1.618~}} |<small><math>\sqrt{3-\phi}</math></small> | rowspan="3" |[[File:Great pentagons rectangle.png|100px]] | rowspan="3" |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]<br>(3600 great rectangles)<br> in 720 <big>𝜙</big> planes | rowspan="3" |4𝝅<br>[[600-cell#Squares and octagrams|{24/5}]]<br>#9 |108° |{{radic|2.𝚽}} |<small><math>\sqrt{\phi^2}</math></small> | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |1.176~ |<small><math>\sqrt{\sqrt{5}/\phi}</math></small> |{{radic|2.618~}} |1.618~ |<small><math>\phi</math></small> |- style="background: yellow;" | |1.176~ |4.353~ |<small><math>\sqrt{2\phi^3\sqrt{5}}\times\zeta</math></small> |1.618~ |5.991~ |<small><math>\phi^3\sqrt{2}\times\zeta</math></small> |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° |{{radic|1.5}} |<small><math>\sqrt{3/2}</math></small> | rowspan="3" |[[File:Great 5-cell digons rectangle.png|100px]] | rowspan="3" |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]<br>(600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Pentagram|{5/2}]]<br>#8 |104.5~° |{{radic|2.5}} |<small><math>\sqrt{5/2}</math></small> | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |1.224~ | |{{radic|2.5}} |1.581~ | |- style="background: palegreen;" | |1.224~ |4.535~ |<small><math>\phi^2\sqrt{3}\times\zeta</math></small> |1.581~ |5.854~ |<small><math>\sqrt{5\phi^4}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° |{{radic|1.69~}} |<small><math>\sqrt{\tfrac{1}{4}(9-\sqrt{5})}</math></small> | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |98.9~° |{{radic|2.31~}} | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |1.300~ |<small><math>\tfrac{1}{2}\sqrt{9-\sqrt{5}}</math></small> |{{radic|2.309~}} |1.520~ | |- style="background: gainsboro;" | |1.300~ |4.815~ |<small><math>\text{‡}\times\zeta</math></small> |1.520~ |5.626~ |<small><math>\sqrt{\psi\phi^5}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° |{{radic|0.81~}} |<small><math>\sqrt{\tfrac{2\phi\sqrt{5}}{4}}</math></small> | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |95.5~° |{{radic|2.19~}} |<small><math>\sqrt{\tfrac{11-\sqrt{5}}{4}}</math></small> | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |1.345~ | |{{radic|2.191~}} |1.480~ | |- style="background: gainsboro;" | |1.345~ |4.980~ |<small><math>\sqrt{\phi^5\sqrt{5}}\times\zeta</math></small> |1.480~ |5.480~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |[[File:Great square rectangle.png|100px]] | rowspan="3" |4050 [[600-cell#Squares|great squares]]<br> in 4050 <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:30-gon#Triacontagram|{30/7}]]<br>#7 |90° |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |1.414~ | |{{radic|2}} |1.414~ | |- style="background: seashell;" | |1.414~ |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |1.414~ |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |} == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>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</math> The chord ratio <math>r_3=\sqrt{2}+1</math> 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: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. 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: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> 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: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ 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]] <small><math>\{3,3,4\}</math></small>. 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 <math>\sqrt{2}</math> 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: :<math>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</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> 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 <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, 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 <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> 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 <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> 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 <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> 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° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> 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 <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> 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 <small><math>\{4,3,3\}</math></small> 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]] <small><math>\{4,3,3\}</math></small>. 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 <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. 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: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> 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 <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> 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: :<math>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}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. The <math>r_1</math> and <math>r_5</math> 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 <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ 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 <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, 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. [[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]] The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> 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.]] [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of <math>r_7</math> edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference <math>6\pi</math> over <math>r_7</math> chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges. [[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of <math>r_9</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges. [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of <math>r_4</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_4</math> chords. Successive <math>r_4</math> chords are edges of different 24-cells. The rotational curve over each <math>r_4</math> chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges. In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 <math>\sqrt{1}</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == The 5-point (5-cell) 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> 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 & 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 & 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}} 2u3ytmdf4wwr1jwt3xwv8s7i0ud7nx8 2814961 2814960 2026-06-10T01:17:12Z Dc.samizdat 2856930 /* Complementary chord pairs */ 2814961 wikitext text/x-wiki = 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}} <blockquote>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.</blockquote> == 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="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "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]]."}} 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="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{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 5² 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & 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 <math>\phi^6</math> 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="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == Complementary chord pairs == The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 kinds of polyhedral section |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Irregular great hexagons of the 120-cell.png|100px]] | rowspan="3" |400 irregular great hexagons<br> (600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}} |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:25.2° × 154.8° chords great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:Triacontagon#Triacontagram|{30/13}]]<br>#13 |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Great decagon rectangle.png|100px]] | rowspan="3" |720 great decagons<br>(3600 great rectangles)<br>in 720 <big>𝜙</big> planes | rowspan="3" |5𝝅<br>[[600-cell#Decagons and pentadecagrams|{15/2}]]<br>#5 |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" |[[File:√0.5 × √3.5 great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Irregular great dodecagon.png|100px]] | rowspan="3" |200 irregular great dodecagons<br>(600 great rectangles)<br>in 200 △ planes | rowspan="3" | |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° |{{radic|0.69~}} |<small><math>\sqrt{\sqrt{5}/{2\phi}}</math></small> | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |130.9~° |{{radic|3.31~}} |<small><math>\sqrt{4 - \sqrt{5}/{2\phi}}</math></small> | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |0.831~ | |{{radic|3.309~}} |1.819~ | |- style="background: gainsboro;" | |0.831~ |3.078~ |<small><math>\sqrt{\phi^3\sqrt{5}}\times\zeta</math></small> |1.819~ |6.735~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° |{{radic|0.88~}} |<small><math>\sqrt{\psi/{2\phi}}</math></small> | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |124° |{{radic|3.12~}} |<small><math>\sqrt{4 - \psi/{2\phi}}</math></small> | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |0.939~ | |{{radic|3.118~}} |1.766~ | |- style="background: gainsboro;" | |0.939~ |3.477~ |<small><math>\sqrt{\psi\phi^3}\times\zeta</math></small> |1.766~ |6.538~ |<small><math>\sqrt{\chi\phi^5}\times\zeta</math></small>{{Sfn|Coxeter|1973|pp=300-301|loc=Table V (v) Simplified sections of {5,3,3} beginning with a vertex (see footnote ✼)|ps=:<br> {{indent|4}}<math>11/\chi = \psi</math> <br> {{indent|4}}<math>\chi=(3\sqrt{5}+1)/2 \approx 3.854~</math> {{indent|4}}<math>\psi=(3\sqrt{5}-1)/2 \approx 2.854~</math>}} |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° |{{radic|1}} |<small><math>\sqrt{1}</math></small> | rowspan="3" |[[File:Great hexagon.png|100px]] | rowspan="3" |400 regular [[600-cell#Hexagons|great hexagons]]<br> (1200 great rectangles)<br>in 200 △ planes | rowspan="3" |4𝝅<br>[[600-cell#Hexagons and hexagrams|2{10/3}]]<br>#4 |120° |{{radic|3}} |<small><math>\sqrt{3}</math></small> | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |1 | |{{radic|3}} |1.732~ | |- style="background: palegreen;" | |1 |3.702~ |<small><math>\phi^2\sqrt{2}\times\zeta</math></small> |1.732~ |6.413~ |<small><math>\phi^2\sqrt{6}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° |{{radic|1.19~}} |<small><math>\sqrt{\chi/2\phi}</math></small> | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |113.9~° |{{radic|2.81~}} |<small><math>\sqrt{4 - \chi/2\phi}</math></small> | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |1.091~ | |{{radic|2.809~}} |1.676~ | |- style="background: gainsboro;" | |1.091~ |4.041~ |<small><math>\sqrt{\chi/\phi^3}\times\zeta</math></small> |1.676~ |6.205~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° |{{radic|1.31~}} |<small><math>\sqrt{\phi^2/2}</math></small> | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |110.2~° |{{radic|2.69~}} |<small><math>\sqrt{4 - \phi^2/2}</math></small> | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |1.144~ |<small><math>\phi/\sqrt{2}</math></small> |{{radic|2.691~}} |1.640~ | |- style="background: gainsboro;" | |1.144~ |4.236~ |<small><math>\phi^3\times\zeta</math></small> |1.640~ |6.074~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° |{{radic|1.618~}} |<small><math>\sqrt{3-\phi}</math></small> | rowspan="3" |[[File:Great pentagons rectangle.png|100px]] | rowspan="3" |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]<br>(3600 great rectangles)<br> in 720 <big>𝜙</big> planes | rowspan="3" |4𝝅<br>[[600-cell#Squares and octagrams|{24/5}]]<br>#9 |108° |{{radic|2.𝚽}} |<small><math>\sqrt{\phi^2}</math></small> | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |1.176~ |<small><math>\sqrt{\sqrt{5}/\phi}</math></small> |{{radic|2.618~}} |1.618~ |<small><math>\phi</math></small> |- style="background: yellow;" | |1.176~ |4.353~ |<small><math>\sqrt{2\phi^3\sqrt{5}}\times\zeta</math></small> |1.618~ |5.991~ |<small><math>\phi^3\sqrt{2}\times\zeta</math></small> |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° |{{radic|1.5}} |<small><math>\sqrt{3/2}</math></small> | rowspan="3" |[[File:Great 5-cell digons rectangle.png|100px]] | rowspan="3" |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]<br>(600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Pentagram|{5/2}]]<br>#8 |104.5~° |{{radic|2.5}} |<small><math>\sqrt{5/2}</math></small> | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |1.224~ | |{{radic|2.5}} |1.581~ | |- style="background: palegreen;" | |1.224~ |4.535~ |<small><math>\phi^2\sqrt{3}\times\zeta</math></small> |1.581~ |5.854~ |<small><math>\sqrt{5\phi^4}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° |{{radic|1.69~}} |<small><math>\sqrt{\tfrac{1}{4}(9-\sqrt{5})}</math></small> | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |98.9~° |{{radic|2.31~}} | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |1.300~ |<small><math>\tfrac{1}{2}\sqrt{9-\sqrt{5}}</math></small> |{{radic|2.309~}} |1.520~ | |- style="background: gainsboro;" | |1.300~ |4.815~ |<small><math>\text{‡}\times\zeta</math></small> |1.520~ |5.626~ |<small><math>\sqrt{\psi\phi^5}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° |{{radic|0.81~}} |<small><math>\sqrt{\tfrac{2\phi\sqrt{5}}{4}}</math></small> | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |95.5~° |{{radic|2.19~}} |<small><math>\sqrt{\tfrac{11-\sqrt{5}}{4}}</math></small> | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |1.345~ | |{{radic|2.191~}} |1.480~ | |- style="background: gainsboro;" | |1.345~ |4.980~ |<small><math>\sqrt{\phi^5\sqrt{5}}\times\zeta</math></small> |1.480~ |5.480~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |[[File:Great square rectangle.png|100px]] | rowspan="3" |4050 [[600-cell#Squares|great squares]]<br> in 4050 <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:30-gon#Triacontagram|{30/7}]]<br>#7 |90° |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |1.414~ | |{{radic|2}} |1.414~ | |- style="background: seashell;" | |1.414~ |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |1.414~ |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |} == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>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</math> The chord ratio <math>r_3=\sqrt{2}+1</math> 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: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. 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: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> 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: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ 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]] <small><math>\{3,3,4\}</math></small>. 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 <math>\sqrt{2}</math> 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: :<math>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</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> 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 <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, 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 <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> 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 <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> 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 <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> 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° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> 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 <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> 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 <small><math>\{4,3,3\}</math></small> 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]] <small><math>\{4,3,3\}</math></small>. 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 <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. 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: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> 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 <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> 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: :<math>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}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. The <math>r_1</math> and <math>r_5</math> 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 <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ 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 <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, 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. [[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]] The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> 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.]] [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of <math>r_7</math> edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference <math>6\pi</math> over <math>r_7</math> chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges. [[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of <math>r_9</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges. [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of <math>r_4</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_4</math> chords. Successive <math>r_4</math> chords are edges of different 24-cells. The rotational curve over each <math>r_4</math> chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges. In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 <math>\sqrt{1}</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == The 5-point (5-cell) 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> 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 & 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 & 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}} 2g6fqln53h3ecv2q6ru8afiqoyiqkin 2814962 2814961 2026-06-10T01:19:33Z Dc.samizdat 2856930 /* Complementary chord pairs */ 2814962 wikitext text/x-wiki = 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}} <blockquote>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.</blockquote> == 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="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "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]]."}} 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="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{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 5² 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & 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 <math>\phi^6</math> 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="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == Complementary chord pairs == The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 kinds of polyhedral section |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Irregular great hexagons of the 120-cell.png|100px]] | rowspan="3" |400 irregular great hexagons<br> (600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}} |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:25.2° × 154.8° chords great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:Triacontagon#Triacontagram|{30/13}]]<br>#13 |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Great decagon rectangle.png|100px]] | rowspan="3" |720 great decagons<br>(3600 great rectangles)<br>in 720 <big>𝜙</big> planes | rowspan="3" |5𝝅<br>[[600-cell#Decagons and pentadecagrams|{15/2}]]<br>#5 |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" |[[File:√0.5 × √3.5 great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Irregular great dodecagon.png|100px]] | rowspan="3" |200 irregular great dodecagons<br>(600 great rectangles)<br>in 200 △ planes | rowspan="3" | |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° |{{radic|1}} |<small><math>\sqrt{1}</math></small> | rowspan="3" |[[File:Great hexagon.png|100px]] | rowspan="3" |400 regular [[600-cell#Hexagons|great hexagons]]<br> (1200 great rectangles)<br>in 200 △ planes | rowspan="3" |4𝝅<br>[[600-cell#Hexagons and hexagrams|2{10/3}]]<br>#4 |120° |{{radic|3}} |<small><math>\sqrt{3}</math></small> | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |1 | |{{radic|3}} |1.732~ | |- style="background: palegreen;" | |1 |3.702~ |<small><math>\phi^2\sqrt{2}\times\zeta</math></small> |1.732~ |6.413~ |<small><math>\phi^2\sqrt{6}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° |{{radic|1.19~}} |<small><math>\sqrt{\chi/2\phi}</math></small> | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |113.9~° |{{radic|2.81~}} |<small><math>\sqrt{4 - \chi/2\phi}</math></small> | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |1.091~ | |{{radic|2.809~}} |1.676~ | |- style="background: gainsboro;" | |1.091~ |4.041~ |<small><math>\sqrt{\chi/\phi^3}\times\zeta</math></small> |1.676~ |6.205~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° |{{radic|1.31~}} |<small><math>\sqrt{\phi^2/2}</math></small> | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |110.2~° |{{radic|2.69~}} |<small><math>\sqrt{4 - \phi^2/2}</math></small> | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |1.144~ |<small><math>\phi/\sqrt{2}</math></small> |{{radic|2.691~}} |1.640~ | |- style="background: gainsboro;" | |1.144~ |4.236~ |<small><math>\phi^3\times\zeta</math></small> |1.640~ |6.074~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° |{{radic|1.618~}} |<small><math>\sqrt{3-\phi}</math></small> | rowspan="3" |[[File:Great pentagons rectangle.png|100px]] | rowspan="3" |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]<br>(3600 great rectangles)<br> in 720 <big>𝜙</big> planes | rowspan="3" |4𝝅<br>[[600-cell#Squares and octagrams|{24/5}]]<br>#9 |108° |{{radic|2.𝚽}} |<small><math>\sqrt{\phi^2}</math></small> | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |1.176~ |<small><math>\sqrt{\sqrt{5}/\phi}</math></small> |{{radic|2.618~}} |1.618~ |<small><math>\phi</math></small> |- style="background: yellow;" | |1.176~ |4.353~ |<small><math>\sqrt{2\phi^3\sqrt{5}}\times\zeta</math></small> |1.618~ |5.991~ |<small><math>\phi^3\sqrt{2}\times\zeta</math></small> |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° |{{radic|1.5}} |<small><math>\sqrt{3/2}</math></small> | rowspan="3" |[[File:Great 5-cell digons rectangle.png|100px]] | rowspan="3" |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]<br>(600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Pentagram|{5/2}]]<br>#8 |104.5~° |{{radic|2.5}} |<small><math>\sqrt{5/2}</math></small> | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |1.224~ | |{{radic|2.5}} |1.581~ | |- style="background: palegreen;" | |1.224~ |4.535~ |<small><math>\phi^2\sqrt{3}\times\zeta</math></small> |1.581~ |5.854~ |<small><math>\sqrt{5\phi^4}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° |{{radic|1.69~}} |<small><math>\sqrt{\tfrac{1}{4}(9-\sqrt{5})}</math></small> | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |98.9~° |{{radic|2.31~}} | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |1.300~ |<small><math>\tfrac{1}{2}\sqrt{9-\sqrt{5}}</math></small> |{{radic|2.309~}} |1.520~ | |- style="background: gainsboro;" | |1.300~ |4.815~ |<small><math>\text{‡}\times\zeta</math></small> |1.520~ |5.626~ |<small><math>\sqrt{\psi\phi^5}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° |{{radic|0.81~}} |<small><math>\sqrt{\tfrac{2\phi\sqrt{5}}{4}}</math></small> | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |95.5~° |{{radic|2.19~}} |<small><math>\sqrt{\tfrac{11-\sqrt{5}}{4}}</math></small> | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |1.345~ | |{{radic|2.191~}} |1.480~ | |- style="background: gainsboro;" | |1.345~ |4.980~ |<small><math>\sqrt{\phi^5\sqrt{5}}\times\zeta</math></small> |1.480~ |5.480~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |[[File:Great square rectangle.png|100px]] | rowspan="3" |4050 [[600-cell#Squares|great squares]]<br> in 4050 <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:30-gon#Triacontagram|{30/7}]]<br>#7 |90° |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |1.414~ | |{{radic|2}} |1.414~ | |- style="background: seashell;" | |1.414~ |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |1.414~ |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |} == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>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</math> The chord ratio <math>r_3=\sqrt{2}+1</math> 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: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. 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: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> 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: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ 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]] <small><math>\{3,3,4\}</math></small>. 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 <math>\sqrt{2}</math> 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: :<math>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</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> 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 <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, 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 <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> 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 <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> 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 <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> 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° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> 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 <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> 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 <small><math>\{4,3,3\}</math></small> 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]] <small><math>\{4,3,3\}</math></small>. 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 <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. 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: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> 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 <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> 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: :<math>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}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. The <math>r_1</math> and <math>r_5</math> 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 <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ 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 <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, 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. [[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]] The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> 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.]] [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of <math>r_7</math> edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference <math>6\pi</math> over <math>r_7</math> chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges. [[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of <math>r_9</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges. [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of <math>r_4</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_4</math> chords. Successive <math>r_4</math> chords are edges of different 24-cells. The rotational curve over each <math>r_4</math> chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges. In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 <math>\sqrt{1}</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == The 5-point (5-cell) 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> 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 & 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 & 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}} kwacrzjvrzxwcr9u9avzfl4yey6nezj 2814963 2814962 2026-06-10T01:21:52Z Dc.samizdat 2856930 /* Complementary chord pairs */ 2814963 wikitext text/x-wiki = 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}} <blockquote>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.</blockquote> == 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="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "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]]."}} 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="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{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 5² 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & 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 <math>\phi^6</math> 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="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == Complementary chord pairs == The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 kinds of polyhedral section |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Irregular great hexagons of the 120-cell.png|100px]] | rowspan="3" |400 irregular great hexagons<br> (600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}} |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:25.2° × 154.8° chords great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:Triacontagon#Triacontagram|{30/13}]]<br>#13 |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Great decagon rectangle.png|100px]] | rowspan="3" |720 great decagons<br>(3600 great rectangles)<br>in 720 <big>𝜙</big> planes | rowspan="3" |5𝝅<br>[[600-cell#Decagons and pentadecagrams|{15/2}]]<br>#5 |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" |[[File:√0.5 × √3.5 great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Irregular great dodecagon.png|100px]] | rowspan="3" |200 irregular great dodecagons<br>(600 great rectangles)<br>in 200 △ planes | rowspan="3" | |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Great hexagon.png|100px]] | rowspan="3" |400 regular [[600-cell#Hexagons|great hexagons]]<br> (1200 great rectangles)<br>in 200 △ planes | rowspan="3" |4𝝅<br>[[600-cell#Hexagons and hexagrams|2{10/3}]]<br>#4 |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |1.144~ |<small><math>\phi/\sqrt{2}</math></small> |{{radic|2.691~}} |1.640~ | |- style="background: gainsboro;" | |1.144~ |4.236~ |<small><math>\phi^3\times\zeta</math></small> |1.640~ |6.074~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° |{{radic|1.618~}} |<small><math>\sqrt{3-\phi}</math></small> | rowspan="3" |[[File:Great pentagons rectangle.png|100px]] | rowspan="3" |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]<br>(3600 great rectangles)<br> in 720 <big>𝜙</big> planes | rowspan="3" |4𝝅<br>[[600-cell#Squares and octagrams|{24/5}]]<br>#9 |108° |{{radic|2.𝚽}} |<small><math>\sqrt{\phi^2}</math></small> | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |1.176~ |<small><math>\sqrt{\sqrt{5}/\phi}</math></small> |{{radic|2.618~}} |1.618~ |<small><math>\phi</math></small> |- style="background: yellow;" | |1.176~ |4.353~ |<small><math>\sqrt{2\phi^3\sqrt{5}}\times\zeta</math></small> |1.618~ |5.991~ |<small><math>\phi^3\sqrt{2}\times\zeta</math></small> |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° |{{radic|1.5}} |<small><math>\sqrt{3/2}</math></small> | rowspan="3" |[[File:Great 5-cell digons rectangle.png|100px]] | rowspan="3" |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]<br>(600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Pentagram|{5/2}]]<br>#8 |104.5~° |{{radic|2.5}} |<small><math>\sqrt{5/2}</math></small> | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |1.224~ | |{{radic|2.5}} |1.581~ | |- style="background: palegreen;" | |1.224~ |4.535~ |<small><math>\phi^2\sqrt{3}\times\zeta</math></small> |1.581~ |5.854~ |<small><math>\sqrt{5\phi^4}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° |{{radic|1.69~}} |<small><math>\sqrt{\tfrac{1}{4}(9-\sqrt{5})}</math></small> | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |98.9~° |{{radic|2.31~}} | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |1.300~ |<small><math>\tfrac{1}{2}\sqrt{9-\sqrt{5}}</math></small> |{{radic|2.309~}} |1.520~ | |- style="background: gainsboro;" | |1.300~ |4.815~ |<small><math>\text{‡}\times\zeta</math></small> |1.520~ |5.626~ |<small><math>\sqrt{\psi\phi^5}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° |{{radic|0.81~}} |<small><math>\sqrt{\tfrac{2\phi\sqrt{5}}{4}}</math></small> | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |95.5~° |{{radic|2.19~}} |<small><math>\sqrt{\tfrac{11-\sqrt{5}}{4}}</math></small> | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |1.345~ | |{{radic|2.191~}} |1.480~ | |- style="background: gainsboro;" | |1.345~ |4.980~ |<small><math>\sqrt{\phi^5\sqrt{5}}\times\zeta</math></small> |1.480~ |5.480~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |[[File:Great square rectangle.png|100px]] | rowspan="3" |4050 [[600-cell#Squares|great squares]]<br> in 4050 <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:30-gon#Triacontagram|{30/7}]]<br>#7 |90° |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |1.414~ | |{{radic|2}} |1.414~ | |- style="background: seashell;" | |1.414~ |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |1.414~ |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |} == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>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</math> The chord ratio <math>r_3=\sqrt{2}+1</math> 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: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. 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: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> 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: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ 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]] <small><math>\{3,3,4\}</math></small>. 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 <math>\sqrt{2}</math> 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: :<math>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</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> 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 <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, 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 <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> 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 <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> 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 <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> 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° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> 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 <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> 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 <small><math>\{4,3,3\}</math></small> 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]] <small><math>\{4,3,3\}</math></small>. 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 <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. 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: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> 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 <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> 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: :<math>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}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. The <math>r_1</math> and <math>r_5</math> 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 <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ 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 <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, 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. [[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]] The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> 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.]] [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of <math>r_7</math> edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference <math>6\pi</math> over <math>r_7</math> chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges. [[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of <math>r_9</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges. [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of <math>r_4</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_4</math> chords. Successive <math>r_4</math> chords are edges of different 24-cells. The rotational curve over each <math>r_4</math> chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges. In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 <math>\sqrt{1}</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == The 5-point (5-cell) 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> 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 & 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 & 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}} 7r20akopkumls0vn717sh6ccjmkgbp6 2814964 2814963 2026-06-10T01:23:09Z Dc.samizdat 2856930 /* Complementary chord pairs */ 2814964 wikitext text/x-wiki = 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}} <blockquote>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.</blockquote> == 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="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "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]]."}} 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="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{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 5² 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 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (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 & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & 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 <math>\phi^6</math> 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="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == Complementary chord pairs == The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>c_1</math> is a dodecahedron cell, and the largest section of radius <math>c_{15}</math> is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance <math>c_{29}</math>. ... {| class="wikitable" style="white-space:nowrap;text-align:center" ! colspan="11" |30 chords (15 180° pairs) make 15 kinds of polyhedral section |- ! colspan="5" |Short chord ! Polyhedron ! colspan="5" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | | | rowspan="3" | | rowspan="3" |600 vertices<br>(300 axes) | rowspan="3" | |180° | | | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} | | |{{radic|4}} | | |- style="background: palegreen;" | |0 | | |2 | | |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | | | rowspan="3" |[[File:Irregular great hexagons of the 120-cell.png|100px]] | rowspan="3" |400 irregular great hexagons<br> (600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Triacontagon#Triacontagram|{15/4}]]{{Efn|name=#4 isocline chord}} |164.5~° | | | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} | | |{{radic|3.927~}} | | |- style="background: palegreen;" | |0.270~ | | |1.982~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | | | rowspan="3" |[[File:25.2° × 154.8° chords great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:Triacontagon#Triacontagram|{30/13}]]<br>#13 |154.8~° | | | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} | | |{{radic|3.809~}} | | |- style="background: gainsboro;" | |0.437~ | | |1.952~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | | | rowspan="3" |[[File:Great decagon rectangle.png|100px]] | rowspan="3" |720 great decagons<br>(3600 great rectangles)<br>in 720 <big>𝜙</big> planes | rowspan="3" |5𝝅<br>[[600-cell#Decagons and pentadecagrams|{15/2}]]<br>#5 |144° | | | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} | | |{{radic|3.618~}} | | |- style="background: yellow;" | |0.618~ | | |1.902~ | | |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | | | rowspan="3" |[[File:√0.5 × √3.5 great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |138.6~° | | | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} | | |{{radic|3.5}} | | |- style="background: gainsboro;" | |0.707~ | | |1.871~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | | | rowspan="3" |[[File:Irregular great dodecagon.png|100px]] | rowspan="3" |200 irregular great dodecagons<br>(600 great rectangles)<br>in 200 △ planes | rowspan="3" | |135.5~° | | | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} | | |{{radic|3.427~}} | | |- style="background: palegreen;" | |0.757~ | | |1.851~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | | | rowspan="3" |[[File:49.1° × 130.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |130.9~° | | | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} | | |{{radic|3.309~}} | | |- style="background: gainsboro;" | |0.831~ | | |1.819~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | | | rowspan="3" |[[File:56° × 124° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br>in <big>☐</big> planes | rowspan="3" | |124° | | | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} | | |{{radic|3.118~}} | | |- style="background: gainsboro;" | |0.939~ | | |1.766~ | | |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | | | rowspan="3" |[[File:Great hexagon.png|100px]] | rowspan="3" |400 regular [[600-cell#Hexagons|great hexagons]]<br> (1200 great rectangles)<br>in 200 △ planes | rowspan="3" |4𝝅<br>[[600-cell#Hexagons and hexagrams|2{10/3}]]<br>#4 |120° | | | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} | | |{{radic|3}} | | |- style="background: palegreen;" | |1 | | |1.732~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | | | rowspan="3" |[[File:66.1° × 113.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |113.9~° | | | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} | | |{{radic|2.809~}} | | |- style="background: gainsboro;" | |1.091~ | | |1.676~ | | |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | | | rowspan="3" |[[File:69.8° × 110.2° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |110.2~° | | | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} | | |{{radic|2.691~}} | | |- style="background: gainsboro;" | |1.144~ | | |1.640~ | | |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | | | rowspan="3" |[[File:Great pentagons rectangle.png|100px]] | rowspan="3" |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]<br>(3600 great rectangles)<br> in 720 <big>𝜙</big> planes | rowspan="3" |4𝝅<br>[[600-cell#Squares and octagrams|{24/5}]]<br>#9 |108° | | | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |1.176~ |<small><math>\sqrt{\sqrt{5}/\phi}</math></small> |{{radic|2.618~}} |1.618~ |<small><math>\phi</math></small> |- style="background: yellow;" | |1.176~ |4.353~ |<small><math>\sqrt{2\phi^3\sqrt{5}}\times\zeta</math></small> |1.618~ |5.991~ |<small><math>\phi^3\sqrt{2}\times\zeta</math></small> |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° |{{radic|1.5}} |<small><math>\sqrt{3/2}</math></small> | rowspan="3" |[[File:Great 5-cell digons rectangle.png|100px]] | rowspan="3" |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]<br>(600 great rectangles)<br> in 200 △ planes | rowspan="3" |4𝝅{{Efn|name=isocline circumference}}<br>[[W:Pentagram|{5/2}]]<br>#8 |104.5~° |{{radic|2.5}} |<small><math>\sqrt{5/2}</math></small> | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |1.224~ | |{{radic|2.5}} |1.581~ | |- style="background: palegreen;" | |1.224~ |4.535~ |<small><math>\phi^2\sqrt{3}\times\zeta</math></small> |1.581~ |5.854~ |<small><math>\sqrt{5\phi^4}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° |{{radic|1.69~}} |<small><math>\sqrt{\tfrac{1}{4}(9-\sqrt{5})}</math></small> | rowspan="3" |[[File:81.1° × 98.9° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |98.9~° |{{radic|2.31~}} | | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |1.300~ |<small><math>\tfrac{1}{2}\sqrt{9-\sqrt{5}}</math></small> |{{radic|2.309~}} |1.520~ | |- style="background: gainsboro;" | |1.300~ |4.815~ |<small><math>\text{‡}\times\zeta</math></small> |1.520~ |5.626~ |<small><math>\sqrt{\psi\phi^5}\times\zeta</math></small> |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° |{{radic|0.81~}} |<small><math>\sqrt{\tfrac{2\phi\sqrt{5}}{4}}</math></small> | rowspan="3" |[[File:84.5° × 95.5° great rectangle.png|100px]] | rowspan="3" |Great rectangles<br> in <big>☐</big> planes | rowspan="3" | |95.5~° |{{radic|2.19~}} |<small><math>\sqrt{\tfrac{11-\sqrt{5}}{4}}</math></small> | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |1.345~ | |{{radic|2.191~}} |1.480~ | |- style="background: gainsboro;" | |1.345~ |4.980~ |<small><math>\sqrt{\phi^5\sqrt{5}}\times\zeta</math></small> |1.480~ |5.480~ |<small><math>\text{‡}\times\zeta</math></small> |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |[[File:Great square rectangle.png|100px]] | rowspan="3" |4050 [[600-cell#Squares|great squares]]<br> in 4050 <big>☐</big> planes | rowspan="3" |4𝝅<br>[[W:30-gon#Triacontagram|{30/7}]]<br>#7 |90° |{{radic|2}} |<small><math>\sqrt{2}</math></small> | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |1.414~ | |{{radic|2}} |1.414~ | |- style="background: seashell;" | |1.414~ |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |1.414~ |5.236~ |<small><math>2\phi^2\times\zeta</math></small> |} == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>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</math> The chord ratio <math>r_3=\sqrt{2}+1</math> 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: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. 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: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> 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: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> 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 <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ 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]] <small><math>\{3,3,4\}</math></small>. 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 <math>\sqrt{2}</math> 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: :<math>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</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> 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 <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, 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 <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> 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 <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> 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 <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> 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 <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> 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° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> 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 <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> 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 <small><math>\{4,3,3\}</math></small> 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]] <small><math>\{4,3,3\}</math></small>. 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 <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular double helix which visits each vertex once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. 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: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> 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 <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> 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: :<math>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}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. The <math>r_1</math> and <math>r_5</math> 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 <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ 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 <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of <math>\sqrt{2}</math> great square edges in the 24-cell.]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of <math>\sqrt{3}</math> great hexagon diagonals in the 24-cell.]] We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of <math>r_5</math> chords. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math> is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, 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. [[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]] The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> 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.]] [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of <math>r_7</math> edges.]] We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference <math>6\pi</math> over <math>r_7</math> chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the <math>r_7</math> chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges. [[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of <math>r_9</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference <math>10\pi</math> over <math>\sqrt{3}</math> chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.] Fontaine and Hurley's rotation over the <math>r_9</math> chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges. [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of <math>r_4</math> edges.]] We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of <math>r_4</math> chords. Successive <math>r_4</math> chords are edges of different 24-cells. The rotational curve over each <math>r_4</math> chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>\sqrt{1}</math> chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the <math>r_4</math> star polygon {30/8}=2{15/4} which constructs <math>1/r_4</math> is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges. In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 <math>\sqrt{1}</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. == The 5-point (5-cell) 4-simplex == In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... == Finally the 120-cell == The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. ... == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> 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 & 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 & 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}} lq4iell3pfhbjsskioes3uadtlkert4 0 330050 2814850 2814634 2026-06-09T12:29:39Z MathXplore 2888076 added [[Category:Economics]] using [[Help:Gadget-HotCat|HotCat]] 2814850 wikitext text/x-wiki See also [[OpenStax]] == <big>OpenStax</big> <big>Principles of</big> <big>Economics 3e</big> == == Summary == ''Principles of Economics 3e'' covers the scope and sequence of most introductory economics courses. The third edition takes a balanced approach to the theory and application of economics concepts. The text uses conversational language and ample illustrations to explore economic theories, and provides a wide array of examples using both fictional and real-world scenarios. The third edition has been carefully and thoroughly updated to reflect current data and understanding, as well as to provide a deeper background in diverse contributors and their impacts on economic thought and analysis. For example, the third edition highlights the research and views of a broader group of economists. Brief references and deeply explored socio-political examples have been updated to showcase the critical – and sometimes unnoticed – ties between economic developments and topics relevant to students. * [https://openstax.org/details/books/principles-economics-3e OpenStax Principles of Economics 3e] (original content). Available as pdf or web view. * [https://audileo.com/audiobooks/openstax/principles-of-economics-3e/ OpenStax Principles of Economics 3e audiobook] Available as audio textbook. [[Category:Economics]] gcbmj9gm9znlp2mwaud66xzkokepaa8 2 330058 2814932 2814693 2026-06-10T00:04:16Z AUBSTRAWBS 3060598 2814932 wikitext text/x-wiki {{userboxtop}} {{Gate anime}} {{userboxbottom}} Hello and welcome to my user page. I'm AUBSTRAWBS, I like history, art, aerospace and physics and especially any subject that somehow manages to combine them all. Also i really like anime which will be made obvious later on. == About me == I love divulging my personal info online lol. === Stuff === === Sports === === Projects === === Online presence === == People I think are cool == (Disclaimer I have not fully researched these people it is possible they have done bad things that I don't agree with. But with everything I have read about them it seems they be good.) === Marc Isambar(d/t) Brunel === I learnt about this man when i was looking for a subject for a presentation in school. He did many super cool things such as automating the production of pulley blocks, making super beautiful designs for the US Capitol building and building the first tunnel under the river Thames. Also he was the father of Isambard Kingdom Brunel and he has been unfortunately hella overshadowed by his son. [[File:CapitolBuildingMarcBrunel.jpg|thumb|center|250px|Marc Isambard Brunel's design of the capitol building]] == Jokes == === Stutter === Wanna hear a joke, a a a a man walks into a bar. The Bar tender: couldn't have said it better myself. === Cards === Wanna hear another joke, a horse walks into a bar, the bartender asks do you wanna hear a joke. The horse says yes, so the bartender pulls out the 2 of diamonds 3 of diamonds 4 of diamonds 5 of diamonds 6 of diamonds 7 of diamonds 8 of diamonds 9 of diamonds 10 of diamonds jack of diamonds queen of diamonds king of diamonds and the ace of diamonds then the 2 of clubs 3 of clubs 4 of clubs 5 of clubs 6 of clubs 7 of clubs 8 of clubs 9 of clubs 10 of clubs jack of clubs queen of clubs king of clubs and the ace of clubs then the 2 of spades 3 of spades 4 of spades 5 of spades 6 of spades 7 of spades 8 of spades 9 of spades 10 of spades jack of spades queen of spades king of spades and the ace of spades then the 2 of hearts 3 of hearts 4 of hearts 5 of hearts 6 of hearts 7 of hearts 8 of hearts 9 of hearts 10 of hearts jack of hearts queen of hearts king of hearts but I'm missing the ace of hearts can I have your heart to complete my set. The horse says neigh and walks away. [[File:Cat with open mouth.jpg|thumb|User's cat for user page]] {{WikiCookie}} gunvpxh4zgil2kavi2ee1tqncpvrz9v 3 330059 2814919 2814689 2026-06-09T20:08:51Z IanVG 2918363 2814919 wikitext text/x-wiki ==Welcome== {{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]], AUBSTRAWBS!'''|width=100%}} <div style="{{Robelbox/pad}}"> You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:Atcovi|me personally]] if you would like some [[Help:Contents|help]]. Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple. We invite you to [[Wikiversity:Be bold|be bold]] and [[Wikiversity|assume good faith]]. Please abide by our [[Wikiversity:Civility|civility]], [[Wikiversity:Privacy policy|privacy]], and [[Foundation:Terms of Use|terms of use]] policies. 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See you around Wikiversity! --—[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:43, 8 June 2026 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} === Welcome as well! === Great to see new and enthusiastic users! I'm working on [[Portal:Engineering and Technology|engineering]] and [[School:Mathematics|math]] topics if you are interested in contributing anywhere in those schools!--[[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 20:08, 9 June 2026 (UTC) pojuqmuhh1c1cbvqbi3641e8s2ipc4r 2814924 2814919 2026-06-09T23:07:40Z AUBSTRAWBS 3060598 /* Welcome as well! */ Reply 2814924 wikitext text/x-wiki ==Welcome== {{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]], AUBSTRAWBS!'''|width=100%}} <div style="{{Robelbox/pad}}"> You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:Atcovi|me personally]] if you would like some [[Help:Contents|help]]. Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple. We invite you to [[Wikiversity:Be bold|be bold]] and [[Wikiversity|assume good faith]]. Please abide by our [[Wikiversity:Civility|civility]], [[Wikiversity:Privacy policy|privacy]], and [[Foundation:Terms of Use|terms of use]] policies. To find your way around, check out: <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Wikiversity:Introduction|Introduction to Wikiversity]] * [[Help:Guides|Take a guided tour]] and learn [[Help:Editing|how to edit]] * [[Wikiversity:Browse|Browse]] or visit an educational level portal:<br>[[Portal:Pre-school Education|pre-school]] | [[Portal:Primary Education|primary]] | [[Portal:Secondary Education|secondary]] | [[Portal:Tertiary Education|tertiary]] | [[Portal:Non-formal Education|non-formal]] * [[Wikiversity:Introduction explore|Explore]] links in left-hand navigation menu </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] * Learn [[Help:How to write an educational resource|how to write an educational resource]] * Find out about [[Wikiversity:Research|research]] activities * Give [[Wikiversity:Feedback|feedback]] about your observations * Discuss issues or ask questions at the [[Wikiversity:Colloquium|colloquium]] </div> <br clear="both"/> To get started, experiment in the [[wikiversity:sandbox|sandbox]] or on [[special:mypage|your userpage]]. See you around Wikiversity! --—[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:43, 8 June 2026 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} === Welcome as well! === Great to see new and enthusiastic users! I'm working on [[Portal:Engineering and Technology|engineering]] and [[School:Mathematics|math]] topics if you are interested in contributing anywhere in those schools!--[[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 20:08, 9 June 2026 (UTC) :Im very exited to start editing and i also like engineering and maths might be a week until i do anything serious as have finals this week and i also spilt a drink down my computer which has had an effect on its performance. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 23:07, 9 June 2026 (UTC) bl3wgpjb3pvnz4vuycty5izrespqim0 10 330087 2814855 2026-06-09T12:44:27Z Atcovi 276019 Create. 2814855 wikitext text/x-wiki {{userbox | border-c = #093A80 | id = [[File:Rubin University-College Icon (rubin-University-College).jpg|45px]] | id-c = white | id-fc = #8cf | id-s = 19 | info = This user is a '''[[University students|university student]]'''. | info-c = #8CF | info-op = text-align:center; | nocat = {{{nocat|}}} }}<noinclude> [[Category:School user templates|Canberra]] </noinclude> b2il7xgle8nvt1h04bn4og2njld507s 0 330088 2814858 2026-06-09T13:26:22Z Projet PEP 3002502 Created page with "== Starting activity == * Map out your linguistic background on a timeline, noting when you learnt or came across each language or variety you know. * From there, identify which of the languages or varieties listed you use in your daily life (to a greater or lesser extent) and specify the contexts in which you use them (private, public, professional, digital, cultural, religious…). This will give you your current ''[[Dominant language constellation]]'' (DLC). * Now th..." 2814858 wikitext text/x-wiki == Starting activity == * Map out your linguistic background on a timeline, noting when you learnt or came across each language or variety you know. * From there, identify which of the languages or varieties listed you use in your daily life (to a greater or lesser extent) and specify the contexts in which you use them (private, public, professional, digital, cultural, religious…). This will give you your current ''[[Dominant language constellation]]'' (DLC). * Now think about the following questions: ** Is there one (or more) language or variety that clearly dominates? ** Are there contexts where you switch between languages or varieties, or mix them together? ** Compare your current DLC with your DLC from an earlier point in time. What has changed? Are there any languages/varieties that you once learned but hardly ever use or no longer use at all? * Given all these formalised elements, what is your view (or views) of your plurilingualism? == Objectives == By the end of this section, you should be able to: * recognise that there are different views on plurilingualism; * understand the different stages of the multilingual turn; * understand the implications that this shift towards multilingualism has on language teaching and learning. == Keywords == monolingualism – plurilingualism – multilingualism – approaches to plurilingualism == Introduction == The multilingual turn should be seen as a major development in the way languages and their uses have been viewed since the 1980s. The term ‘turn’ (May, 2013) is used in the sense that plurilingual approaches to teaching have moved away from a compartmentalised, static view of languages – which was previously emphasised – towards a more dynamic, integrated and individualised conception of the linguistic and cultural diversity of individuals and communities. == History == For a long time, in language teaching theory underpinned by research, monolingualism (with a dominant language) was contrasted with bilingualism (conceived as perfect and symmetrical mastery of two languages, for example, as described by Bloomfield, 1993, Penfield & Roberts, 1959, and the code-switching theory) or multilingualism (the presence of several languages within a society or in an individual’s practice, but in a compartmentalised manner). The multilingual turn, which reached its peak in the 2010s, marked a transition from approaches emphasising the need to isolate languages for learning (monolingual immersion was the norm) to the development of a more open, inclusive perspective that takes into account the plurality – particularly the linguistic diversity – of learners. The ideal of the native speaker has thus been set aside as a central element of various teaching approaches and methods (the direct method, the audio-oral method, etc.), in favour of a greater focus on the plurilingual learner. We are therefore at a turning point that forms part of a critical perspective on language teaching and learning. In support of the concept of plurilingual and inter/transcultural competence developed notably under the auspices of the Council of Europe through the Common European Framework of Reference for Languages (CEFR; Council of Europe, 2001), the multilingual turn has become institutionalised, initially nourished by the work of pioneers such as Grosjean (1982, 1993) and Lüdi & Py (1986). As Macaire (2025, p. 315) points out, « le tournant plurilingue se décline selon une triple orientation en éducation. La première explore une compétence unifiante ; la deuxième en discute les enjeux pour l’éducation ; la troisième s’intéresse à l’apprenant-sujet dans l’écologie des diversités ».  – “The shift towards multilingualism takes three distinct forms in education. The first explores a unifying competency; the second examines the implications for education; the third focuses on the individual learner within the context of diversity.” [own translation] == Conceptualisation of the multilingual turn == This turning point marks a conceptual shift centred on four key dimensions, which completely transform the way in which we view multilingualism and the associated multiculturalism. '''An integrated view of languages''': The multilingual turn recognises that an individual’s languages and language varieties are not compartmentalised, but form a single, evolving repertoire from which they draw depending on context and needs. We are moving towards a unifying approach to language competencies. '''Dynamic dimension''': Languages are used in a complementary manner, alternately, and sometimes mixed together depending on the social, educational or family contexts, or on the needs of speakers in various areas of use (see the concept of multiliteracy; Cook, 1991). '''Recognition of partial language competence''': One of the hallmarks of the shift towards plurilingualism lies in the recognition of all language competencies, whether partial or receptive, moving away from a view of plurilingualism that aims to achieve ‘native’ proficiency in every language. Being plurilingual should no longer be equated with being equally competent in each language one knows or uses. '''Relationship with cultural diversity''': Another dimension is involved, namely the cultural dimension, through the recognition of multiple cultural identities and the ability to navigate between different cultures. The multicultural dimension is valued. == Educational and social implications == '''A conceptual paradigm shift''': As the term suggests, the multilingual turn represents a paradigm shift, in the sense that it offers a new perspective on linguistic and cultural diversity. We are thus moving away from viewing multilingualism as an obstacle towards recognising it as a source of richness and an asset for learning languages and citizenship. This conceptual shift inevitably leads to changes in educational practices, such as recognising the value of learners' linguistic repertoires and developing innovative teaching methods. '''Valuing linguistic repertoires''': The multilingual turn is prompting schools and society to recognise and actively foster learners’ linguistic repertoires, enabling learners  to draw on and develop them – especially those from plurilingual families, including migrant backgrounds. This includes activities built around [[Language biography and identity texts|language biographies]] (Molinié, 2006; Auger, 2024), which contribute to this promotion. '''Innovative teaching approaches''': Taking this into account makes it necessary to implement new teaching approaches, particularly within the framework of [[Pluralistic approach|pluralistic approaches]] (Integrated didactic approach to languages, Awakening to languages, intercomprehension, and the Intercultural approach). Teaching approaches based on  [[translanguaging]] (García & Lin, 2017) are also fully part of this shift. The first four approaches can incorporate students’ home languages or other languages they know – whether learnt as foreign or second languages, or acquired in other ways – in order to build on them,  for example by comparing languages, and to promote practices aligned with [[language inclusion]] (Auger, 2020). '''Institutional recognition''': Despite this shift and the arguments associated with it, the promotion of plurilingualism sometimes struggles to take shape within school systems, for various reasons. Consequently, the idea of a pluralistic society is often challenged by political and/or individual ideologies, as well as the strongly held, identity-based views of certain governments. The value placed on [[English as a Lingua Franca (ELF)|English as a lingua franca]] may also contribute to this difficulty in recognition. Even when states are willing to promote this diversity, implementation is complex, due to a lack of teacher training and suitable educational resources and tools. == Take-home messages == The multilingual turn refers to a shift from an additive, compartmentalised, and static view of languages to a dynamic, integrated and value-affirming conception of individuals’ and societies’ linguistic and cultural diversity. It invites us to rethink education, citizenship, and language policies in an era of globalisation and mobility. The multilingual turn also challenges the traditional notions of mother tongue/first language and foreign language, encouraging us to focus more on linguistic repertoires and less on individual languages. == Self-assessment == '''A- The multilingual turn holds that:''' # plurilingualism is a “handicap” for language learning; # plurilingualism is an obstacle to the development of citizenship; # plurilingualism is positive only if each language is mastered equally; # plurilingualism is an asset for language learning. '''B- From the perspective of pedagogical practices in language learning, following the multilingual turn, it is considered:''' # counterproductive to emphasize learners’ home languages ; # necessary to build on the languages learners already know; # that a foreign language should only be learnt through full immersion; # inadvisable to rely on translation or on simply linking to other languages. '''C- With the multilingual turn, in the context of language teaching and learning, we have moved:''' # from taking learners’ languages into account to focusing solely on the target language; # from a compartmentalised view of languages to a unified view; # from the necessary consideration of learners’ languages to a purely immersive target-language approach; # from a holistic view of languages to a compartmentalised one. '''D- To implement the educational guidelines resulting from the multilingual turn, it is necessary''' # that all teachers are familiar with all their students’ languages; # that learners of the same origin be grouped in the same classes; # that teachers be trained to implement pluralistic approaches; # that teachers use AI-based automatic translators. == Resources to go further == * Conteh, J., & Meier, G, (Eds.) (2014). ''The multilingual turn in language education: Opportunities and challenges''. Multilingual Matters. * Cook, V. J. (1991). The poverty-of-the-stimulus argument and multi-competence. ''Second Language Research'', ''7''(2), 103–17. * Finex, N., & Makalela, L. (2021). ''Decolonising multilingualism in Africa: Recentering silenced voices from the global south''. Multilingual Matters. https://doi.org/10.21832/NDHLOV3354 * Kramsch, C. (2022). Afterword: The multilingual turn in language teacher education. ''Language and Education'', ''36''(5), 467–471. https://doi.org/10.1080/09500782.2022.2118542   == Bibliography == Auger, N. (2020). Enseigner à des élèves plurilingues: vers une didactique inclusive. In C. Mendonça-Dias, B. Azaoui, & F. Chnane-Davin (Eds.), ''Allophonie. Inclusion et langues des enfants migrants à l’école'' (pp. 171–183). Éditions Lambert-Lucas. Auger, N. (2024). ''Le diamant langagier''. French, English subtitles: (1h47) https://www.youtube.com/watch?v=artl-xcTDJc     Bloomfield, L. (1933). ''Language''. Holt. Coste, D., Moore, D., & Zarate, G. (1997, 2009). ''Compétence plurilingue et pluriculturelle, Vers un Cadre Européen Commun de Référence pour l'enseignement et l'apprentissage des langues vivantes: études préparatoires.'' Conseil de l’Europe, Division des politiques linguistiques. https://rm.coe.int/168069d29c Council of Europe (Ed.). (2001). ''Common European framework of reference for languages: Learning, teaching, assessment''. Council of Europe Publishing. https://rm.coe.int/16802fc1bf García, O., & Lin, A.M.Y. (2017). Translanguaging in bilingual education. In O. García, A.M.Y. Lin., & S. May (Ed.). ''Bilingual and multilingual education''. Springer. https://doi.org/10.1007/978-3-319-02258-1_9 Grosjean, F (1982). ''Life with two Languages: An introduction to bilingualism''. Harvard University Press. Grosjean, F. (1993). Le bilinguisme et le biculturalisme. Essai de définition. ''Travaux neuchâtelois de linguistique'', (19), 13–41. https://doi.org/10.26034/tranel.1993.2342 Lüdi, G., & Py, B. (2003). ''Être bilingue'' (3rd revised edition). Lang. Lüdi, G., & Py, B. (2009). To be or not to be... a plurilingual speaker. ''International Journal of Multilingualism'', ''6''(2), 154–167. https://doi.org/10.1080/14790710902846715 Macaire, D. (2025). Le tournant plurilingue. In C. Ollivier & S. Melo-Pfeifer (Eds.), ''Encyclopédie de l'éducation plurilingue'' (pp. 305–307).  Peter Lang. https://doi.org/10.3726/b23455   May, S. (2013). ''The multilingual turn : Implications for SLA, TESOL, and bilingual education.'' Taylor & Francis Group. Meier, G. S. (2017).  The multilingual turn as a critical movement in education: assumptions, challenges and a need for reflection. ''Applied Linguistics Review'', ''8''(1), 131–161. https://doi.org/10.1515/applirev-2016-2010 Melo-Pfeifer, S. (2018). The multilingual turn in foreign language education. Facts and fallacies. In A. Bonnet & P. Siemund (Eds.), ''Foreign language education in multilingual classrooms'' (pp. 191–210). John Benjamins. https://doi.org/10.1075/hsld.7 Molinié, M. (2006). Une approche biographique des trajectoires linguistiques et culturelles. In M. Molinié (Ed.), Biographie ''langagière et apprentissage plurilingue. Le français dans le monde. Recherches et applications, 39'', 8–11. https://halshs.archives-ouvertes.fr/halshs-01224908/document Montagne-Macaire, D. (2008). D’une didactique des langues à une didactique des plurilinguismes ? Réflexions pour la recherche. ''Recherches en didactique des langues et des cultures'', ''5''(1), 2–40. https://doi.org/10.4000/rdlc.6245 Moore, D., & Gajo, L. (2009). Introduction. French voices on plurilingualism and pluriculturalism: Significance and perspectives. ''International Journal of Multilingualism'', ''6''(2), 137–153. https://doi.org/10.1080/14790710902846707 Penfield, W., & Roberts, L. (1959). ''Speech and brain mechanisms''. Princeton University Press. 669hujzihsq3itpl3qkahd4i2r1n28c 2814859 2814858 2026-06-09T13:31:50Z Projet PEP 3002502 2814859 wikitext text/x-wiki == Starting activity == * Map out your linguistic background on a timeline, noting when you learnt or came across each language or variety you know. * From there, identify which of the languages or varieties listed you use in your daily life (to a greater or lesser extent) and specify the contexts in which you use them (private, public, professional, digital, cultural, religious…). This will give you your current ''[[Dominant language constellation]]'' (DLC). * Now think about the following questions: ** Is there one (or more) language or variety that clearly dominates? ** Are there contexts where you switch between languages or varieties, or mix them together? ** Compare your current DLC with your DLC from an earlier point in time. What has changed? Are there any languages/varieties that you once learned but hardly ever use or no longer use at all? * Given all these formalised elements, what is your view (or views) of your plurilingualism? == Objectives == By the end of this section, you should be able to: * recognise that there are different views on plurilingualism; * understand the different stages of the multilingual turn; * understand the implications that this shift towards multilingualism has on language teaching and learning. == Keywords == monolingualism – plurilingualism – multilingualism – approaches to plurilingualism == Introduction == The multilingual turn should be seen as a major development in the way languages and their uses have been viewed since the 1980s. The term ‘turn’ (May, 2013) is used in the sense that plurilingual approaches to teaching have moved away from a compartmentalised, static view of languages – which was previously emphasised – towards a more dynamic, integrated and individualised conception of the linguistic and cultural diversity of individuals and communities. == History == For a long time, in language teaching theory underpinned by research, monolingualism (with a dominant language) was contrasted with bilingualism (conceived as perfect and symmetrical mastery of two languages, for example, as described by Bloomfield, 1993, Penfield & Roberts, 1959, and the code-switching theory) or multilingualism (the presence of several languages within a society or in an individual’s practice, but in a compartmentalised manner). The multilingual turn, which reached its peak in the 2010s, marked a transition from approaches emphasising the need to isolate languages for learning (monolingual immersion was the norm) to the development of a more open, inclusive perspective that takes into account the plurality – particularly the linguistic diversity – of learners. The ideal of the native speaker has thus been set aside as a central element of various teaching approaches and methods (the direct method, the audio-oral method, etc.), in favour of a greater focus on the plurilingual learner. We are therefore at a turning point that forms part of a critical perspective on language teaching and learning. In support of the concept of plurilingual and inter/transcultural competence developed notably under the auspices of the Council of Europe through the Common European Framework of Reference for Languages (CEFR; Council of Europe, 2001), the multilingual turn has become institutionalised, initially nourished by the work of pioneers such as Grosjean (1982, 1993) and Lüdi & Py (1986). As Macaire (2025, p. 315) points out, « le tournant plurilingue se décline selon une triple orientation en éducation. La première explore une compétence unifiante ; la deuxième en discute les enjeux pour l’éducation ; la troisième s’intéresse à l’apprenant-sujet dans l’écologie des diversités ».  – “The shift towards multilingualism takes three distinct forms in education. The first explores a unifying competency; the second examines the implications for education; the third focuses on the individual learner within the context of diversity.” [own translation] == Conceptualisation of the multilingual turn == This turning point marks a conceptual shift centred on four key dimensions, which completely transform the way in which we view multilingualism and the associated multiculturalism. '''An integrated view of languages''': The multilingual turn recognises that an individual’s languages and language varieties are not compartmentalised, but form a single, evolving repertoire from which they draw depending on context and needs. We are moving towards a unifying approach to language competencies. '''Dynamic dimension''': Languages are used in a complementary manner, alternately, and sometimes mixed together depending on the social, educational or family contexts, or on the needs of speakers in various areas of use (see the concept of multiliteracy; Cook, 1991). '''Recognition of partial language competence''': One of the hallmarks of the shift towards plurilingualism lies in the recognition of all language competencies, whether partial or receptive, moving away from a view of plurilingualism that aims to achieve ‘native’ proficiency in every language. Being plurilingual should no longer be equated with being equally competent in each language one knows or uses. '''Relationship with cultural diversity''': Another dimension is involved, namely the cultural dimension, through the recognition of multiple cultural identities and the ability to navigate between different cultures. The multicultural dimension is valued. == Educational and social implications == '''A conceptual paradigm shift''': As the term suggests, the multilingual turn represents a paradigm shift, in the sense that it offers a new perspective on linguistic and cultural diversity. We are thus moving away from viewing multilingualism as an obstacle towards recognising it as a source of richness and an asset for learning languages and citizenship. This conceptual shift inevitably leads to changes in educational practices, such as recognising the value of learners' linguistic repertoires and developing innovative teaching methods. '''Valuing linguistic repertoires''': The multilingual turn is prompting schools and society to recognise and actively foster learners’ linguistic repertoires, enabling learners  to draw on and develop them – especially those from plurilingual families, including migrant backgrounds. This includes activities built around [[Language biography and identity texts|language biographies]] (Molinié, 2006; Auger, 2024), which contribute to this promotion. '''Innovative teaching approaches''': Taking this into account makes it necessary to implement new teaching approaches, particularly within the framework of [[Pluralistic approach|pluralistic approaches]] (Integrated didactic approach to languages, Awakening to languages, intercomprehension, and the Intercultural approach). Teaching approaches based on  [[translanguaging]] (García & Lin, 2017) are also fully part of this shift. The first four approaches can incorporate students’ home languages or other languages they know – whether learnt as foreign or second languages, or acquired in other ways – in order to build on them,  for example by comparing languages, and to promote practices aligned with [[language inclusion]] (Auger, 2020). '''Institutional recognition''': Despite this shift and the arguments associated with it, the promotion of plurilingualism sometimes struggles to take shape within school systems, for various reasons. Consequently, the idea of a pluralistic society is often challenged by political and/or individual ideologies, as well as the strongly held, identity-based views of certain governments. The value placed on [[English as a Lingua Franca (ELF)|English as a lingua franca]] may also contribute to this difficulty in recognition. Even when states are willing to promote this diversity, implementation is complex, due to a lack of teacher training and suitable educational resources and tools. == Take-home messages == The multilingual turn refers to a shift from an additive, compartmentalised, and static view of languages to a dynamic, integrated and value-affirming conception of individuals’ and societies’ linguistic and cultural diversity. It invites us to rethink education, citizenship, and language policies in an era of globalisation and mobility. The multilingual turn also challenges the traditional notions of mother tongue/first language and foreign language, encouraging us to focus more on linguistic repertoires and less on individual languages. == Self-assessment == <quiz> {The multilingual turn holds that:} - plurilingualism is a “handicap” for language learning; - plurilingualism is an obstacle to the development of citizenship; - plurilingualism is positive only if each language is mastered equally; + plurilingualism is an asset for language learning. {From the perspective of pedagogical practices in language learning, following the multilingual turn, it is considered:} - counterproductive to emphasize learners’ home languages ; + necessary to build on the languages learners already know; - that a foreign language should only be learnt through full immersion; + inadvisable to rely on translation or on simply linking to other languages. {With the multilingual turn, in the context of language teaching and learning, we have moved:} - from taking learners’ languages into account to focusing solely on the target language; + from a compartmentalised view of languages to a unified view; - from the necessary consideration of learners’ languages to a purely immersive target-language approach; - from a holistic view of languages to a compartmentalised one. {To implement the educational guidelines resulting from the multilingual turn, it is necessary} - that all teachers are familiar with all their students’ languages; + that learners of the same origin be grouped in the same classes; - that teachers be trained to implement pluralistic approaches; - that teachers use AI-based automatic translators. </quiz> == Resources to go further == * Conteh, J., & Meier, G, (Eds.) (2014). ''The multilingual turn in language education: Opportunities and challenges''. Multilingual Matters. * Cook, V. J. (1991). The poverty-of-the-stimulus argument and multi-competence. ''Second Language Research'', ''7''(2), 103–17. * Finex, N., & Makalela, L. (2021). ''Decolonising multilingualism in Africa: Recentering silenced voices from the global south''. Multilingual Matters. https://doi.org/10.21832/NDHLOV3354 * Kramsch, C. (2022). Afterword: The multilingual turn in language teacher education. ''Language and Education'', ''36''(5), 467–471. https://doi.org/10.1080/09500782.2022.2118542   == Bibliography == Auger, N. (2020). Enseigner à des élèves plurilingues: vers une didactique inclusive. In C. Mendonça-Dias, B. Azaoui, & F. Chnane-Davin (Eds.), ''Allophonie. Inclusion et langues des enfants migrants à l’école'' (pp. 171–183). Éditions Lambert-Lucas. Auger, N. (2024). ''Le diamant langagier''. French, English subtitles: (1h47) https://www.youtube.com/watch?v=artl-xcTDJc     Bloomfield, L. (1933). ''Language''. Holt. Coste, D., Moore, D., & Zarate, G. (1997, 2009). ''Compétence plurilingue et pluriculturelle, Vers un Cadre Européen Commun de Référence pour l'enseignement et l'apprentissage des langues vivantes: études préparatoires.'' Conseil de l’Europe, Division des politiques linguistiques. https://rm.coe.int/168069d29c Council of Europe (Ed.). (2001). ''Common European framework of reference for languages: Learning, teaching, assessment''. Council of Europe Publishing. https://rm.coe.int/16802fc1bf García, O., & Lin, A.M.Y. (2017). Translanguaging in bilingual education. In O. García, A.M.Y. Lin., & S. May (Ed.). ''Bilingual and multilingual education''. Springer. https://doi.org/10.1007/978-3-319-02258-1_9 Grosjean, F (1982). ''Life with two Languages: An introduction to bilingualism''. Harvard University Press. Grosjean, F. (1993). Le bilinguisme et le biculturalisme. Essai de définition. ''Travaux neuchâtelois de linguistique'', (19), 13–41. https://doi.org/10.26034/tranel.1993.2342 Lüdi, G., & Py, B. (2003). ''Être bilingue'' (3rd revised edition). Lang. Lüdi, G., & Py, B. (2009). To be or not to be... a plurilingual speaker. ''International Journal of Multilingualism'', ''6''(2), 154–167. https://doi.org/10.1080/14790710902846715 Macaire, D. (2025). Le tournant plurilingue. In C. Ollivier & S. Melo-Pfeifer (Eds.), ''Encyclopédie de l'éducation plurilingue'' (pp. 305–307).  Peter Lang. https://doi.org/10.3726/b23455   May, S. (2013). ''The multilingual turn : Implications for SLA, TESOL, and bilingual education.'' Taylor & Francis Group. Meier, G. S. (2017).  The multilingual turn as a critical movement in education: assumptions, challenges and a need for reflection. ''Applied Linguistics Review'', ''8''(1), 131–161. https://doi.org/10.1515/applirev-2016-2010 Melo-Pfeifer, S. (2018). The multilingual turn in foreign language education. Facts and fallacies. In A. Bonnet & P. Siemund (Eds.), ''Foreign language education in multilingual classrooms'' (pp. 191–210). John Benjamins. https://doi.org/10.1075/hsld.7 Molinié, M. (2006). Une approche biographique des trajectoires linguistiques et culturelles. In M. Molinié (Ed.), Biographie ''langagière et apprentissage plurilingue. Le français dans le monde. Recherches et applications, 39'', 8–11. https://halshs.archives-ouvertes.fr/halshs-01224908/document Montagne-Macaire, D. (2008). D’une didactique des langues à une didactique des plurilinguismes ? Réflexions pour la recherche. ''Recherches en didactique des langues et des cultures'', ''5''(1), 2–40. https://doi.org/10.4000/rdlc.6245 Moore, D., & Gajo, L. (2009). Introduction. French voices on plurilingualism and pluriculturalism: Significance and perspectives. ''International Journal of Multilingualism'', ''6''(2), 137–153. https://doi.org/10.1080/14790710902846707 Penfield, W., & Roberts, L. (1959). ''Speech and brain mechanisms''. Princeton University Press. 5z7m2ah74cnifsx76qqri5yqk9akfdi 2814860 2814859 2026-06-09T13:32:29Z Projet PEP 3002502 2814860 wikitext text/x-wiki {{Portal|Plurilingual education|Logo PEP.jpg}} {{Education}} == Starting activity == * Map out your linguistic background on a timeline, noting when you learnt or came across each language or variety you know. * From there, identify which of the languages or varieties listed you use in your daily life (to a greater or lesser extent) and specify the contexts in which you use them (private, public, professional, digital, cultural, religious…). This will give you your current ''[[Dominant language constellation]]'' (DLC). * Now think about the following questions: ** Is there one (or more) language or variety that clearly dominates? ** Are there contexts where you switch between languages or varieties, or mix them together? ** Compare your current DLC with your DLC from an earlier point in time. What has changed? Are there any languages/varieties that you once learned but hardly ever use or no longer use at all? * Given all these formalised elements, what is your view (or views) of your plurilingualism? == Objectives == By the end of this section, you should be able to: * recognise that there are different views on plurilingualism; * understand the different stages of the multilingual turn; * understand the implications that this shift towards multilingualism has on language teaching and learning. == Keywords == monolingualism – plurilingualism – multilingualism – approaches to plurilingualism == Introduction == The multilingual turn should be seen as a major development in the way languages and their uses have been viewed since the 1980s. The term ‘turn’ (May, 2013) is used in the sense that plurilingual approaches to teaching have moved away from a compartmentalised, static view of languages – which was previously emphasised – towards a more dynamic, integrated and individualised conception of the linguistic and cultural diversity of individuals and communities. == History == For a long time, in language teaching theory underpinned by research, monolingualism (with a dominant language) was contrasted with bilingualism (conceived as perfect and symmetrical mastery of two languages, for example, as described by Bloomfield, 1993, Penfield & Roberts, 1959, and the code-switching theory) or multilingualism (the presence of several languages within a society or in an individual’s practice, but in a compartmentalised manner). The multilingual turn, which reached its peak in the 2010s, marked a transition from approaches emphasising the need to isolate languages for learning (monolingual immersion was the norm) to the development of a more open, inclusive perspective that takes into account the plurality – particularly the linguistic diversity – of learners. The ideal of the native speaker has thus been set aside as a central element of various teaching approaches and methods (the direct method, the audio-oral method, etc.), in favour of a greater focus on the plurilingual learner. We are therefore at a turning point that forms part of a critical perspective on language teaching and learning. In support of the concept of plurilingual and inter/transcultural competence developed notably under the auspices of the Council of Europe through the Common European Framework of Reference for Languages (CEFR; Council of Europe, 2001), the multilingual turn has become institutionalised, initially nourished by the work of pioneers such as Grosjean (1982, 1993) and Lüdi & Py (1986). As Macaire (2025, p. 315) points out, « le tournant plurilingue se décline selon une triple orientation en éducation. La première explore une compétence unifiante ; la deuxième en discute les enjeux pour l’éducation ; la troisième s’intéresse à l’apprenant-sujet dans l’écologie des diversités ».  – “The shift towards multilingualism takes three distinct forms in education. The first explores a unifying competency; the second examines the implications for education; the third focuses on the individual learner within the context of diversity.” [own translation] == Conceptualisation of the multilingual turn == This turning point marks a conceptual shift centred on four key dimensions, which completely transform the way in which we view multilingualism and the associated multiculturalism. '''An integrated view of languages''': The multilingual turn recognises that an individual’s languages and language varieties are not compartmentalised, but form a single, evolving repertoire from which they draw depending on context and needs. We are moving towards a unifying approach to language competencies. '''Dynamic dimension''': Languages are used in a complementary manner, alternately, and sometimes mixed together depending on the social, educational or family contexts, or on the needs of speakers in various areas of use (see the concept of multiliteracy; Cook, 1991). '''Recognition of partial language competence''': One of the hallmarks of the shift towards plurilingualism lies in the recognition of all language competencies, whether partial or receptive, moving away from a view of plurilingualism that aims to achieve ‘native’ proficiency in every language. Being plurilingual should no longer be equated with being equally competent in each language one knows or uses. '''Relationship with cultural diversity''': Another dimension is involved, namely the cultural dimension, through the recognition of multiple cultural identities and the ability to navigate between different cultures. The multicultural dimension is valued. == Educational and social implications == '''A conceptual paradigm shift''': As the term suggests, the multilingual turn represents a paradigm shift, in the sense that it offers a new perspective on linguistic and cultural diversity. We are thus moving away from viewing multilingualism as an obstacle towards recognising it as a source of richness and an asset for learning languages and citizenship. This conceptual shift inevitably leads to changes in educational practices, such as recognising the value of learners' linguistic repertoires and developing innovative teaching methods. '''Valuing linguistic repertoires''': The multilingual turn is prompting schools and society to recognise and actively foster learners’ linguistic repertoires, enabling learners  to draw on and develop them – especially those from plurilingual families, including migrant backgrounds. This includes activities built around [[Language biography and identity texts|language biographies]] (Molinié, 2006; Auger, 2024), which contribute to this promotion. '''Innovative teaching approaches''': Taking this into account makes it necessary to implement new teaching approaches, particularly within the framework of [[Pluralistic approach|pluralistic approaches]] (Integrated didactic approach to languages, Awakening to languages, intercomprehension, and the Intercultural approach). Teaching approaches based on  [[translanguaging]] (García & Lin, 2017) are also fully part of this shift. The first four approaches can incorporate students’ home languages or other languages they know – whether learnt as foreign or second languages, or acquired in other ways – in order to build on them,  for example by comparing languages, and to promote practices aligned with [[language inclusion]] (Auger, 2020). '''Institutional recognition''': Despite this shift and the arguments associated with it, the promotion of plurilingualism sometimes struggles to take shape within school systems, for various reasons. Consequently, the idea of a pluralistic society is often challenged by political and/or individual ideologies, as well as the strongly held, identity-based views of certain governments. The value placed on [[English as a Lingua Franca (ELF)|English as a lingua franca]] may also contribute to this difficulty in recognition. Even when states are willing to promote this diversity, implementation is complex, due to a lack of teacher training and suitable educational resources and tools. == Take-home messages == The multilingual turn refers to a shift from an additive, compartmentalised, and static view of languages to a dynamic, integrated and value-affirming conception of individuals’ and societies’ linguistic and cultural diversity. It invites us to rethink education, citizenship, and language policies in an era of globalisation and mobility. The multilingual turn also challenges the traditional notions of mother tongue/first language and foreign language, encouraging us to focus more on linguistic repertoires and less on individual languages. == Self-assessment == <quiz> {The multilingual turn holds that:} - plurilingualism is a “handicap” for language learning; - plurilingualism is an obstacle to the development of citizenship; - plurilingualism is positive only if each language is mastered equally; + plurilingualism is an asset for language learning. {From the perspective of pedagogical practices in language learning, following the multilingual turn, it is considered:} - counterproductive to emphasize learners’ home languages ; + necessary to build on the languages learners already know; - that a foreign language should only be learnt through full immersion; + inadvisable to rely on translation or on simply linking to other languages. {With the multilingual turn, in the context of language teaching and learning, we have moved:} - from taking learners’ languages into account to focusing solely on the target language; + from a compartmentalised view of languages to a unified view; - from the necessary consideration of learners’ languages to a purely immersive target-language approach; - from a holistic view of languages to a compartmentalised one. {To implement the educational guidelines resulting from the multilingual turn, it is necessary} - that all teachers are familiar with all their students’ languages; + that learners of the same origin be grouped in the same classes; - that teachers be trained to implement pluralistic approaches; - that teachers use AI-based automatic translators. </quiz> == Resources to go further == * Conteh, J., & Meier, G, (Eds.) (2014). ''The multilingual turn in language education: Opportunities and challenges''. Multilingual Matters. * Cook, V. J. (1991). The poverty-of-the-stimulus argument and multi-competence. ''Second Language Research'', ''7''(2), 103–17. * Finex, N., & Makalela, L. (2021). ''Decolonising multilingualism in Africa: Recentering silenced voices from the global south''. Multilingual Matters. https://doi.org/10.21832/NDHLOV3354 * Kramsch, C. (2022). Afterword: The multilingual turn in language teacher education. ''Language and Education'', ''36''(5), 467–471. https://doi.org/10.1080/09500782.2022.2118542   == Bibliography == Auger, N. (2020). Enseigner à des élèves plurilingues: vers une didactique inclusive. In C. Mendonça-Dias, B. Azaoui, & F. Chnane-Davin (Eds.), ''Allophonie. Inclusion et langues des enfants migrants à l’école'' (pp. 171–183). Éditions Lambert-Lucas. Auger, N. (2024). ''Le diamant langagier''. French, English subtitles: (1h47) https://www.youtube.com/watch?v=artl-xcTDJc     Bloomfield, L. (1933). ''Language''. Holt. Coste, D., Moore, D., & Zarate, G. (1997, 2009). ''Compétence plurilingue et pluriculturelle, Vers un Cadre Européen Commun de Référence pour l'enseignement et l'apprentissage des langues vivantes: études préparatoires.'' Conseil de l’Europe, Division des politiques linguistiques. https://rm.coe.int/168069d29c Council of Europe (Ed.). (2001). ''Common European framework of reference for languages: Learning, teaching, assessment''. Council of Europe Publishing. https://rm.coe.int/16802fc1bf García, O., & Lin, A.M.Y. (2017). Translanguaging in bilingual education. In O. García, A.M.Y. Lin., & S. May (Ed.). ''Bilingual and multilingual education''. Springer. https://doi.org/10.1007/978-3-319-02258-1_9 Grosjean, F (1982). ''Life with two Languages: An introduction to bilingualism''. Harvard University Press. Grosjean, F. (1993). Le bilinguisme et le biculturalisme. Essai de définition. ''Travaux neuchâtelois de linguistique'', (19), 13–41. https://doi.org/10.26034/tranel.1993.2342 Lüdi, G., & Py, B. (2003). ''Être bilingue'' (3rd revised edition). Lang. Lüdi, G., & Py, B. (2009). To be or not to be... a plurilingual speaker. ''International Journal of Multilingualism'', ''6''(2), 154–167. https://doi.org/10.1080/14790710902846715 Macaire, D. (2025). Le tournant plurilingue. In C. Ollivier & S. Melo-Pfeifer (Eds.), ''Encyclopédie de l'éducation plurilingue'' (pp. 305–307).  Peter Lang. https://doi.org/10.3726/b23455   May, S. (2013). ''The multilingual turn : Implications for SLA, TESOL, and bilingual education.'' Taylor & Francis Group. Meier, G. S. (2017).  The multilingual turn as a critical movement in education: assumptions, challenges and a need for reflection. ''Applied Linguistics Review'', ''8''(1), 131–161. https://doi.org/10.1515/applirev-2016-2010 Melo-Pfeifer, S. (2018). The multilingual turn in foreign language education. Facts and fallacies. In A. Bonnet & P. Siemund (Eds.), ''Foreign language education in multilingual classrooms'' (pp. 191–210). John Benjamins. https://doi.org/10.1075/hsld.7 Molinié, M. (2006). Une approche biographique des trajectoires linguistiques et culturelles. In M. Molinié (Ed.), Biographie ''langagière et apprentissage plurilingue. Le français dans le monde. Recherches et applications, 39'', 8–11. https://halshs.archives-ouvertes.fr/halshs-01224908/document Montagne-Macaire, D. (2008). D’une didactique des langues à une didactique des plurilinguismes ? Réflexions pour la recherche. ''Recherches en didactique des langues et des cultures'', ''5''(1), 2–40. https://doi.org/10.4000/rdlc.6245 Moore, D., & Gajo, L. (2009). Introduction. French voices on plurilingualism and pluriculturalism: Significance and perspectives. ''International Journal of Multilingualism'', ''6''(2), 137–153. https://doi.org/10.1080/14790710902846707 Penfield, W., & Roberts, L. (1959). ''Speech and brain mechanisms''. Princeton University Press. ml8jtokzzlcvoexy5zu3tb6jll0q01w 6 330089 2814864 2026-06-09T14:04:25Z Young1lim 21186 {{Information |Description=Carry Lookahead Adders 2A traditional (20260609 - 20260608) |Source={{own|Young1lim}} |Date=2026-06-09 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2814864 wikitext text/x-wiki == Summary == {{Information |Description=Carry Lookahead Adders 2A traditional (20260609 - 20260608) |Source={{own|Young1lim}} |Date=2026-06-09 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} mbeeq5o1ujp8sjprt69pvgi8qfm0owa 6 330090 2814865 2026-06-09T14:05:06Z Young1lim 21186 {{Information |Description=Carry Lookahead Adders 2B simplified (20260609 - 20260608) |Source={{own|Young1lim}} |Date=2026-06-09 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2814865 wikitext text/x-wiki == Summary == {{Information |Description=Carry Lookahead Adders 2B simplified (20260609 - 20260608) |Source={{own|Young1lim}} |Date=2026-06-09 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 0chwwiybhn2158m9dliu1cw5uadqanv 6 330091 2814867 2026-06-09T14:13:09Z Young1lim 21186 {{Information |Description=C04.SA0: Address and Dereference Operators (20260609 - 20260608) |Source={{own|Young1lim}} |Date=2026-06-09 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2814867 wikitext text/x-wiki == Summary == {{Information |Description=C04.SA0: Address and Dereference Operators (20260609 - 20260608) |Source={{own|Young1lim}} |Date=2026-06-09 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 1kfuyrhywbibpzwjbx1gjqti21nsbgh 6 330092 2814869 2026-06-09T14:21:36Z Young1lim 21186 {{Information |Description=Laurent.5: Permutation 6C (20260608 - 20260606) |Source={{own|Young1lim}} |Date=2026-06-09 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2814869 wikitext text/x-wiki == Summary == {{Information |Description=Laurent.5: Permutation 6C (20260608 - 20260606) |Source={{own|Young1lim}} |Date=2026-06-09 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 5keu57hg4eyco8haxepxsm50vjb4k7y 6 330093 2814874 2026-06-09T14:27:41Z Young1lim 21186 {{Information |Description=Laurent.5: Permutation 6C (20260609 - 20260608) |Source={{own|Young1lim}} |Date=2026-06-09 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2814874 wikitext text/x-wiki == Summary == {{Information |Description=Laurent.5: Permutation 6C (20260609 - 20260608) |Source={{own|Young1lim}} |Date=2026-06-09 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 5mkx1lw7ajr7oolb8etjsjzg5uv0epa 0 330094 2814875 2026-06-09T14:29:33Z Projet PEP 3002502 Created page with "== Initial activity == Examine the following language profiles: * '''Melissa''', born in Berlin to an Albanian father and an Italian mother. At home, she speaks Albanian with her father and Italian with her mother. With her sister, she mainly speaks German, the language in which they have been educated since kindergarten. At school, she learns English and French, which are part of the curriculum. * '''Teresa''', who is Portuguese, has been living in France for 18 years,..." 2814875 wikitext text/x-wiki == Initial activity == Examine the following language profiles: * '''Melissa''', born in Berlin to an Albanian father and an Italian mother. At home, she speaks Albanian with her father and Italian with her mother. With her sister, she mainly speaks German, the language in which they have been educated since kindergarten. At school, she learns English and French, which are part of the curriculum. * '''Teresa''', who is Portuguese, has been living in France for 18 years, having emigrated there as an adult. She learned French as an adult through daily interactions. She learned German and English at school before emigrating. She speaks Portuguese with her children. Try to answer these questions: * What is Melissa's mother tongue? * What is the difference in the status of German in the two language profiles? * What is the status of English in the two profiles? == Objectives == By the end of this section, you should be able to... * understand the difficulty of classifying and naming the status of languages present in individuals' language biographies; * understand that classifications of relationships to languages are more sociolinguistic and subjective than based on stable linguistic and scientific criteria- == Keywords == Language biography - Mother tongue - L1 - Foreign language - Host language - Language of origin - Border language - Minority language - Linguistic profiles. == Prerequisites == Prior knowledge of the concept of [[Language biography and identity texts|language biography]]. == Introduction == Concepts are not neutral and imply worldviews, either implicitly or explicitly. More specifically, linguistic concepts, although they may seem neutral and imbued with objectivity and scientificity, are polysemic and lend themselves to different interpretations depending on the context, even among researchers. Concepts may have a linguistic basis, based on the order of acquisition, for example, as in the case of L1, or they may refer to the sociolinguistic contexts in which they are learned/acquired, as in the case of “language of origin” or “host language,” which refer to situations of immigration, whether forced or not. Other classifications refer to the geographical contexts in which linguistic contacts occur (“border languages”), or to power relations (“minority language”, “majority language”), and still others to classifications related to typology (such as “languages of the same family”). In other words, when we talk about languages, we use concepts that draw on different disciplines, discourses, representations of what a language is, and even what a speaker is, their skills, and their communication needs. Thus, when we use the term “mother tongue” in the singular, we tend to imagine the designated speaker as a native speaker with advanced skills in a given language. But this concept draws on ideologies related to the very process of language transmission, from mother to child. In this section, we will learn how to designate different relationships between subjects and languages, and we will realize that the designations that subjects give to the languages they come into contact with do not always correspond to criteria considered scientific and “linguistic”. Given the complexity of addressing all potential designations and polysemies, we will focus on the most common ones in discourses on  plurilingualism and  plurilingual education, emphasizing the most consensual conceptions in research (Lisboa, data). == Concepts == Concepts of language are not static. The same could be said of concepts used to describe different relationships with languages. These concepts are functional and dynamic in nature, continually being revised, redefined, and updated in response to the demands and advances of research. They are also closely linked to sociolinguistic and identity contexts, which, by their very nature, are constantly evolving. In this process, new concepts emerge, old ones are rethought, while others become inoperative or unsuitable for certain contexts or research frameworks (Lisboa, 2024). Conceptions of languages can be analyzed using, for example, dictionaries specific to disciplinary fields (Cuq, 2003; Galisson & Coste, 1976; Surkamp, 2010) or meta-analyses of scientific texts in these fields, attempting to understand the common uses associated with each term and the variations. This is what Lisboa (2024) did in a study in which he analyzed published scientific articles. The author concludes: "The terminology of fields dealing with the interrelationships between languages, individuals, and societies, such as language teaching and language policy, is permeated with concepts such as ''mother tongue'', ''first language'', ''non-mother tongue'', ''additional language'', ''official language'', ''foreign language'', ''second language'', ''language of origin'', and ''host language'', among many others. These concepts are used to classify, describe, represent, and specify the roles played by languages, the social functions attributed to them, and the relationships between different languages in the same sociolinguistic space and in individuals' linguistic repertoires." (Lisboa, 2025, our translation) For this section, we have translated and reproduced the definitions from the meta-analytical work mentioned above (Lisboa, 2024), although some contextualization is still needed, as not all terms have the same academic traditions. For the definition of the host language, which does not appear in the entries produced by Lisboa (2024), we turn to Lopez & Diniz (2018). In any case, it should be noted that meta-analytic work reflects the practices of researchers, which may vary between epistemological, research, and linguistic traditions. Thus, Lisboa refers to the concepts most commonly used in the Brazilian context. For example, “non-native language” is often used in Portuguese, mainly because of the difficulties in differentiating the relationships and statuses of languages that come after the so-called ‘first’ or “native” languages. This concept is largely absent from English-language research. {| class="wikitable" |Concept |Definition |- |Mother tongue |“The language first acquired around the age of five” (p. 260), i.e., during the early years of primary socialization. It can also be understood as “the language of greatest mastery and affirmation of an individual's identity” (p. 271). From a sociolinguistic point of view, this designation can also refer to the majority language in a given linguistic community (p. 273). Regarding this attribution, the author adds: “This meaning is oriented towards an imaginary of national, regional, cultural, and/or ethnic identity and unity. It is also closely linked to the concepts of national language and majority language” (p. 273). |- |Non-native language |“A language acquired after the acquisition of the native language, whether in a formal or informal context, immersive or non-immersive, and regardless of the sociopolitical status of the language” (p. 275). |- |L1 |“The language first acquired up to the age of about five” (p. 313). This may be the “language of greatest proficiency and affirmation of an individual's identity” (p. 316), although the degree of proficiency may change over the course of a lifetime. |- |Heritage language |“Language acquired in early childhood, usually within the family, which, due to migration, usually during childhood, becomes a minority language in relation to the majority language of the host society and, as a result, restricted in terms of contexts and opportunities for use and spaces of circulation, which can gradually lead to a weakening of linguistic proficiency or even to the individual forgetting the language” (p. 237). It can also refer to “the language with which the individual maintains a sense of belonging and cultural and identity ties, regardless of the context or order of acquisition, the minority or majority status of the language, or the level of proficiency” (p. 245). The author adds “a non-native language taught, learned, and/or used in non-immersion contexts, i.e., when the language in question is not the majority language in the place of teaching, use, and/or learning and/or does not have official sociopolitical status” (p. 252; for a third, more political-administrative meaning, see p. 257). |- |Host language |A subdomain of additional language that refers to the language acquired or to be acquired by migrants, with an emphasis on forced displacement, who are in a situation of vulnerability (Lopez & Diniz, 2018, n/p.). |- |L2 (or Second Language) |“A non-native or non-majority language that has been given (co)official status in bilingual or multilingual countries and is therefore the language or one of the languages of public administration, education, and common means of communication in the country” (p. 290). It can also refer to “a non-native language taught, learned, and/or used in language immersion contexts, i.e., when the language in question is the majority language in the place of teaching, use, and/or learning and/or has official sociopolitical status” (p. 300). A third concept, which does not enjoy international consensus, particularly in the German context (see definitions in Surkamp, 2010), refers to “a language acquired after the acquisition of the mother tongue, regardless of the formal, informal, immersive or non-immersive context, the socio-political status of the language or its position in the chronological order of language acquisition” (p. 307) or “a language acquired first after the partial or complete acquisition of the mother tongue, i.e., from the point of view of the chronology of language acquisition, it is the second language to constitute the speaker's linguistic repertoire” (p. 309). |- |Foreign Language |This is a language learned in an institutional context and in a guided manner. “A language acquired after the acquisition of the mother tongue, whether in a formal or informal context, immersive or non-immersive, and regardless of the sociopolitical status of the language” (p. 250). |- |Additional language |“A language acquired after the acquisition of the mother tongue, regardless of the formal or informal context, immersive or non-immersive, the sociopolitical status of the language, or the chronology of acquisition” (Lisboa, 2024, p. 231). It can also refer to “a non-native language acquired after the acquisition of the mother tongue and at least one other non-native language” (p. 236). |- |Language(s) of schooling |Languages of schooling cover two dimensions: languages taught as a subject (such as foreign languages in the curriculum) and languages used for and in teaching. These are often the national and majority languages of a country (Portuguese in Portugal) or regional or minority languages, depending on local education and language policies. (<nowiki>https://www.coe.int/fr/web/language-policy/languages-of-schooling</nowiki>) |} The language of origin is also often referred to as the “home language”, “identity language”, or “family language” (Stratilaki-Klein, 2025). However, speakers of a language of origin do not always consider that this language best defines their identity. Sometimes families choose not to speak their language of origin at home. Furthermore, a speaker may have two languages of origin, as in the case of Melissa, presented in the initial activity. The term “neighboring language” has both linguistic and geographical connotations and may coincide. From a linguistic point of view, it is a language belonging to the same language family (for example, Romance or Slavic languages, to name just two examples). From a geographical point of view, these are languages of neighboring territories, with more or less fluid borders (the French-German border or between Brazil and Argentina, for example). The concept of linguistic proximity or “border language” (Araújo e Sá et al, 2024) is used to refer to languages that coexist in border regions. Regardless of the connotation chosen, we can clearly see how important this concept is for thinking about plurilingual education: either it generally refers to teaching approaches that aim to develop receptive skills in several languages of the same language family (see entry “intercomprehension” in this Wikiversity), or it refers to cross-border linguistic contacts aimed at developing plurilingual and intercultural competence (Putsche, 2022). It is important to remember that most language designations take the so-called native speaker as the implicit norm (Bono, 2025), below which speakers of other designations are ranked, giving rise to situations of linguistic prejudice and discrimination in society in general, and in schools in particular. == Take away messages == * Language classifications can be based on linguistic, sociolinguistic, geographical, economic, curricular, and individual aspects. * The concepts used to name languages are polysemous and are not neutral, but rather convey discourses and ideologies that must be (re)constructed in order to understand the objectives set in plurilingual education. * Language designations and their associated connotations may depend on the linguistic and academic traditions in which they circulate. == Self-assessment == In the first situation presented in the initial activity, Melissa's mother tongue is German: * true * false In the second linguistic profile presented in the initial activity, Portuguese is the language of origin of Teresa's children. * true * false The host language generally refers to: * the language that people in situations of mobility acquire in order to meet their most basic and urgent needs. * the language that parents and educators in situations of mobility (forced or otherwise) pass on to their children in order to maintain a link with their country of origin. * the language in which learners feel welcome when on vacation abroad and which makes them feel “at home”. Which of these concepts refers to the language that parents and educators in situations of migration and mobility (forced or otherwise) pass on to their children in the host country, in a situation where that language is a minority language? * Host language. * Mother tongue. * Language of origin. == Resources to go further == * Araújo e Sá, M.H., Carinhas, R., Carlos, V., Kremer, M., Lourenço, M., Pinto, S., Simões, C., Teixeira, M., & Ulhôa, A. (2024). ''Linguistic and cultural management in border regions''. https://www.eu-bold.com/linguistic-and-cultural-management-in-border-regions/ * Dabène, L. (1994). ''Repères sociolinguistiques pour l’enseignement des langues''. Hachette. * Dewaele, J.-M.; Bak, T.; Ortega, L. (2022). Why the mythical “native speaker” has mud on its face. In N. Slavkov, S. Melo-Pfeifer & N. Kerschhofer-Puhalo (Eds.), ''The Changing Face of the “Native Speaker”. Perspectives from Multilingualism and Globalization'' (pp. 25-45). De Gruyter. * Lisboa, J. (2024). ''Vocabulário terminológico de conceitos de Língua''. Tese de doutoramento. Universididade da Uberlândia. URL: https://repositorio.ufu.br/bitstream/123456789/43065/1/VocabularioConceitosLingua.pdf == Bibliography == Bono, M. (2025). Mother tongue, native language, first language, heritage language. In Ch. Ollivier & S. Melo-Pfeifer (Eds.), ''Encyclopédie de l’éducation plurilingue / Encyclopaedia of plurilingual education''. Peter Lang. Cuq, J.-P. (dir.) (2003). ''Dictionnaire de didactique du français langue étrangère et seconde''. Cle International. Galisson, R., & Coste, D. (1976). ''Dictionnaire de Didactique des Langues''. Hachette. Lisboa, J. (2024). ''Vocabulário terminológico de conceitos de Língua''. Tese de doutoramento. Universidade da Uberlândia. URL: https://repositorio.ufu.br/bitstream/123456789/43065/1/VocabularioConceitosLingua.pdf Lisboa, J. (2025). A criação do Vocabulário Terminológico de Conceitos de Língua. ''Domínios de Lingu@gem'', 19, p. e019025. DOI: 10.14393/DLv19a2025-25. URL:  https://seer.ufu.br/index.php/dominiosdelinguagem/article/view/76903. Lopez, A.P.A., & Diniz, L. R. (2018). Iniciativas jurídicas e académicas para o acolhimento no Brasil de deslocados forçados. Revista da Sociedade Internacional do Português Língua Estrangeira, 9, s/p.  https://www.researchgate.net/publication/330635043_Iniciativas_Juridicas_e_Academicas_Brasileiras_para_o_Acolhimento_de_Imigrantes_Deslocados_Forcados Putsche, J. (2022). ''L’interculturel contextualisé : franco-allemand, frontalier, transculturel – Une approche sociodidactique''. [Dossier presentado para habilitación a la dirección de investigadores, Université de Lorraine]. https://hal.science/tel-03882905/document Stratilaki-Klein, S. (2025). La galaxie des langues et leurs désignations. In Ch. Ollivier & S. Melo-Pfeifer (Eds.), ''Encyclopédie de l’éducation plurilingue / Encyclopaedia of plurilingual education''. Peter Lang. Surkamp, C. (Ed.) (2010). ''Metzler Lexikon Fremdsprachendidaktik''. J.B. Metzler. hydz8fsfmz4ngws5ch542d6zqkzchjr 2814876 2814875 2026-06-09T14:33:21Z Projet PEP 3002502 2814876 wikitext text/x-wiki == Initial activity == Examine the following language profiles: * '''Melissa''', born in Berlin to an Albanian father and an Italian mother. At home, she speaks Albanian with her father and Italian with her mother. With her sister, she mainly speaks German, the language in which they have been educated since kindergarten. At school, she learns English and French, which are part of the curriculum. * '''Teresa''', who is Portuguese, has been living in France for 18 years, having emigrated there as an adult. She learned French as an adult through daily interactions. She learned German and English at school before emigrating. She speaks Portuguese with her children. Try to answer these questions: * What is Melissa's mother tongue? * What is the difference in the status of German in the two language profiles? * What is the status of English in the two profiles? == Objectives == By the end of this section, you should be able to... * understand the difficulty of classifying and naming the status of languages present in individuals' language biographies; * understand that classifications of relationships to languages are more sociolinguistic and subjective than based on stable linguistic and scientific criteria- == Keywords == Language biography - Mother tongue - L1 - Foreign language - Host language - Language of origin - Border language - Minority language - Linguistic profiles. == Prerequisites == Prior knowledge of the concept of [[Language biography and identity texts|language biography]]. == Introduction == Concepts are not neutral and imply worldviews, either implicitly or explicitly. More specifically, linguistic concepts, although they may seem neutral and imbued with objectivity and scientificity, are polysemic and lend themselves to different interpretations depending on the context, even among researchers. Concepts may have a linguistic basis, based on the order of acquisition, for example, as in the case of L1, or they may refer to the sociolinguistic contexts in which they are learned/acquired, as in the case of “language of origin” or “host language,” which refer to situations of immigration, whether forced or not. Other classifications refer to the geographical contexts in which linguistic contacts occur (“border languages”), or to power relations (“minority language”, “majority language”), and still others to classifications related to typology (such as “languages of the same family”). In other words, when we talk about languages, we use concepts that draw on different disciplines, discourses, representations of what a language is, and even what a speaker is, their skills, and their communication needs. Thus, when we use the term “mother tongue” in the singular, we tend to imagine the designated speaker as a native speaker with advanced skills in a given language. But this concept draws on ideologies related to the very process of language transmission, from mother to child. In this section, we will learn how to designate different relationships between subjects and languages, and we will realize that the designations that subjects give to the languages they come into contact with do not always correspond to criteria considered scientific and “linguistic”. Given the complexity of addressing all potential designations and polysemies, we will focus on the most common ones in discourses on  plurilingualism and  plurilingual education, emphasizing the most consensual conceptions in research (Lisboa, data). == Concepts == Concepts of language are not static. The same could be said of concepts used to describe different relationships with languages. These concepts are functional and dynamic in nature, continually being revised, redefined, and updated in response to the demands and advances of research. They are also closely linked to sociolinguistic and identity contexts, which, by their very nature, are constantly evolving. In this process, new concepts emerge, old ones are rethought, while others become inoperative or unsuitable for certain contexts or research frameworks (Lisboa, 2024). Conceptions of languages can be analyzed using, for example, dictionaries specific to disciplinary fields (Cuq, 2003; Galisson & Coste, 1976; Surkamp, 2010) or meta-analyses of scientific texts in these fields, attempting to understand the common uses associated with each term and the variations. This is what Lisboa (2024) did in a study in which he analyzed published scientific articles. The author concludes: "The terminology of fields dealing with the interrelationships between languages, individuals, and societies, such as language teaching and language policy, is permeated with concepts such as ''mother tongue'', ''first language'', ''non-mother tongue'', ''additional language'', ''official language'', ''foreign language'', ''second language'', ''language of origin'', and ''host language'', among many others. These concepts are used to classify, describe, represent, and specify the roles played by languages, the social functions attributed to them, and the relationships between different languages in the same sociolinguistic space and in individuals' linguistic repertoires." (Lisboa, 2025, our translation) For this section, we have translated and reproduced the definitions from the meta-analytical work mentioned above (Lisboa, 2024), although some contextualization is still needed, as not all terms have the same academic traditions. For the definition of the host language, which does not appear in the entries produced by Lisboa (2024), we turn to Lopez & Diniz (2018). In any case, it should be noted that meta-analytic work reflects the practices of researchers, which may vary between epistemological, research, and linguistic traditions. Thus, Lisboa refers to the concepts most commonly used in the Brazilian context. For example, “non-native language” is often used in Portuguese, mainly because of the difficulties in differentiating the relationships and statuses of languages that come after the so-called ‘first’ or “native” languages. This concept is largely absent from English-language research. {| class="wikitable" |Concept |Definition |- |Mother tongue |“The language first acquired around the age of five” (p. 260), i.e., during the early years of primary socialization. It can also be understood as “the language of greatest mastery and affirmation of an individual's identity” (p. 271). From a sociolinguistic point of view, this designation can also refer to the majority language in a given linguistic community (p. 273). Regarding this attribution, the author adds: “This meaning is oriented towards an imaginary of national, regional, cultural, and/or ethnic identity and unity. It is also closely linked to the concepts of national language and majority language” (p. 273). |- |Non-native language |“A language acquired after the acquisition of the native language, whether in a formal or informal context, immersive or non-immersive, and regardless of the sociopolitical status of the language” (p. 275). |- |L1 |“The language first acquired up to the age of about five” (p. 313). This may be the “language of greatest proficiency and affirmation of an individual's identity” (p. 316), although the degree of proficiency may change over the course of a lifetime. |- |Heritage language |“Language acquired in early childhood, usually within the family, which, due to migration, usually during childhood, becomes a minority language in relation to the majority language of the host society and, as a result, restricted in terms of contexts and opportunities for use and spaces of circulation, which can gradually lead to a weakening of linguistic proficiency or even to the individual forgetting the language” (p. 237). It can also refer to “the language with which the individual maintains a sense of belonging and cultural and identity ties, regardless of the context or order of acquisition, the minority or majority status of the language, or the level of proficiency” (p. 245). The author adds “a non-native language taught, learned, and/or used in non-immersion contexts, i.e., when the language in question is not the majority language in the place of teaching, use, and/or learning and/or does not have official sociopolitical status” (p. 252; for a third, more political-administrative meaning, see p. 257). |- |Host language |A subdomain of additional language that refers to the language acquired or to be acquired by migrants, with an emphasis on forced displacement, who are in a situation of vulnerability (Lopez & Diniz, 2018, n/p.). |- |L2 (or Second Language) |“A non-native or non-majority language that has been given (co)official status in bilingual or multilingual countries and is therefore the language or one of the languages of public administration, education, and common means of communication in the country” (p. 290). It can also refer to “a non-native language taught, learned, and/or used in language immersion contexts, i.e., when the language in question is the majority language in the place of teaching, use, and/or learning and/or has official sociopolitical status” (p. 300). A third concept, which does not enjoy international consensus, particularly in the German context (see definitions in Surkamp, 2010), refers to “a language acquired after the acquisition of the mother tongue, regardless of the formal, informal, immersive or non-immersive context, the socio-political status of the language or its position in the chronological order of language acquisition” (p. 307) or “a language acquired first after the partial or complete acquisition of the mother tongue, i.e., from the point of view of the chronology of language acquisition, it is the second language to constitute the speaker's linguistic repertoire” (p. 309). |- |Foreign Language |This is a language learned in an institutional context and in a guided manner. “A language acquired after the acquisition of the mother tongue, whether in a formal or informal context, immersive or non-immersive, and regardless of the sociopolitical status of the language” (p. 250). |- |Additional language |“A language acquired after the acquisition of the mother tongue, regardless of the formal or informal context, immersive or non-immersive, the sociopolitical status of the language, or the chronology of acquisition” (Lisboa, 2024, p. 231). It can also refer to “a non-native language acquired after the acquisition of the mother tongue and at least one other non-native language” (p. 236). |- |Language(s) of schooling |Languages of schooling cover two dimensions: languages taught as a subject (such as foreign languages in the curriculum) and languages used for and in teaching. These are often the national and majority languages of a country (Portuguese in Portugal) or regional or minority languages, depending on local education and language policies. (<nowiki>https://www.coe.int/fr/web/language-policy/languages-of-schooling</nowiki>) |} The language of origin is also often referred to as the “home language”, “identity language”, or “family language” (Stratilaki-Klein, 2025). However, speakers of a language of origin do not always consider that this language best defines their identity. Sometimes families choose not to speak their language of origin at home. Furthermore, a speaker may have two languages of origin, as in the case of Melissa, presented in the initial activity. The term “neighboring language” has both linguistic and geographical connotations and may coincide. From a linguistic point of view, it is a language belonging to the same language family (for example, Romance or Slavic languages, to name just two examples). From a geographical point of view, these are languages of neighboring territories, with more or less fluid borders (the French-German border or between Brazil and Argentina, for example). The concept of linguistic proximity or “border language” (Araújo e Sá et al, 2024) is used to refer to languages that coexist in border regions. Regardless of the connotation chosen, we can clearly see how important this concept is for thinking about plurilingual education: either it generally refers to teaching approaches that aim to develop receptive skills in several languages of the same language family (see entry “intercomprehension” in this Wikiversity), or it refers to cross-border linguistic contacts aimed at developing plurilingual and intercultural competence (Putsche, 2022). It is important to remember that most language designations take the so-called native speaker as the implicit norm (Bono, 2025), below which speakers of other designations are ranked, giving rise to situations of linguistic prejudice and discrimination in society in general, and in schools in particular. == Take away messages == * Language classifications can be based on linguistic, sociolinguistic, geographical, economic, curricular, and individual aspects. * The concepts used to name languages are polysemous and are not neutral, but rather convey discourses and ideologies that must be (re)constructed in order to understand the objectives set in plurilingual education. * Language designations and their associated connotations may depend on the linguistic and academic traditions in which they circulate. == Self-assessment == <quiz> {In the first situation presented in the initial activity, Melissa's mother tongue is German:} -true +false {In the second linguistic profile presented in the initial activity, Portuguese is the language of origin of Teresa's children.} +true -false {The host language generally refers to:} +the language that people in situations of mobility acquire in order to meet their most basic and urgent needs. -the language that parents and educators in situations of mobility (forced or otherwise) pass on to their children in order to maintain a link with their country of origin. -the language in which learners feel welcome when on vacation abroad and which makes them feel “at home”. {Which of these concepts refers to the language that parents and educators in situations of migration and mobility (forced or otherwise) pass on to their children in the host country, in a situation where that language is a minority language?} -Host language. -Mother tongue. +Language of origin. </quiz> == Resources to go further == * Araújo e Sá, M.H., Carinhas, R., Carlos, V., Kremer, M., Lourenço, M., Pinto, S., Simões, C., Teixeira, M., & Ulhôa, A. (2024). ''Linguistic and cultural management in border regions''. https://www.eu-bold.com/linguistic-and-cultural-management-in-border-regions/ * Dabène, L. (1994). ''Repères sociolinguistiques pour l’enseignement des langues''. Hachette. * Dewaele, J.-M.; Bak, T.; Ortega, L. (2022). Why the mythical “native speaker” has mud on its face. In N. Slavkov, S. Melo-Pfeifer & N. Kerschhofer-Puhalo (Eds.), ''The Changing Face of the “Native Speaker”. Perspectives from Multilingualism and Globalization'' (pp. 25-45). De Gruyter. * Lisboa, J. (2024). ''Vocabulário terminológico de conceitos de Língua''. Tese de doutoramento. Universididade da Uberlândia. URL: https://repositorio.ufu.br/bitstream/123456789/43065/1/VocabularioConceitosLingua.pdf == Bibliography == Bono, M. (2025). Mother tongue, native language, first language, heritage language. In Ch. Ollivier & S. Melo-Pfeifer (Eds.), ''Encyclopédie de l’éducation plurilingue / Encyclopaedia of plurilingual education''. Peter Lang. Cuq, J.-P. (dir.) (2003). ''Dictionnaire de didactique du français langue étrangère et seconde''. Cle International. Galisson, R., & Coste, D. (1976). ''Dictionnaire de Didactique des Langues''. Hachette. Lisboa, J. (2024). ''Vocabulário terminológico de conceitos de Língua''. Tese de doutoramento. Universidade da Uberlândia. URL: https://repositorio.ufu.br/bitstream/123456789/43065/1/VocabularioConceitosLingua.pdf Lisboa, J. (2025). A criação do Vocabulário Terminológico de Conceitos de Língua. ''Domínios de Lingu@gem'', 19, p. e019025. DOI: 10.14393/DLv19a2025-25. URL:  https://seer.ufu.br/index.php/dominiosdelinguagem/article/view/76903. Lopez, A.P.A., & Diniz, L. R. (2018). Iniciativas jurídicas e académicas para o acolhimento no Brasil de deslocados forçados. Revista da Sociedade Internacional do Português Língua Estrangeira, 9, s/p.  https://www.researchgate.net/publication/330635043_Iniciativas_Juridicas_e_Academicas_Brasileiras_para_o_Acolhimento_de_Imigrantes_Deslocados_Forcados Putsche, J. (2022). ''L’interculturel contextualisé : franco-allemand, frontalier, transculturel – Une approche sociodidactique''. [Dossier presentado para habilitación a la dirección de investigadores, Université de Lorraine]. https://hal.science/tel-03882905/document Stratilaki-Klein, S. (2025). La galaxie des langues et leurs désignations. In Ch. Ollivier & S. Melo-Pfeifer (Eds.), ''Encyclopédie de l’éducation plurilingue / Encyclopaedia of plurilingual education''. Peter Lang. Surkamp, C. (Ed.) (2010). ''Metzler Lexikon Fremdsprachendidaktik''. J.B. Metzler. orqp5xggo4s1ct6jk2a7p9td9m50ns3 2814878 2814876 2026-06-09T14:35:43Z Projet PEP 3002502 2814878 wikitext text/x-wiki {{Portal|Plurilingual education|Logo PEP.jpg}} {{Education}} == Initial activity == Examine the following language profiles: * '''Melissa''', born in Berlin to an Albanian father and an Italian mother. At home, she speaks Albanian with her father and Italian with her mother. With her sister, she mainly speaks German, the language in which they have been educated since kindergarten. At school, she learns English and French, which are part of the curriculum. * '''Teresa''', who is Portuguese, has been living in France for 18 years, having emigrated there as an adult. She learned French as an adult through daily interactions. She learned German and English at school before emigrating. She speaks Portuguese with her children. Try to answer these questions: * What is Melissa's mother tongue? * What is the difference in the status of German in the two language profiles? * What is the status of English in the two profiles? == Objectives == By the end of this section, you should be able to... * understand the difficulty of classifying and naming the status of languages present in individuals' language biographies; * understand that classifications of relationships to languages are more sociolinguistic and subjective than based on stable linguistic and scientific criteria- == Keywords == Language biography - Mother tongue - L1 - Foreign language - Host language - Language of origin - Border language - Minority language - Linguistic profiles. == Prerequisites == Prior knowledge of the concept of [[Language biography and identity texts|language biography]]. == Introduction == Concepts are not neutral and imply worldviews, either implicitly or explicitly. More specifically, linguistic concepts, although they may seem neutral and imbued with objectivity and scientificity, are polysemic and lend themselves to different interpretations depending on the context, even among researchers. Concepts may have a linguistic basis, based on the order of acquisition, for example, as in the case of L1, or they may refer to the sociolinguistic contexts in which they are learned/acquired, as in the case of “language of origin” or “host language,” which refer to situations of immigration, whether forced or not. Other classifications refer to the geographical contexts in which linguistic contacts occur (“border languages”), or to power relations (“minority language”, “majority language”), and still others to classifications related to typology (such as “languages of the same family”). In other words, when we talk about languages, we use concepts that draw on different disciplines, discourses, representations of what a language is, and even what a speaker is, their skills, and their communication needs. Thus, when we use the term “mother tongue” in the singular, we tend to imagine the designated speaker as a native speaker with advanced skills in a given language. But this concept draws on ideologies related to the very process of language transmission, from mother to child. In this section, we will learn how to designate different relationships between subjects and languages, and we will realize that the designations that subjects give to the languages they come into contact with do not always correspond to criteria considered scientific and “linguistic”. Given the complexity of addressing all potential designations and polysemies, we will focus on the most common ones in discourses on  plurilingualism and  plurilingual education, emphasizing the most consensual conceptions in research (Lisboa, data). == Concepts == Concepts of language are not static. The same could be said of concepts used to describe different relationships with languages. These concepts are functional and dynamic in nature, continually being revised, redefined, and updated in response to the demands and advances of research. They are also closely linked to sociolinguistic and identity contexts, which, by their very nature, are constantly evolving. In this process, new concepts emerge, old ones are rethought, while others become inoperative or unsuitable for certain contexts or research frameworks (Lisboa, 2024). Conceptions of languages can be analyzed using, for example, dictionaries specific to disciplinary fields (Cuq, 2003; Galisson & Coste, 1976; Surkamp, 2010) or meta-analyses of scientific texts in these fields, attempting to understand the common uses associated with each term and the variations. This is what Lisboa (2024) did in a study in which he analyzed published scientific articles. The author concludes: "The terminology of fields dealing with the interrelationships between languages, individuals, and societies, such as language teaching and language policy, is permeated with concepts such as ''mother tongue'', ''first language'', ''non-mother tongue'', ''additional language'', ''official language'', ''foreign language'', ''second language'', ''language of origin'', and ''host language'', among many others. These concepts are used to classify, describe, represent, and specify the roles played by languages, the social functions attributed to them, and the relationships between different languages in the same sociolinguistic space and in individuals' linguistic repertoires." (Lisboa, 2025, our translation) For this section, we have translated and reproduced the definitions from the meta-analytical work mentioned above (Lisboa, 2024), although some contextualization is still needed, as not all terms have the same academic traditions. For the definition of the host language, which does not appear in the entries produced by Lisboa (2024), we turn to Lopez & Diniz (2018). In any case, it should be noted that meta-analytic work reflects the practices of researchers, which may vary between epistemological, research, and linguistic traditions. Thus, Lisboa refers to the concepts most commonly used in the Brazilian context. For example, “non-native language” is often used in Portuguese, mainly because of the difficulties in differentiating the relationships and statuses of languages that come after the so-called ‘first’ or “native” languages. This concept is largely absent from English-language research. {| class="wikitable" |Concept |Definition |- |Mother tongue |“The language first acquired around the age of five” (p. 260), i.e., during the early years of primary socialization. It can also be understood as “the language of greatest mastery and affirmation of an individual's identity” (p. 271). From a sociolinguistic point of view, this designation can also refer to the majority language in a given linguistic community (p. 273). Regarding this attribution, the author adds: “This meaning is oriented towards an imaginary of national, regional, cultural, and/or ethnic identity and unity. It is also closely linked to the concepts of national language and majority language” (p. 273). |- |Non-native language |“A language acquired after the acquisition of the native language, whether in a formal or informal context, immersive or non-immersive, and regardless of the sociopolitical status of the language” (p. 275). |- |L1 |“The language first acquired up to the age of about five” (p. 313). This may be the “language of greatest proficiency and affirmation of an individual's identity” (p. 316), although the degree of proficiency may change over the course of a lifetime. |- |Heritage language |“Language acquired in early childhood, usually within the family, which, due to migration, usually during childhood, becomes a minority language in relation to the majority language of the host society and, as a result, restricted in terms of contexts and opportunities for use and spaces of circulation, which can gradually lead to a weakening of linguistic proficiency or even to the individual forgetting the language” (p. 237). It can also refer to “the language with which the individual maintains a sense of belonging and cultural and identity ties, regardless of the context or order of acquisition, the minority or majority status of the language, or the level of proficiency” (p. 245). The author adds “a non-native language taught, learned, and/or used in non-immersion contexts, i.e., when the language in question is not the majority language in the place of teaching, use, and/or learning and/or does not have official sociopolitical status” (p. 252; for a third, more political-administrative meaning, see p. 257). |- |Host language |A subdomain of additional language that refers to the language acquired or to be acquired by migrants, with an emphasis on forced displacement, who are in a situation of vulnerability (Lopez & Diniz, 2018, n/p.). |- |L2 (or Second Language) |“A non-native or non-majority language that has been given (co)official status in bilingual or multilingual countries and is therefore the language or one of the languages of public administration, education, and common means of communication in the country” (p. 290). It can also refer to “a non-native language taught, learned, and/or used in language immersion contexts, i.e., when the language in question is the majority language in the place of teaching, use, and/or learning and/or has official sociopolitical status” (p. 300). A third concept, which does not enjoy international consensus, particularly in the German context (see definitions in Surkamp, 2010), refers to “a language acquired after the acquisition of the mother tongue, regardless of the formal, informal, immersive or non-immersive context, the socio-political status of the language or its position in the chronological order of language acquisition” (p. 307) or “a language acquired first after the partial or complete acquisition of the mother tongue, i.e., from the point of view of the chronology of language acquisition, it is the second language to constitute the speaker's linguistic repertoire” (p. 309). |- |Foreign Language |This is a language learned in an institutional context and in a guided manner. “A language acquired after the acquisition of the mother tongue, whether in a formal or informal context, immersive or non-immersive, and regardless of the sociopolitical status of the language” (p. 250). |- |Additional language |“A language acquired after the acquisition of the mother tongue, regardless of the formal or informal context, immersive or non-immersive, the sociopolitical status of the language, or the chronology of acquisition” (Lisboa, 2024, p. 231). It can also refer to “a non-native language acquired after the acquisition of the mother tongue and at least one other non-native language” (p. 236). |- |Language(s) of schooling |Languages of schooling cover two dimensions: languages taught as a subject (such as foreign languages in the curriculum) and languages used for and in teaching. These are often the national and majority languages of a country (Portuguese in Portugal) or regional or minority languages, depending on local education and language policies. (<nowiki>https://www.coe.int/fr/web/language-policy/languages-of-schooling</nowiki>) |} The language of origin is also often referred to as the “home language”, “identity language”, or “family language” (Stratilaki-Klein, 2025). However, speakers of a language of origin do not always consider that this language best defines their identity. Sometimes families choose not to speak their language of origin at home. Furthermore, a speaker may have two languages of origin, as in the case of Melissa, presented in the initial activity. The term “neighboring language” has both linguistic and geographical connotations and may coincide. From a linguistic point of view, it is a language belonging to the same language family (for example, Romance or Slavic languages, to name just two examples). From a geographical point of view, these are languages of neighboring territories, with more or less fluid borders (the French-German border or between Brazil and Argentina, for example). The concept of linguistic proximity or “border language” (Araújo e Sá et al, 2024) is used to refer to languages that coexist in border regions. Regardless of the connotation chosen, we can clearly see how important this concept is for thinking about plurilingual education: either it generally refers to teaching approaches that aim to develop receptive skills in several languages of the same language family (see entry “intercomprehension” in this Wikiversity), or it refers to cross-border linguistic contacts aimed at developing plurilingual and intercultural competence (Putsche, 2022). It is important to remember that most language designations take the so-called native speaker as the implicit norm (Bono, 2025), below which speakers of other designations are ranked, giving rise to situations of linguistic prejudice and discrimination in society in general, and in schools in particular. == Take away messages == * Language classifications can be based on linguistic, sociolinguistic, geographical, economic, curricular, and individual aspects. * The concepts used to name languages are polysemous and are not neutral, but rather convey discourses and ideologies that must be (re)constructed in order to understand the objectives set in plurilingual education. * Language designations and their associated connotations may depend on the linguistic and academic traditions in which they circulate. == Self-assessment == <quiz> {In the first situation presented in the initial activity, Melissa's mother tongue is German:} -true +false {In the second linguistic profile presented in the initial activity, Portuguese is the language of origin of Teresa's children.} +true -false {The host language generally refers to:} +the language that people in situations of mobility acquire in order to meet their most basic and urgent needs. -the language that parents and educators in situations of mobility (forced or otherwise) pass on to their children in order to maintain a link with their country of origin. -the language in which learners feel welcome when on vacation abroad and which makes them feel “at home”. {Which of these concepts refers to the language that parents and educators in situations of migration and mobility (forced or otherwise) pass on to their children in the host country, in a situation where that language is a minority language?} -Host language. -Mother tongue. +Language of origin. </quiz> == Resources to go further == * Araújo e Sá, M.H., Carinhas, R., Carlos, V., Kremer, M., Lourenço, M., Pinto, S., Simões, C., Teixeira, M., & Ulhôa, A. (2024). ''Linguistic and cultural management in border regions''. https://www.eu-bold.com/linguistic-and-cultural-management-in-border-regions/ * Dabène, L. (1994). ''Repères sociolinguistiques pour l’enseignement des langues''. Hachette. * Dewaele, J.-M.; Bak, T.; Ortega, L. (2022). Why the mythical “native speaker” has mud on its face. In N. Slavkov, S. Melo-Pfeifer & N. Kerschhofer-Puhalo (Eds.), ''The Changing Face of the “Native Speaker”. Perspectives from Multilingualism and Globalization'' (pp. 25-45). De Gruyter. * Lisboa, J. (2024). ''Vocabulário terminológico de conceitos de Língua''. Tese de doutoramento. Universididade da Uberlândia. URL: https://repositorio.ufu.br/bitstream/123456789/43065/1/VocabularioConceitosLingua.pdf == Bibliography == Bono, M. (2025). Mother tongue, native language, first language, heritage language. In Ch. Ollivier & S. Melo-Pfeifer (Eds.), ''Encyclopédie de l’éducation plurilingue / Encyclopaedia of plurilingual education''. Peter Lang. Cuq, J.-P. (dir.) (2003). ''Dictionnaire de didactique du français langue étrangère et seconde''. Cle International. Galisson, R., & Coste, D. (1976). ''Dictionnaire de Didactique des Langues''. Hachette. Lisboa, J. (2024). ''Vocabulário terminológico de conceitos de Língua''. Tese de doutoramento. Universidade da Uberlândia. URL: https://repositorio.ufu.br/bitstream/123456789/43065/1/VocabularioConceitosLingua.pdf Lisboa, J. (2025). A criação do Vocabulário Terminológico de Conceitos de Língua. ''Domínios de Lingu@gem'', 19, p. e019025. DOI: 10.14393/DLv19a2025-25. URL:  https://seer.ufu.br/index.php/dominiosdelinguagem/article/view/76903. Lopez, A.P.A., & Diniz, L. R. (2018). Iniciativas jurídicas e académicas para o acolhimento no Brasil de deslocados forçados. Revista da Sociedade Internacional do Português Língua Estrangeira, 9, s/p.  https://www.researchgate.net/publication/330635043_Iniciativas_Juridicas_e_Academicas_Brasileiras_para_o_Acolhimento_de_Imigrantes_Deslocados_Forcados Putsche, J. (2022). ''L’interculturel contextualisé : franco-allemand, frontalier, transculturel – Une approche sociodidactique''. [Dossier presentado para habilitación a la dirección de investigadores, Université de Lorraine]. https://hal.science/tel-03882905/document Stratilaki-Klein, S. (2025). La galaxie des langues et leurs désignations. In Ch. Ollivier & S. Melo-Pfeifer (Eds.), ''Encyclopédie de l’éducation plurilingue / Encyclopaedia of plurilingual education''. Peter Lang. Surkamp, C. (Ed.) (2010). ''Metzler Lexikon Fremdsprachendidaktik''. J.B. Metzler. my2gjx09qoqaa66barnesdgi0lnjdse 2 330095 2814879 2026-06-09T16:17:09Z Ясамойла 1373064 пачатак 2814879 wikitext text/x-wiki Жыву ў Эўропе. omibgah2t62ygsbw0ranqvd4jjqhvpf 2814880 2814879 2026-06-09T16:18:25Z Ясамойла 1373064 дапаўненьне 2814880 wikitext text/x-wiki {{babel|en-2|be-4|ru-4|cu-4|uk-2|pl-1}} Жыву ў Эўропе. elz374iildgnmrnisbfpje15f6766vg 2 330096 2814881 2026-06-09T16:50:49Z Ясамойла 1373064 пачатак 2814881 wikitext text/x-wiki Можна пісаць допісы. rzgvan3mfo0ghg2tbou23ce8x2kkdat 0 330097 2814884 2026-06-09T17:10:32Z IanVG 2918363 Created page with " == Examples == # Let's consider a first-order nonlinear equation that represents an organism's population: <math>\frac {dP}{dt} = 0.6P(1 - \frac {P}{540})</math> ## a) What are the equilibrium population states? ## b) At what populations is the population increasing? ## c) At what populations is the population decreasing?" 2814884 wikitext text/x-wiki == Examples == # Let's consider a first-order nonlinear equation that represents an organism's population: <math>\frac {dP}{dt} = 0.6P(1 - \frac {P}{540})</math> ## a) What are the equilibrium population states? ## b) At what populations is the population increasing? ## c) At what populations is the population decreasing? s5e2fqwf2l47wr289s1xwgej6adyqce 2814899 2814884 2026-06-09T18:55:03Z IanVG 2918363 2814899 wikitext text/x-wiki == Examples == # Let's consider a first-order nonlinear equation that represents an organism's population: #: #: <math>\frac {dP}{dt} = 0.6P(1 - \frac {P}{540})</math> #: #: a) What are the equilibrium population states? #: b) At what populations is the population increasing? #: c) At what populations is the population decreasing? b5aa91wqan0tvs7v6srvax1uvquc5yx 2814900 2814899 2026-06-09T18:56:28Z IanVG 2918363 /* Examples */ 2814900 wikitext text/x-wiki == Examples == # Let's consider a first-order nonlinear equation that represents an organism's population: #: #: <math>\frac {dP}{dt} = 0.6P(1 - \frac {P}{540})</math> #: #: a) What are the equilibrium population states? #: b) At what populations is the population increasing? #: c) At what populations is the population decreasing? #Consider another nonlinear equation representing population: #:<math>\frac {dP}{dt} = 0.8P(1 - \frac {P}{80})(\frac {P}{30}-1)P</math> #: #: a) What are the equilibrium population states? #: b) At what populations is the population increasing? #: c) At what populations is the population decreasing? 4bvvxbn4bed89sppoy60iuq1mlejrm8 1 330098 2814895 2026-06-09T18:48:32Z IanVG 2918363 /* Problems needed! */ new section 2814895 wikitext text/x-wiki == Problems needed! == If you magically land on this page and wonder how you can assist, please think of adding example problems (and their solutions ideally!) to the course! [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 18:48, 9 June 2026 (UTC) 34y83ey28sfcvdyk32yr4prux4pntbw 2814896 2814895 2026-06-09T18:50:01Z IanVG 2918363 /* Some naming conventions */ new section 2814896 wikitext text/x-wiki == Problems needed! == If you magically land on this page and wonder how you can assist, please think of adding example problems (and their solutions ideally!) to the course! [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 18:48, 9 June 2026 (UTC) == Some naming conventions == Equations of the order that is first, should be written as: '''first-order'''. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 18:50, 9 June 2026 (UTC) 1ol200wloqshc8h1sp1x1pjesjfsmai 2814897 2814896 2026-06-09T18:51:00Z IanVG 2918363 2814897 wikitext text/x-wiki == Problems needed! == If you magically land on this page and wonder how you can assist, please think of adding example problems (and their solutions ideally!) to the course! [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 18:48, 9 June 2026 (UTC) == Some naming conventions == * Equations of the order that is first, should be written as: '''first-order'''. * The titles of the course sub-pages should have the first word capitalized and the following words should be lower-case. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 18:50, 9 June 2026 (UTC) 9h6lfwda6hf9r6qx49xap9etbssgbqk 2 330099 2814918 2026-06-09T20:06:08Z IanVG 2918363 Created page with " == Welcome == {{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]], Insert username here!'''|width=100%}} <div style="{{Robelbox/pad}}"> You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:IanVG|me personally]] if you would like some [[Help:Contents|help]]. Remember to Wikiversity:Signature#How to add..." 2814918 wikitext text/x-wiki == Welcome == {{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]], Insert username here!'''|width=100%}} <div style="{{Robelbox/pad}}"> You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:IanVG|me personally]] if you would like some [[Help:Contents|help]]. Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple. We invite you to [[Wikiversity:Be bold|be bold]] and [[Wikiversity|assume good faith]]. Please abide by our [[Wikiversity:Civility|civility]], [[Wikiversity:Privacy policy|privacy]], and [[Foundation:Terms of Use|terms of use]] policies. To find your way around, check out: <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Wikiversity:Introduction|Introduction to Wikiversity]] * [[Help:Guides|Take a guided tour]] and learn [[Help:Editing|how to edit]] * [[Wikiversity:Browse|Browse]] or visit an educational level portal:<br>[[Portal:Pre-school Education|pre-school]] | [[Portal:Primary Education|primary]] | [[Portal:Secondary Education|secondary]] | [[Portal:Tertiary Education|tertiary]] | [[Portal:Non-formal Education|non-formal]] * [[Wikiversity:Introduction explore|Explore]] links in left-hand navigation menu </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] * Learn [[Help:How to write an educational resource|how to write an educational resource]] * Find out about [[Wikiversity:Research|research]] activities * Give [[Wikiversity:Feedback|feedback]] about your observations * Discuss issues or ask questions at the [[Wikiversity:Colloquium|colloquium]] </div> <br clear="both"/> To get started, experiment in the [[wikiversity:sandbox|sandbox]] or on [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 20:05, 9 June 2026 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} 9rdotfdczb8h79qgrodvwb9a3cydgtj 10 330100 2814925 2026-06-09T23:41:12Z AUBSTRAWBS 3060598 Created page with "{{Userbox | border-c = #000000 | id = [[File:(logo) Gate jieitai kanochi nite, kaku tatakaeri.svg|100x50px]] | id-c = #000000 | id-fc = #000000 | id-s = 14 | info = This user testified in front of the [[National Diet|national diet]] | info-c = #000000 | info-fc = #ffffff | info-s = 8 }} <noinclude></noinclude>" 2814925 wikitext text/x-wiki {{Userbox | border-c = #000000 | id = [[File:(logo) Gate jieitai kanochi nite, kaku tatakaeri.svg|100x50px]] | id-c = #000000 | id-fc = #000000 | id-s = 14 | info = This user testified in front of the [[National Diet|national diet]] | info-c = #000000 | info-fc = #ffffff | info-s = 8 }} <noinclude></noinclude> r6a5pe7sxiyo9erwbmdups8qo415ux1 2814926 2814925 2026-06-09T23:42:44Z AUBSTRAWBS 3060598 AUBSTRAWBS moved page [[User:AUBSTRAWBS/Gate]] to [[Template:Gate-anime]]: move to template 2814925 wikitext text/x-wiki {{Userbox | border-c = #000000 | id = [[File:(logo) Gate jieitai kanochi nite, kaku tatakaeri.svg|100x50px]] | id-c = #000000 | id-fc = #000000 | id-s = 14 | info = This user testified in front of the [[National Diet|national diet]] | info-c = #000000 | info-fc = #ffffff | info-s = 8 }} <noinclude></noinclude> r6a5pe7sxiyo9erwbmdups8qo415ux1 2814928 2814926 2026-06-09T23:47:50Z AUBSTRAWBS 3060598 AUBSTRAWBS moved page [[Template:Gate-anime]] to [[Template:Gate anime]] 2814925 wikitext text/x-wiki {{Userbox | border-c = #000000 | id = [[File:(logo) Gate jieitai kanochi nite, kaku tatakaeri.svg|100x50px]] | id-c = #000000 | id-fc = #000000 | id-s = 14 | info = This user testified in front of the [[National Diet|national diet]] | info-c = #000000 | info-fc = #ffffff | info-s = 8 }} <noinclude></noinclude> r6a5pe7sxiyo9erwbmdups8qo415ux1 2814935 2814928 2026-06-10T00:10:54Z Atcovi 276019 Atcovi moved page [[Template:Gate anime]] to [[Template:User gate anime]] without leaving a redirect: its preferred for userboxes to have "user" as the first word for its title page after the mainspace ("Template") 2814925 wikitext text/x-wiki {{Userbox | border-c = #000000 | id = [[File:(logo) Gate jieitai kanochi nite, kaku tatakaeri.svg|100x50px]] | id-c = #000000 | id-fc = #000000 | id-s = 14 | info = This user testified in front of the [[National Diet|national diet]] | info-c = #000000 | info-fc = #ffffff | info-s = 8 }} <noinclude></noinclude> r6a5pe7sxiyo9erwbmdups8qo415ux1 2814939 2814935 2026-06-10T00:12:13Z Atcovi 276019 cat(s) 2814939 wikitext text/x-wiki {{Userbox | border-c = #000000 | id = [[File:(logo) Gate jieitai kanochi nite, kaku tatakaeri.svg|100x50px]] | id-c = #000000 | id-fc = #000000 | id-s = 14 | info = This user testified in front of the [[National Diet|national diet]] | info-c = #000000 | info-fc = #ffffff | info-s = 8 }} <noinclude>[[Category:Userbox templates]]</noinclude> 1tf9axtd8zqm234mm85ckf4tyr9o2b4 2814940 2814939 2026-06-10T00:12:33Z Atcovi 276019 2814940 wikitext text/x-wiki {{Userbox | border-c = #000000 | id = [[File:(logo) Gate jieitai kanochi nite, kaku tatakaeri.svg|100x50px]] | id-c = #000000 | id-fc = #000000 | id-s = 14 | info = This user testified in front of the [[National Diet|national diet]] | info-c = #000000 | info-fc = #ffffff | info-s = 8 }} <noinclude>[[Category:Userboxes]]</noinclude> j2k1g0usijd7f3e8onyeyg863lekt2o 2 330101 2814927 2026-06-09T23:42:44Z AUBSTRAWBS 3060598 AUBSTRAWBS moved page [[User:AUBSTRAWBS/Gate]] to [[Template:Gate-anime]]: move to template 2814927 wikitext text/x-wiki #REDIRECT [[Template:Gate-anime]] ojkvbyl6n2cjq7ybp4f46dx9roxytx6 10 330102 2814929 2026-06-09T23:47:50Z AUBSTRAWBS 3060598 AUBSTRAWBS moved page [[Template:Gate-anime]] to [[Template:Gate anime]] 2814929 wikitext text/x-wiki #REDIRECT [[Template:Gate anime]] t8w065utq63o0gl8wkhddxt41oa3t00 2814936 2814929 2026-06-10T00:11:12Z Atcovi 276019 update 2814936 wikitext text/x-wiki #REDIRECT [[Template:User gate anime]] h0o56qmcplhjx116q3ywr0ewxf82yqb 118 330103 2814943 2026-06-10T00:16:10Z AUBSTRAWBS 3060598 work in progress 2814943 wikitext text/x-wiki wip ctcv8yk12vildty3p2p57q56gzwocm4 2814946 2814943 2026-06-10T00:20:11Z AUBSTRAWBS 3060598 2814946 wikitext text/x-wiki This page is currently a work in progress j1wd04612mwp7k3q35vtubqeo4xu2er 2814947 2814946 2026-06-10T00:20:39Z AUBSTRAWBS 3060598 AUBSTRAWBS moved page [[Wikiversity:Userboxes/Galleries/alphabetical]] to [[Draft:Userboxes/Galleries/alphabetical]] 2814946 wikitext text/x-wiki This page is currently a work in progress j1wd04612mwp7k3q35vtubqeo4xu2er 118 330104 2814945 2026-06-10T00:18:56Z AUBSTRAWBS 3060598 work in progress 2814945 wikitext text/x-wiki {{Short description|Guide to galleries of userbox templates}} The following is a '''topical guide to galleries of [[Wikipedia:Userboxes|userbox]] [[Help:Template|templates]]'''. You can use the search box at the right to search for available userboxes on a specific topic. For an alphabetical list of all userbox galleries, please see '''[[Wikiversity:Userboxes/Galleries/alphabetical]]'''. For information about using and creating userboxes, please see '''[[Wikipedia:Userboxes]]'''. {{-}} {{infobox|image=<div style="width:45em"> <inputbox> type=fulltext width=28 prefix=Wikipedia:Userboxes/ searchbuttonlabel=Search for a userbox topic break=no </inputbox> </div>}} ==Interests== * [[Wikipedia:Userboxes/Interests]], ''an alphabetical userbox gallery about your personal interests'' ** [[Wikipedia:Userboxes/Animals|Animals]], ''a userbox gallery about animals'' ==Language== * [[Wikipedia:Userboxes/Language]], ''a guide to userbox galleries about languages'' === English language === * [[Wikipedia:Userboxes/Language/English]], ''a userbox gallery about the English language'' *** qkvj5ozwiktfdn5q8hg23b045xgttvn 2814951 2814945 2026-06-10T00:26:02Z AUBSTRAWBS 3060598 improved page 2814951 wikitext text/x-wiki {{Short description|Guide to galleries of userbox templates}} The following is a '''topical guide to galleries of [[Wikipedia:Userboxes|userbox]] [[Help:Template|templates]]'''. You can use the search box at the right to search for available userboxes on a specific topic. For an alphabetical list of all userbox galleries, please see '''[[Wikiversity:Userboxes/Galleries/alphabetical]]'''. For information about using and creating userboxes, please see '''[[Wikipedia:Userboxes]]'''. {{-}} {{infobox|image=<div style="width:45em"> <inputbox> type=fulltext width=28 prefix=Wikipedia:Userboxes/ searchbuttonlabel=Search for a userbox topic break=no </inputbox> </div>}} ==Interests== * [[Wikipedia:Userboxes/Interests]], ''an alphabetical userbox gallery about your personal interests'' ==Language== * [[Wikiversity:Userboxes]], ''userboxes about languages'' ==Key information== * [[Wikiversity:Userboxes]],''userboxes about important information'' 6uukaemm17lgw3rmcbq2p7c4i3hg1iw 2814953 2814951 2026-06-10T00:27:16Z AUBSTRAWBS 3060598 changed wikipedia link to wikiversity 2814953 wikitext text/x-wiki {{Short description|Guide to galleries of userbox templates}} The following is a '''topical guide to galleries of [[Wikipedia:Userboxes|userbox]] [[Help:Template|templates]]'''. You can use the search box at the right to search for available userboxes on a specific topic. For an alphabetical list of all userbox galleries, please see '''[[Wikiversity:Userboxes/Galleries/alphabetical]]'''. For information about using and creating userboxes, please see '''[[Wikipedia:Userboxes]]'''. {{-}} {{infobox|image=<div style="width:45em"> <inputbox> type=fulltext width=28 prefix=Wikipedia:Userboxes/ searchbuttonlabel=Search for a userbox topic break=no </inputbox> </div>}} ==Interests== * [[Wikiversity:Userboxes/Interests]], ''an alphabetical userbox gallery about your personal interests'' ==Language== * [[Wikiversity:Userboxes]], ''userboxes about languages'' ==Key information== * [[Wikiversity:Userboxes]],''userboxes about important information'' 1w7rz71jwt1smzfj0d4aksfea05r76f 2814954 2814953 2026-06-10T00:37:28Z AUBSTRAWBS 3060598 AUBSTRAWBS moved page [[Wikiversity:Userboxes/Galleries]] to [[Draft:Userboxes/Galleries]] 2814953 wikitext text/x-wiki {{Short description|Guide to galleries of userbox templates}} The following is a '''topical guide to galleries of [[Wikipedia:Userboxes|userbox]] [[Help:Template|templates]]'''. You can use the search box at the right to search for available userboxes on a specific topic. For an alphabetical list of all userbox galleries, please see '''[[Wikiversity:Userboxes/Galleries/alphabetical]]'''. For information about using and creating userboxes, please see '''[[Wikipedia:Userboxes]]'''. {{-}} {{infobox|image=<div style="width:45em"> <inputbox> type=fulltext width=28 prefix=Wikipedia:Userboxes/ searchbuttonlabel=Search for a userbox topic break=no </inputbox> </div>}} ==Interests== * [[Wikiversity:Userboxes/Interests]], ''an alphabetical userbox gallery about your personal interests'' ==Language== * [[Wikiversity:Userboxes]], ''userboxes about languages'' ==Key information== * [[Wikiversity:Userboxes]],''userboxes about important information'' 1w7rz71jwt1smzfj0d4aksfea05r76f 4 330105 2814948 2026-06-10T00:20:39Z AUBSTRAWBS 3060598 AUBSTRAWBS moved page [[Wikiversity:Userboxes/Galleries/alphabetical]] to [[Draft:Userboxes/Galleries/alphabetical]] 2814948 wikitext text/x-wiki #REDIRECT [[Draft:Userboxes/Galleries/alphabetical]] scxg2bp5sum6hxg0w0dk4nhwh6i10mg 4 330106 2814955 2026-06-10T00:37:28Z AUBSTRAWBS 3060598 AUBSTRAWBS moved page [[Wikiversity:Userboxes/Galleries]] to [[Draft:Userboxes/Galleries]] 2814955 wikitext text/x-wiki #REDIRECT [[Draft:Userboxes/Galleries]] 6ywplj26egbdb058ag3imwspcnw8zwu 3 330108 2814968 2026-06-10T06:42:56Z Jtneill 10242 Welcome ip 2814968 wikitext text/x-wiki {{#ifeq:{{NAMESPACE}}|User talk||{{error|Error: substitution required. Use <nowiki>{{subst:Welcomeip}}</nowiki> instead.}}[[Category:Template substitution errors]]<div style="display:none;">}}{{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> Hello, and [[Wikiversity:Welcome, newcomers|welcome]] to [[Wikiversity]]. Thank you for your contributions. Currently, you are [[Help:Editing|editing]] without a permanent account. You can continue to do so, as you are not required to log in to Wikiversity to read and edit articles; however, logging in will result in a username being shown instead of a temporary account (which will expire 90 days after first edit). Logging in does not require any personal details, and there are many other '''[[Wikiversity:Why create an account|benefits for logging in]]'''. When you edit pages: * Please [[Wikiversity:Copyrights|respect others' copyrights]]; do not copy and paste the contents from webpages directly. * Please use a [[Wikiversity:Disclosures|neutral point of view]] when editing articles. * If you are testing, please use the [[Wikiversity:Sandbox|Sandbox]] to <span class="plainlinks">[http://en.wikiversity.org/w/index.php?title=Wikiversity:Sandbox&action=edit do so].</span> * Do not add unreasonable contents into any [[Wikiversity:Browse|articles]], such as copyrighted text, advertisement messages, and text that is not related to an areas's subject. Adding such content or editing articles maliciously is considered [[Wikiversity:Blocking policy|vandalism]]. The [[Wikiversity:Introduction|Introduction]] is a good place to start learning about Wikiversity. For now, if you are stuck, you can ask a question on {{#if:|[[user talk:{{{1}}}|my Talk page]]|my Talk page}}. I will answer your questions as far as I can! Thank you again for contributing to Wikiversity. -- -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 06:42, 10 June 2026 (UTC) </div> {{Robelbox/close}} {{#ifeq:{{NAMESPACE}}|User talk||</div>}} j39sq8zlbr6ev2e884a7yc5nlm88nty 14 330109 2814975 2026-06-10T06:58:45Z Jtneill 10242 Created page with "[[Category:Motivation and emotion/Book/Self]]" 2814975 wikitext text/x-wiki [[Category:Motivation and emotion/Book/Self]] ea3826x1qgjek0ssmbyjmbiuwkf1l3r